Classical Mechanics - Quanta: BS/MSc Physics Books

1: The number of independent ways in which a mechanical system can move without violating any constraint is called

a. Constraint

b. Number of freedoms

c. Degrees of freedom

d. Generalized coordinates

2: The generalized momentum Pi need not always of linear momentum

a. Speed

b. Velocity

c. Dimension

d. None

3: A Single particle in a space requires three coordinates, so it has degree of freedom

a. Two

b. Three

c. Four

d. None

4: The sixth coordinates must satisfy a single constraints equation for a · · ·

a. Monoatomic

b. Diatomic

c. Triatomic molecule

d. None

5: In degree of freedom the · · · number of coordinates required to specify the configuration of a dynamical system

a. Minimum

b. Maximum

c. Both are correct

d. None

6: If constrains can be expressed as a function of position then it is called

a. Holonomic

b. Non Holonomic

c. Scleronomic

d. Neither

7: If constrains are stationary then it is called

a. Scleronomic

b. Non Holonomic

c. Rheononomic

d. All of these

8: First difficulty of constraints in the solution of mechanical problems

a. Force of constraints

b. Coordinates r_i, d are not all independent

c. None of these

d. Both ‘a’ and ‘b’

9: Rigid body and gas molecules within the wall of container are example of

a. Holonomic constraints

b. Non Holonomic constraints

c. constraints

d. All of these

10: The square of the period is proportional to semi major axis of

a. cubic orbit

b. vertical orbit

c. central orbit

d. Elliptical orbit

11: Further away from planet is from sun · · · it takes what period reaches orbit to sun

a. Longer

b. shorter

c. None

d. Undefined

12: The area per unit time swept out by a radius vector from sun to planet becomes

a. Constant

b. independent

c. vertical

d. none of these

13: Planets move in · · ·. Around sun

a. Elliptical orbits

b. Cubic orbits

c. Central orbits

d. Horizontal axis

14: the angular momentum of rigid body is L and its K.E is halved. What happens to its angular momentum

a. L

b. 2L

c. L/2

d. L/4

15: If a linear momentum is increased by 50% K.E will increase by what percent?

a. 25%

b. 50%

c. 100%

d. 125%

16: The gravitational force between two masses is

a. Repulsive

b. Attractive

c. Zero

d. Infinity

17: The value of universal gravitational constant G is

a. 4 × 1042 Nm²/Kg²

b. 6.67 × 1011Nm²/Kg²

c. 9.81 cm/sec²

d. 6.67 × 10−11Nm²/Kg²

18: The angular momentum is in a central force field

a. Zero

b. Not conserved

c. infinity

d. conserved

19: The areal velocity of the particle in a central force field is

a. Zero

b. constant

c. infinity

d. Not conserved

20: At the turning point in an arbitrary potential field the radial velocity is

a. Zero

b. 1

c. infinity

d. 1/2

21: For hyperbolic orbit the values of energy E and eccentricity ϵ are

a. E = 0 and ϵ = 1

b. E > 0 and ϵ > 1

c. E > 0 and ϵ = 1

d. E> 0 and ϵ = 0

22: For parabolic orbit the values of energy E and eccentricity ϵ are

a. E = 0 and ϵ = 1

b. E > 0 and ϵ > 1

c. E > 0 and ϵ = 1

d. E > 0 and ϵ = 0

23: For elliptical orbit the values of energy E and eccentricity ϵ are

a. E = 0 and ϵ > 1

b. E > 0 and ϵ > 1

c. E < 0 and ϵ < 1

d. E > 0 and ϵ = 0

24: For circular orbit the value of eccentricity

a. ϵ > 1

b. ϵ ≥ 1

c. ϵ < 1

d. ϵ = 0

25: All the planet moves around the Sun in orbit

a. circular

b. parabolic

c. hyperbolic

d. elliptical

26: In the formation of the cycloidal curves, the circle which rolls with a fixed point without

slipping is called

a. Generating circle

b. Rolling circle

c. Slipping circle

d. Direct circle

27: · · · is a curve generated by a point on the circumference of a circle, which rolls without

slipping along another circle outside it

a. Trochoid

b. Epicycloid

c. Hypotrochoid

d. Involute

28: Type of curve is created by the intersection of a plane parallel to the side of cone?

a. Parabola

b. Hyperbola

c. Ellipse

d. Roulette

29: Type of curve is created by the intersection of a plane with a cone which makes an angle with the axis greater than the angle between the side of the cone and the axis?

a. Parabola

b. Hyperbola

c. Ellipse

d. Roulette

30: Curve generated by a point on the circumference of a circle, which rolls without slipping along outside of another circle is known as

a. Hypocycloid

b. Epicycloid

c. Cycloid

d. Trochoid

31: Curved traced out by a point which moves uniformly both about the center and at the same time away or towards the center is known as

a. Involute

b. Archimedean spiral

c. Cycloid

d. None of above

32: Poisson bracket is used to define a?

a. Poisson function

b. Poisson equation

c. Poisson algebra

d. None of the above

33: The Poisson bracket is an important binary operation in

a. Hamiltonian mechanics

b. lagrangian formalism

c. both a & b

d. None of the above

34: Hamilton’s formulation of classical mechanics made use of a mathematical tool called

a. angular momentum

b. Poisson brackets

c. None of the above

d. lagrange

35: the bracket of p with any reasonably smooth function of q is

a. [p, f(a)] = df/dq

b. [p, f(q)] = da/dq

c. [p, f(q)] = df/fq

d. [p, f(q)] = df/dq

36: Lagrange’s bracket is

a. Canonical invariant

b. Non-invariant

c. Canonical variant

d. None of these

37: Poisson’s bracket is

a. invariant under canonical transformation

b. variant under canonical transformation

c. Both a and b

d. None of these

38: As there are three generalized coordinates, then Hamilton’s canonical equations will be · · · in number

a. Three

b. Four

c. Five

d. Six

39: The generalized momentum is also called

a. conjugate momentum

b. canonical momentum

c. Both a and b

d. None of these

40: Which of the following answer is true for the Lagrange’s bracket

a. -1

b. 0

c. 1

d. 2

41: If the Poisson bracket of a function with the Hamiltonian vanishes

a. the function depends upon time

b. the function is a constant of motion

c. the function is not the constant of motion

d. None of these

42: If we make a canonical transformation from the set of variables (p_k, q_k) to new set of Variables (P_k, Q_k) and the transformed Hamiltonian is identically zero, then

a. the new variables are constant in time

b. the new variables are not constant in time

c. the new variables are not cyclic

d. None of these

43: Hamiltonian H is defined as

a. the total energy of the system

b. the difference in energy of the system

c. the product of energy of the system

d. All of these

44: Whenever the Lagrangian for a system does not contain a coordinate explicitly.

a. p_k is cyclic coordinate

b. p_k the generalized momentum is a constant of motion

c. q_k is always zero

d. None of these

45: The dimensions of generalized momentum

a. are always those of linear momentum

b. may be those of angular momentum

c. may be those of linear momentum

d. Both b and c are true

46: If the Lagrangian does not depend on time explicitly

a. the Hamiltonian is constant

b. Hamiltonian not constant

c. the kinetic energy is constant

d. the potential energy is constant

47: Any · · · on the freedom of movement of a system of particles in the form equation is

called Constraints

a. Restrictions

b. Points

c. Both

d. resistance

48: Example of the Constraints is

a. Rigid bodies

b. a particle on the surface

c. A bead of abacus

d. all

49: In the Constraints the particles can move

a. On the surface

b. in the surface

c. Over the surface

d. none

50: There are only · · · types of constraints

a. One

b. two

c. Three

d. four

51: The constraints which can be expressed as a function of positions and time are called

a. Holonomic

b. Non holonomic

c. Both

d. rheonomic

52: Example of the Scleronomic constraints

a. Pendulum

b. rigid body

c. Particles on the surface of solid sphere

d. Beads of abacus

53: If the Constraints depends on time explicitly then it is called

a. Rheonomic

b. Scleronomic

c. Non holonomic

d. None

54: The constraints which cannot be expressed as a function of position and time are called

a. Non holonomic

b. Rehonomic

c. Scleronomic

d. none

55: In Lagrangian approach, the flow parcels follows

a. Pressure field

b. Velocity field

c. Temperature field

d. Density field

56: Each parcel in the Lagrangian formulation is tagged using

a. Time-dependent position vector

b. Time -independent position vector

c. Time-dependent velocity vector

d. Time-independent velocity vector

57: Which of these is an acceptable tag for Lagrangian parcels?

a. Parcels center of mass at instantaneous time

b. Parcels center of pressure at instantaneous time

c. Parcels center of mass at initial time

d. Parcels center of pressure at initial time

58: The Lagrangian equation of motion is order differential equations.

a. First

b. Second

c.Third

d. Forth

59: In Lagrange’s equation virtual displacement does not involve

a. Space

b. Time

c. N number of particle

d. None

60: In Lagrange’s equation if there are N number of particle and so the generalized coordinated are

a. n = N − K

b. n = 3N

c. n = 3N − K

d. n = 3n − k

61: In a simple one-constraints Lagrange multiplier setup, the constraint has to be always one dimension lesser than the objective function.

a. True

b. False

c. None

d. May be true

62: Whenever the Lagrangian for a system does not contain a coordinate explicitly

a. P_k is cyclic coordinates

b. P_k the generalized momentum is a constant of motion

c. q_k is always zero
d. None of these

63: Euler equation govern flows

a Viscous adiabatic flows

b Inviscid flows

c. Adiabatic and inviscid flow

d. Adiabatic flows

64: In Euler form of energy equations, which of these terms is not present?

a. Rate of charge of energy

b. Heat radiation

c. Heat source

d. Thermal conductivity

65: The gravitational force between two masses is

a. Repulsive

b. Attractive

c. Zero

d. Infinite

66: All the planet moves around the sun in orbit?

a. Circular

b. Parabolic

c. Elliptical

d. Hyperbolic

67: The combination of total and partial derivative that enters into Euler equation is called

a. Variations derivative

b. Degree of freedom

c. Actual length

d. None of these

68: One of the possible paths systems will travel through a path which is

a. Long

b. Shorter

c. Stationary

d. None of these

69: A particle of unit mass moves in a potential $V (x) = ax^2+\frac{b}{x^2}$ , where a and b are positive constants. The angular frequency of small oscillation about the minimum of the potential

a. $\sqrt{8b}$

b. $\mathbf{\sqrt{8a}}$

c. $\sqrt{\frac{8a}{b}}$

d. $\sqrt{\frac{8b}{a}}$

70: The acceleration due to gravity (g) on the surface of Earth is approximately 2.6 times that on the surface of Mars. Given that the radius of Mars is about one half the radius of Earth, the ration of the escape velocity on Earth to that on Mars is approximately

a. 1.1

b. 1.3

c. 2.3

d 5.2

71: A constant force F is applied to a relativistic particle of rest mass m. If the particle starts from rest at t = 0, its speed after time t is

a. $\frac{Ft}{m}$

b. $c\tan h(\frac{Ft}{mc})$

c. $c(1-e^{-\frac{Ft}{mc}})$

d. $\mathbf{\frac{Fct}{\sqrt{F^2t^2+m^2c^2}}}$

72: The potential of a diatomic molecule as a function of the distance r between the atoms is given by $\mathbf{V (r) = -\frac{d}{r^6}+\frac{b}{r^{12}}}$ . The value of the potential at equilibrium separation between the atom is

a. $-\frac{4a^2}{b}$

b. $-\frac{2a^2}{b}$

c. $-\frac{a^2}{2b}$

d. $\mathbf{-\frac{a^2}{4b}}$

73: Two particles of identical mass moves in circular orbits under a central potential $\mathbf{V\left(r\right)=\frac{1}{2}kR^2}$ . Let l₁ and l₂ be the angular momenta and r₁ and r₂ be the radii of the orbits respectively. If $\frac{l_1}{l_2} = 2$ . the value of r₁/r₂ is

a. $\sqrt\mathbf{2}$

b. $\frac{1}{\sqrt2}$

c. 2

d. 1/2

74: A planet of mass m moves in the inverse square central force field of the sun of mass M. If the semi-major and semi-minor axes of the orbit are a and b, respectively, the total energy of the planet is

a. $\mathbf{-\frac{GMm}{a+b}}$

b. $-GMm(\frac{1}{a}+\frac{1}{b})$

c. $-\frac{GMm}{a+b}(\frac{1}{b}-\frac{1}{a})$

d. $-GMm(\frac{a-b}{\left(a+b\right)^2})$

75: An annulus of mass M made of a material of uniform of uniform density has inner and outer radii a and b respectively. Its principle moment of inertia along the axis of symmetry perpendicular to the plane of the annulus is

a. $\frac{1}{1}M\frac{(b^4+a^4)}{b^2-a^2}$

b. $\frac{1}{2}M\pi(b^2-a^2)$

c. $\frac{1}{\ 2}M(b^2-a^2)$

d. $\mathbf{\frac{1}{2}M(b^2+a^2)}$

76: Two events separated by a (spatial) distance 9⁹ m, are simultaneous in one inertial frame. The time interval between these two events is a frame moving with a constant speed 0.8 c (where speed of light c = 3 × 10⁸ m/s) is

a. 60 s

b. 40 s

c. 20 s

d. 0 s

77: If the Lagrangian of a particle moving in one dimension is given by L = x²/2 − V (x) the Hamiltonian is

a. $\mathbf{\frac{1}{2}xp^2+V(x)}$

b. $\frac{x^2}{2x}+V(x)$

c. $\frac{1}{2}x^2+V(x)$

d. $\frac{p^2}{2x}+V(x)$

78: A horizontal circular platform mutes with a constant angular velocity Ω directed vertically upward. A person seated at the center shoots a bullet of mass m horizontally with speed v. The acceleration of the bullet, in the reference frame of the shooter, is

a.2vΩ to his right

b. 2vΩ to his left

c. vΩ to his right

d. vΩ to his left

79: The Poisson bracket ${\left|\vec{r}\right|,|\vec{p|}}$ has the value

a. $\left|\vec{r}\right|,|\vec{p|}$

b. $\mathbf{\hat{r}.\hat{p} }$

c. 3

d. 1

80: What is proper time interval between the occurrence of two events i f in one inertial frame events are separated by 7.5 × 10⁸ m and occur 6.5 s apart?

a. 6.50 s

b. 6.00 s

c. 5.7 s

d. 5.00 s

81: A solid cylinder of height H, radius R and density ρ, floats vertically on the surface of a liquid of density ρ◦. The cylinder will be set into oscillatory motion when a small instantaneous downward force is applied. The frequency of oscillation is

a. $\frac{\rho g}{\rho_oH}$

b. $\frac{\rho}{\rho_o}\sqrt{\frac{g}{H}\ }$

c. $\sqrt{\frac{\rho g}{\rho_oH}}$

d. $\mathbf{\sqrt{\frac{\rho_og}{\rho H}}}$

82: Two bodies of equal mass m are connected by a massless rigid rod of length l lying in the xy−plane with the center of the rod at the origin. If this system is rotating about the z−axis with a frequency ω, its angular momentum is

a. $\frac{ml^2\omega}{4}$

b. $\mathbf{\frac{ml^2\omega}{2}}$

c. $ml^2\omega$

d. $2ml^2\omega$

83: The muon has mass 105 MeV/c² and mean lifetime 2.2 µs in its frame. The mean distance traversed by a muon of energy 315 MeV before decaying is approximately

a. 3 × 105 km

b. 2.2 cm

c. 6.6 µm

d. 1.98 km

84: A planet of mass m and an angular momentum L moves in a circular orbit in a potential, $V(r) =-\frac{k}{r},$ , where k is a constant. If it is slightly perturbed radially, the angular frequency of radial oscillation is

a. $\frac{mk^2}{\sqrt2L^3}$

b $\frac{mk^2}{L^3}$

c. $\frac{\sqrt2mk^2}{L^3}$

d. $\frac{\sqrt3mk^2}{L^3}$

85: The number of degrees of freedom of a rigid body in d space-dimension is

a. 2 d

b. 6

c. $\mathbf{\frac{d\left(d+1\right)}{2}}$

d. d!

86: Let A, B and C be function of phase space variables (coordinates and momenta of a mechanical system). If {◦} represents the Poisson bracket, the value of {A, {B, C}} – {{A, B}, C} is given by

a. 0

b. {B, {C, A}}

c. {A, {C, B}}

d. {{C, A}, B}

87: A particle moves in a potential V = x² + y² + z²/2 . Which components(c) of the angular momentum is/are constant(s) of motion?

a. None

b. L_x, L_y and L_z

c. only L_x and L_y

d only L_z

88: A particle of mass m and coordinate q has the Lagrangian $L =\frac{1}{2}mq^2-\frac{\lambda}{2}qq^2$ , where λ is a constant. The Hamiltonian for the system is given by

a. $\frac{p^2}{2m}+\frac{\lambda q p^2}{2m^2}$

b. $\frac{p^2}{2(m-\lambda q)}$

c. $\frac{p^2}{2m}+\frac{\lambda q p^2}{2\left(m-\lambda q\right)^2}$

d. $\frac{pq}{2}$ ˙

89: A canonical transformation relates the old coordinates (q, p) to the new ones (Q, P) by the relation Q = q² and P = pq/2. The corresponding time independent generating function is

a. $\frac{p}{q^2}$

b. $\mathbf{q^2P}$

c. $\frac{q^2}{P}$

d. $qP^2$

90: The radius of Earth is approximately 6400 km. The height h at which the acceleration due to Earth’s gravity differs from g at the Earth’s surface by approximately 1% is

a. 64 km

b. 48 km

c. 32 km

d. 16 km

91: The Hamiltonian of a classical particle moving in one dimension is $H = \frac{p^2}{2m}+\alpha q^4$ where α is a positive constant and p and q are its momentum and position respectively. Given that its total energy $E\le E_o$ the available volume of phase space depends on Eο as

a. $\mathbf{E_o^{34}}$

b. E_◦

c. $\sqrt E_o$

d. is independent of E_◦

92: A mechanical system is described by the Hamiltonian $H (q, p) =\frac{p^2}{2m}+\frac{1}{2}m\omega^2q^2$ . As aresult of the canonical transformation generated by $F (q.Q) = -\frac{Q}{a}$ , the Hamiltonian in the new coordinate Q and momentum P becomes

a. $\frac{1}{2m}Q^2P^2+\frac{m\omega^2}{2}Q^2$

b. $\frac{1}{2m}Q^2P^2+\frac{m\omega^2}{2}P^2$

c. $\frac{1}{2m}P^2+\frac{m\omega^2}{2}Q^2$

d. $\mathbf{\frac{1}{2m}Q^2P^4+\frac{m\omega^2}{2}P^{-2}}$

93: A particle moves in two dimensions on the ellipse $\mathbf{x^2\ +\ 4y^2\ =\ 8}$ . At a particular instant it is at the point (x, y) = (2,1) and the x-component of its velocity is 6 (in suitable units). Then the y−component of its velocity is

a. −3

b. −2

c. 1

d. 4

94. Consider three inertial frames of reference A, B and C. The frame B moves with a velocity c/2 with respect to A, and C moves with a velocity c/10 with respect to B in the same direction. The velocity of C as measured in A is

a. 37c

b. 47c

c. 7c

d. √73c

95: If the Lagrangian of a dynamical system in two dimensions is $L\ =\ \frac{1}{2}m{\dot{x}}^2\ +\ m\dot{x}\dot{y}$ ˙, then its Hamiltonian is

a. $H=\frac{1}{m}P_xP_y+\frac{1}{2m}P_y^2$

b. $H=\frac{1}{m}P_xP_y+\frac{1}{2m}P_x^2$

c. $\mathbf{H=\frac{1}{m}P_xP_y-\frac{1}{2m}P_y^2}$

d. $\frac{1}{m\ }P_xP_y-\frac{1}{2m}P_x^2$

96: A particle of mass m moves in the one-dimensional potential $V\ \left(x\right)=\frac{\alpha}{3}x^3\ +\frac{\beta}{4}x^4$ , where α, β > 0. One of the equilibrium points is x=0 . The angular frequency of small oscillation about the other equilibrium point is

a. $\frac{2\alpha}{\sqrt{3m\beta}}$

b. $\frac{\alpha}{\sqrt{m\beta}}$

c. $\frac{\alpha}{\sqrt{12m\beta}}$

d. $\frac{\alpha}{\sqrt{24m\beta}}$

97: A particle of unit mass moves in the xy−plane in such a way that ˙ x(t) = y(t) and y˙(t) = -x(t). We can conclude that it is in a conservative force-field which can be derived from the potential

a. $\mathbf{\frac{1}{2}(x^2+y^2)}$

b. $\frac{1}{2}(x^2-y^2)$

c. x+y

d. x-y

98: Consider a particle of mass m moving with a speed v. If T_R denotes the relativistic kinetic energy and T_N its non-relativistic approximation, then the value of $\frac{T^R-T^N\ }{{T_T}_R}$ for v =0.01 c, is

a. $1.25\ \times\ {10}^{-5}$

b. $5.0\ \times\ {10}^{-5}$

c. $7.5\ \times\ {10}^{-5}$

d. None of these

99: A particle in two dimension is in a potential V (x, y) = x+2y . Which of the following (apart from the total energy of the particle) is also a constant of motion?

a. $\mathbf{p_y\ -\ 2p_x}$

b. $p_x\ -\ 2p_y$

c. $p_x\ +\ 2p_y$

d. $p_y\ +\ 2p_x$

100: After a perfectly elastic collision of two identical balls, one of which was initially at rest, the velocities of both of the balls ar e non-zero. The angle θ the final, velocities (in the lab frame) is

a. $\theta=\frac{\pi}{2}$

b. $\theta=\pi$

c. $0<\ \theta\le\frac{\pi}{2}$

d. $\frac{\pi}{2}<\ \theta\ \le\ \pi$

101: The total angular momentum of system of particles is constant if:

a. Total force is zero

b. Total energy is zero

c. Total torque is zero

d. Total momentum is zero

102: The ……….. force deflects air in the northern hemisphere to right producing cycloid motion:

a. Centripetal

b. Coriolis

c. Euler

d. Electric

103: Acceleration of an object depends on:

a. Force

b. Mass

c. a & b

d. Unpredictable

103: For a system of many particles the kinetic energy consists of:

a. One part

b. Two parts

c. Three parts

d. Many parts

104: Gravitational force is:

a. Central force

b. Attractive force

c. Conservative

d. All of these

105: If total external force is zero then total linear momentum is:

a. Zero

b. Conserved

c. Maximum

d. Minimum

106: The sum of all external forces on a system of many particles is zero. Which of the following must be true for the system:

a. Total mechanical energy is constant

b. Total linear momentum is constant

c. Total potential energy is constant

d. Total kinetic energy is constant

107: If work done is independent of path followed then force is:

a. Frictional

b. Non-conservative

c. Conservative

d. Imaginary

108: Everybody continues to be in a state of rest or of uniform motion in a straight line unless it is compelled by an external force. this statement is called:

a. Newton’s 3^rd law of motion

b. Newton’s 2^nd law of motion

c. Newton’s 1^st law of motion

d. Newton’s law of gravitation

109: Newton’s laws are valid in:

a. Inertial frame

b. Non inertial frame

c. a & b

d. None

109: The branch of physics which deals with ordinary materials is called:

a. Quantum

b. Classical

c. Solid state physics

d. Condensed matter

110: If forces acting on a particle are conservative then:

a. $T+V=0$

b. $L+T=0$

c. $T+V=constant$

d. $L-T=constant$

111: Angular momentum is given by:

a. $L=m_i\left(rv\right)$

b. $\mathbf{m_i\left(r_i\times v_i\right)}$

c. $L=m_i(r_i+v_i)$

d. $L=m_fv_f-m_iv_i$

112: Rate of change linear momentum gives:

a. Impulse

b. Torque

c. Force

d. Velocity

113: A particle of mass M moving with speed v collides with a stationary particle of equal mass. After collision both particles move. Let β be the angle between two velocity vectors. If collision is inelastic then:

a. β is always equal to 90^o

b. β is always less than 90^o

c. β is always greater than 90^o

d. Insufficient data to calculate

114: Newtonian mechanics is not applicable to:

a. Particles moving with speed of light

b. Particles moving with speed comparable to speed of light

c. Objects of atomic size

d. All of these

115: For conservative force:

a. $\oint_{C}\vec{F}.d\vec{r}=0$

b. $\vec{\nabla}\times\vec{F}=0$

c. a & b

d. Cannot be answered

116: Newton’s 1^st law of motion is applicable to only:

a. Bound particles

b. Free particles

c. a & b

d. None

117: A body keeps moving once set in motion. This property is called:

a. Momentum

b. Force

c. Inertia

d. Torque

118: A particle of mass M moving with speed v collides with a stationary particle of equal mass. After collision both particles move. Let β be the angle between two velocity vectors. If collision is inelastic then:

a. β is always equal to 90^o

b. β is always less than 90^o

c. β is always greater than 90^o

d. Insufficient data to calculate

119: A body is moving in a straight line by a machine delivering constant power. The distance d moved by body in time t is:

a. $t^\frac{1}{2}$

b. $\mathbf{t^\frac{3}{2}}$

c. $t^2$

d. $t^\frac{3}{4}$

120: A tube of length l is completely filled with an incompressible liquid of mass M and closed at both ends. The tube is then rotated in a horizontal plane about one of its ends with a uniform angular velocity ω. The force exerted by liquid at the other end is:

a. $F=m\omega^2l$

b. $F=\frac{1}{2}m\omega^2l$

c. $\mathbf{F=\frac{1}{4}m\omega^2l }$

d. $\frac{1}{2}m\omega^2l^3$

121: Mutual interaction force between two particles can change:

a. Linear momentum but not kinetic energy

b. Linear momentum as well as kinetic energy

c. Kinetic energy but not linear momentum

d. All of these

122: Angular momentum of particles is defined as:

a. Kinetic energy

b. Linear momentum

c. Torque

d. Mass

123: The work energy theorem states that the work done is equal to the change in:

a. Potential energy

b. Kinetic energy

c. Torque

d. Force

124: Rate of change of angular momentum is called:

a. Linear momentum

b. Force

c. Torque

d. Impulse

125: Forces of constraints do ………… work for principle of virtual work done:

a. Negative

b. Positive

c. Zero

d. Maximum

126: Constraints that can be expressed as equations of coordinates and time i.e. $f\left(r_1,r_2,r_3,\ldots\ldots r_N,t\right)=0$ are said to be:

a. Holonomic

b. Non-Holonomic

c. Cruciform

d. Simple

127: Lagrangian is a ………. Quantity:

a. Vector

b. Scalar

c. Dyadic

d. All of these

128: Lagrange is a function of:

a. Variable energies

b. Conserved energies

c. Difference of energies

d. Sum of energies

129: The term $Q_j=\sum_{i}{F_i^a.\frac{\partial r_i}{\partial q_i}}$ is called:

a. Generalized work

b. Generalized force

c. Force

d. Virtual work

130: Generalized displacement of a rigid body is represented by …….. motion:

a. Vibrational

b. Rotational

c. Translational

d. a & c

131: Lagrangian is a function of:

a. Generalized coordinates

b. Generalized velocity

c. Time

d. All of these

132: A rigid body is moving in space has degree of freedom:

a. Three

b. Five

c. Nine

d. Six

133: Constraints of rigid body has:

a. Holonomic

b. Rheonomic

c. Scleronomic

d. a & c

134: Generalized coordinates are:

a. Necessarily spherical coordinates

b. Necessarily Cartesian coordinates

c. Depend on each other

d. Independent of each other

135: A particle constrained to move along any curve is an example of:

a. Non-holonomic constraints

b. Holonomic constraints

c. Isolated constraints

d. Scleronomic constraints

136: In case of Atwood machine the number of independent coordinates are:

a. 2

b. 4

c. 3

d. 1

137: At equilibrium the virtual work of applied force is:

a. Maximum

b. Zero

c. Minimum

d. None

138:The degrees of freedom of spherical pendulum is:

a. 2

b. 4

c. 3

d. 1

139: In case of simple pendulum the number of independent coordinates are:

a. 1

b. 4

c. 3

d. 2

140: Configuration space is also known as:

a. 3N-dimensional space

b. 2N-dimesnional space

c. N-dimensional space

d. None

141: If a particle is constrained to move on surface the force of constraint is then:

a. Perpendicular to surface

b. Parallel to surface

c. Antiparallel to surface

d. Tangent to surface

142: A system is in equilibrium if total work done by the applied force is zero then it is called

a. Principle of virtual work

b. Hamiltonian’s principle

c. Principle of static equilibrium

d. None

143: The constraint on a bead on a uniformly rotating wire in a force free space is:

a. Scleronomic

b. Holonomic

c. Rheonomic

d. All of these

144: If generalized coordinate is then corresponding generalized force has dimensions of:

a. Energy

b. Torque

c. Momentum

d. Impulse

145: If generalized coordinate has dimension of velocity then generalized velocity has the dimensions of:

a. Force

b. Displacement

c. Acceleration

d. Momentum

146: The number of independent coordinates required to described a system are called:

a. Generalized coordinates

b. Cartesian coordinates

c. a & b

d. None

147: Generalized coordinates are:

a. Dependent quantity

b. Independent quantity

c. Measurable quantity

d. None

148: D’ Alembert principle states that a dynamic system is in equilibrium if virtual work done by the actual force plus reverse effective force is :

a. Negative

b. Positive

c. Zero

d. Maximum

149: Any set of parameters that can be conveniently used to specify the configuration of system are called as:

a. Configuration space

b. Generalized coordinates

c. Position coordinates

d. None

150: Principle of virtual work is valid for:

a. Sliding motion

b. Rolling contact with slipping

c. Frictionless surface

d. a & b

151: For D’ Alembert principle constraint force is……… to virtual displacement :

a. Parallel

b. Perpendicular

c. Equal

d. Antiparallel

152: A particle constrained to move along the inner surface of a fixed hemispherical bowl. The number of degrees of freedom of particle is:

a. Three

b. One

c. Two

d. Unknown

152: The shortest distance between two points in space will be:

a. Parabola

b. Catenary

c. Straight line

d. Circle

153: A usual expression for the conserved angular momentum in central force problem is:

a. $l=mr^2\theta$

b. $\mathbf{ l=mr^2\dot{\theta}}$

c. $l=mr^2\theta^2$

d. $l=mr^2{\dot{\theta}}^2$

154: The derivation of Euler-Lagrange equation is a problem of:

a. Differentiation

b. Integration

c. Calculus of variation

d. All of these

154: The homogeneity of time leads to law of conservation of:

a. Angular momentum

b. Linear momentum

c. Energy

d. Parity

155: Choose the correct statement:

a. In ∆-variation, time as well as position coordinates are allowed vary

b. $\delta-variation$ dose not involves time

c. a & b

d. None

156: Conservation of areal velocity ………………. For planetary motion only

a. Limited case

b. Unlimited case

c. Holds

d. None

157: A particle is moving under central force about a fixed center of force. Choose the correct statement:

a. Motion of particle is always on circular path

b. Its angular momentum is conserved

c. Motion of particle takes place in plane

d. b & c

158: Central force is:

a. Infinite

b. Conservative

c. Non-conservative

d. None

159: Area covered by planet in equal intervals of time is:

a. Variable

b. Constant

c. Zero

d. Stationary

160: In planetary motion orbits are elliptical with axis:

a. Central axis

b. Perpendicular axis

c. Major axis and minor axis

d. All of these

161: Force between Sun and Planet is:

a. Electrostatic force

b. Gravitational force

c. Electromagnetic force

d. None

162: Two particles of masses m and 2m interacting via gravitational force are rotating about common center of mass with angular velocity w at a fixed distance r. If the particle of mass 2m is taken at origin O. Then

a. The force between them can be represented by $F=\mu\omega^2r$

b. In the inertial frame the origin $O$ is moving on a circular path of radius $\frac{r}{3}$

c. a & b

d. None of these

163: In central force problem, the conservation of angular momentum is equivalent to saying that:

a. Total energy is constant

b. Linear momentum is constant

c. Effective potential is constant

d. Areal velocity is constant

165: The force of attraction between Sun and planet is a central force and is given by:

a. Coulomb’s law

b. Newton’s 2^nd law

c. Gravitational law

d. Kepler’s law

166: The ratio of number of particles scattered into solid angle per unit time per incident intensity is known as:

a. Cross-section of scattering

b. Scattered particle length

c. Solid angle

d. Radian

167: A particle is moving on elliptical path under inverse square law force of form $f\left(r\right)=-\frac{k}{r^2}$ . The eccentricity of orbit is:

a. Function of total energy

b. Function of angular momentum

c. a & b

d. Independent of angular momentum

168: In the Rutherford scattering formula both incident and target were:

a. Negative charge

b. Same charge

c. Positive charge

d. Opposite charge

169: The force due to deflection of body is called:

a. Pseudo force

b. Coriolis force

c. Deflecting force

d. Apparent force

170: The maximum and minimum velocities of a satellite are $v_1$ and $v_2$ respectively. The eccentricity of orbit of satellite is:

a. $e=\frac{v_1}{v_2}$

b. $e=\frac{v_2}{v_1}$

c. $e=\frac{v_1+v_2}{v_1-v_2}$

d. $e=\frac{v_1-v_2}{v_1+v_2}$

171: The areal velocity of planet remains constant is known as Kepler’s :

a. 1^st law

b. 2^nd law

c. 3^rd law

d. None

172: The value of eccentricity for an elliptical orbit is:

a. $e=\frac{v_1-v_2}{v_1+v_2}$

b. $\in=1$

c. $\in=0$

d. $\mathbf{0}<\in<\mathbf{1}$

173: Rutherford differential cross-section has dimensions of:

a. Solid angle

b. Area

c. Length

d. Volume

174: Kepler’s 2^nd law of planetary motion directly follows from:

a. Homogeneity of time

b. Homogeneity of space

c. Law of conservation of angular momentum

d. Law of conservation of linear momentum

175: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This statement is called:

a. Kepler’s 1^st law

b. Kepler’s 2^nd law

c. Kepler’s 3^rd law

d. None

176: Consider a comet of mass moving in a parabolic orbit around the Sun. The closest distance between the comet and Sun is , the mass of Sun is and the universal gravitational constant is . The angular momentum of comet is:

a. $\mathbit{m}\sqrt{\mathbf{2}\mathbit{GMs}}$

b. $M\sqrt{2Gms}$

c. $s\sqrt{GmM}$

d. $G\sqrt{Mms}$

177: A minimum distance of incident particle from the target nucleus at which incident particle is scattered considerably is known as:

a. Radial distance

b. Impact parameter

c. Transverse distance

d. None

178: In motion under central force which of the following is true:

a. Acceleration is constant

b. Linear momentum is conserved

c. Angular momentum is conserved

d. All of these

179: The Lagrangian for Sun-Earth system is:

a. $\mathbf{L}=\frac{\mathbf{1}}{\mathbf{2}}\mathbf{m}{\dot{\mathbf{r}}}^\mathbf{2}+\frac{\mathbf{1}}{\mathbf{2}}\mathbf{m}{\dot{\mathbf{r}}}^\mathbf{2}{\dot{\mathbf{\theta}}}^\mathbf{2}+\frac{\mathbf{GMm}}{\mathbf{r}}$

b. $L=\frac{1}{2}m{\dot{r}}^2+\frac{1}{2}m{\dot{r}}^2{\dot{\theta}}^2-\frac{GMm}{r}$

c. $L=\frac{3}{2}m{\dot{r}}^2+\frac{GMm}{r}$

d. $L=\frac{1}{2}m{\dot{r}}^2{\dot{\theta}}^2-\frac{GMm}{r}$

180: In Rutherford scattering experiment the path of scattered particle is:

a. Parabola

b. Hyperbola

c. Ellipse

d. Straight line

189; A system with respect to which centre of mass of a particle is at rest is called:

a. Laboratory coordinate system

b. Cartesian coordinate system

c. Centre of mass coordinate system

d. Polar coordinate system

190: If a particle is moving under a potential of form $V\left(r\right)=-\frac{k}{r}$ then trajectory made by particle is a:

a. Helix

b. Great circle

c. Straight line

d. Conic section

191: A system that possesses …………. Symmetry, conserve angular momentum:

a. Translational

b. Spherical

c. Space reflection

d. Time reversal

192: Canonical equations of motion represent ……… relations:

a. $n$

b. $\mathbf{2n}$

c. n+1

d. 2n+1

193: Hamilton’s equations of motion are…….. equations:

a. Linear

b. Quadratic

c. 1^st order differential

d. 2^nd order differential

194: Poisson brackets play the same role in classical mechanics as ………. Play in quantum mechanics:

a. Commutation relations

b. Operator

c. Wave function

d. Probability

195: Jacobi identity and anti-symmetry property of Poisson brackets together define:

a. Linear Algebra

b. Lagrangian formalism

c. Hamiltonian formalism

d. None

196: The product of generalized coordinates and its conjugate momentum has dimensions of:

a. Angular momentum

b. Linear momentum

c. Energy

d. Force

197: Hamilton function can be represented by:

a. $f\left(V,p,t\right)$

b. $\mathbf{f}\left(\mathbf{p},\mathbf{q},\mathbf{t}\right)$

c. $f\left(q,p,t\right)$

d. $f(c,V,t)$

198: The Lagrangian of a particle moving in a plane under the influence of a central potential is given by $L=\frac{1}{2}m\left({\dot{r}}^2+r^2{\dot{\theta}}^2\right)-V\left(r\right)$ . The generalized momenta corresponding to r and \theta are given:

a. $m{\dot{r}}^2,\ mr^2\dot{\theta}$

b. $m{\dot{r}}^2,\ mr^2{\dot{\theta}}^2$

c. $\mathbf{m\dot{r},\ mr^2\dot{\theta}}$

d. $Unpredictable$

199: Legendre transformation creates a new function :

a. With different transformation but same variable

b. With same transformation but different variables

c. No transformation

d. None

200: We apply Legendre transformation to the potential energy and transform it into:

a. Entropy

b. Pressure

c. Volume

d. Enthalpy

201: The dimensions of generalized momentum are:

a. Always those of angular momentum

b. Always those of linear momentum

c. Maybe those of angular momentum

d. None

202: Isotropy of space means that the system is invariant under:

a. Translation in time

b. Rotation in space

c. Translation in space

d. All of these

203: For a system of N-particles the dimension of phase space is:

a. 2N

b. 3N

c. $\mathbf{6N}$

d. 9N

204: The Hamiltonian can be constructed from the Lagrangian using the formula:

a. $H=\dot{p_i}\dot{q_i}-L$

b. $\mathbf{H=p_i\dot{q_i}-L}$

c. $H=\frac{\partial L}{\partial\dot{q_i}}$

d. $H=\frac{1}{L}$

205: Hamiltonian is a function of:

a. Variable energies

b. Conserved energies

c. Sum of energies

d. Different energies

206: Hamilton canonical equations of motion for a conservative system are:

a. $\dot{q_i}=-\frac{\partial H}{\partial\dot{p_i}}\ and\ \dot{p_i}=\frac{\partial H}{\partial\dot{q_i}}$

b. $\dot{q_i}=\frac{\partial H}{\partial\dot{q_i}}\ and\ \dot{p_i}=-\frac{\partial H}{\partial\dot{p_i}}$

c. $\dot{q_i}=-\frac{\partial H}{\partial\dot{q_i}}\ and\ \dot{p_i}=\frac{\partial H}{\partial\dot{p_i}}$

d. $\dot{\mathbf{q}\mathbf{i}}=\frac{\partial H}{\partial\dot{\mathbf{p}\mathbf{i}}}\ \mathbf{and}\ \dot{\mathbf{p}\mathbf{i}}=-\frac{\partial H}{\partial\dot{\mathbf{q}\mathbf{i}}}$

207: Law of conservation of linear momentum is a consequence of:

a. Homogeneity of time

b. Homogeneity of space

c. Isotropy of space

d. All of these

208: Law of conservation of energy is a consequence of:

a. Homogeneity of space

b. Isotropy of space

c. Homogeneity of time

d. a & b

209: Whenever the Lagrangian for a system does not contain a coordinate explicity:

a. $q_k$ is cyclic coordinate

b. $p_k$ is cyclic coordinate

c. $p_k$ , the generalized momentum is a constant of motion

d. a & c

210: Law of conservation of angular momentum is a consequence of:

a. Homogeneity of space

b. Homogeneity of time

c. Isotropy of space

d. a & b

211: If Lagrangian of a closed system is invariant under rotation then the ……… of a system is constant vector in time:

a. Linear momentum

b. Angular momentum

c. Force

d. Kinetic energy

212: The generalized momentum p_xof a particle of mass m with velocity in an electromagnetic field is:

a. $p_x=mv_x$

b. $p_x=qv_xA_x$

c. $p_x=mv_x-qA_x$

d. $\mathbf{p_x=mv_x+qA_x}$

213: The angular momentum of coordinate moving under central force is:

a. Zero

b. Variable

c. Negative

d. Constant

214: Position and momentum space in combined form is called:

a. Polar space

b. Cartesian space

c. Phase space

d. Hilbert space

215: Results of Lagrangian and Hamiltonian analysis are always:

a. Different

b. Same

c. Sometimes same sometimes different

d. None

216: The product of generalized momentum and associated coordinate have dimensions of:

a. Angular momentum

b. Linear momentum

c. Force

d. Energy

217: Choose the correct statement:

a. The angular momentum is conserved for a system possessing rotational symmetry

b. If Lagrangian of a system is invariant under translation along a direction the corresponding linear momentum is conserved

c. For a conservative system the Hamiltonian is equal to sum of kinetic energy and potential energy

d. All of these

218: In absence of a given component of applied force the corresponding component of linear momentum is:

a. Not conserved

b. Conserved

c. Equal to force

d. None

219: Whenever the Lagrangian function does not contain a coordinate explicity , the generalized momentum is:

a. Holonomic

b. Derivative of motion

c. Integral of motion

d. None

220: If the Lagrangian is cyclic in then:

a. $p_i$ is not conserved

b. $\mathbf{p}_\mathbf{i}$ is conserved

c. $q_i$ appears in Lagrangian

d. All of these

221: If A and B are any two constants of motion their Poisson bracket is:

a. Invariant

b. Zero

c. Constant of motion

d. Covariant

231: The transformation $\left(q,p\right)\rightarrow\left(Q,P\right)$ is canonical if:

a. $\left{q,P\right}=1$

b. $\left{Q,p\right}=1$

c. ${P,P}=1$

d. $\left{\mathbf{Q},\mathbf{P}\right}=\mathbf{1}$

232: The Poisson bracket of two dynamical variables f & g obeys:

a. $\left{\mathbf{f},\mathbf{g}\right}={\mathbf{g},\mathbf{f}}$

b. $\left{f,g\right}={g,g}$

c. $\left{f,g\right}=fg$

d. $\left{f,g\right}=-{g,f}$

233: A dynamical variable A $\left(q_i,p_i\right)$ is a constant of motion if its Poisson bracket commutes with:

a. Lagrangian

b. Hamiltonian

c. Angular momentum

d. Energy

234: If the Poisson bracket of a function with Hamiltonian vanishes then function is:

a. Constant of motion

b. Does not depend on time explicity

c. a & b

d. None

235: The correct relations for Poisson brackets are:

a. $\left[\ q_i,\ q_j\right]=\ \delta_{ij}$

b. $\left[\ q_i,\ p_j\right]=1$

c. $\left[\ q_i,\ q_j\right]=0$

d. b & c

236: For the Lagrange brackets which of the following is correct:

a. $\left{q_i,\ q_j\right}=\ \delta_{ij}$

b. $\left{\ q_i,\ p_j\right}=\ \delta_{ij}$

c. $\left{\ p_i,\ p_j\right}= 0$

d. b & c

237: Poisson brackets for angular momentum components $L_x,\ L_y,\ L_z$ satisfy the relation:

a. $\left[L_x,p_x\right]=0$

b. $\left[L_x,p_z\right]=-p_y$

c. $\left[L_y,L_z\right]=L_x$

d. All of these

238: If p_i and q_i(i=1,2,3) represent the momentum and position coordinates respectively for a particle:

a. The configuration space is three dimensional

b. The phase space is six dimensional

c. a & b

d. The configuration space is six dimensional

239: The phase space refers to:

a. Position coordinates

b. Momentum coordinates

c. Position and momentum coordinates

d. All of these

240: If we make a canonical transformation from set of variables $\left(P_i,\ Q_i\right)$ to new set of variables $\left(p_i,\ q_i\right)$ and the transformed Hamiltonian is identically zero then the:

a. New variables are constant in time

b. New variables are cyclic

c. Old variables remain constant in time

d. a & b

241: For a one dimensional harmonic oscillator the representative point in two dimensional phase space traces the:

a. Hyperbola

b. Parabola

c. Ellipse

d. Straight line

242: In case of canonical transformations:

a. Hamilton’s principle is satisfied in old as well as new variables

b. The form of Hamilton equations is preserved

c. a & b

d. None

243: If the generating function has form $F=\left(q_i,\ p_i,t\right)$ then:

a. $\mathbf{p_i=\frac{\partial F}{\partial q_i}\ and\ Q_i=\frac{\partial F}{\partial P_i}}$

b. $p_i=-\frac{\partial F}{\partial q_i}\ and\ Q_i=\frac{\partial F}{\partial P_i}$

c. $p_i=\frac{\partial F}{\partial q_i}\ and\ Q_i=-\frac{\partial F}{\partial P_i}$

d. $p_i=-\frac{\partial F}{\partial q_i}\ and\ Q_i=-\frac{\partial F}{\partial P_i}$

244: Choose the correct statement:

a. The generating function $F=\sum_{i}{q_iP_i}$ generates the identity transformation

b. The generating function $F=-\sum_{i}{q_iP_i}$ generates the identity transformation $p_i=-P_i and {Q_i=-q}_i$

c. The generating function $F=\sum_{i}{q_iP_i}$ cannot generate the identity transformation

d. a & b

245: Which one of the following is known as Newton’s law?

a. Law of inertia

b. Action ad reaction

c. Law of force

d. a & b

246: For a conservative force curl of must be equal to ____?

a. 1

b. Infinity

c. Zero

d. None

247: Laws of newton are not applicable to____?

a. Quantum mechanics

b. Classical mechanics

c. None

248: The strong form of newton’s law is valid if two forces have?

a. Same line of action

b. Different line of action

c. Same direction

d. a & c

249: Which one of the following assumption we made about the inertial space during the motion of free particle in it?

a. Homogenous

b. Isotropic

c. Heterogeneous

d. a & b

250: The mass determining the acceleration of a body under the action of force is called ____?

a. Inertial mass

b. Accelerated mass

c. Gravitational mass

251: ____ are the factors that restrict the motion of a body?

a. Constraints

b. Forces

c. Friction

d. None

252: Scleronomic and rheonomic are the types of _____ constraint?

a. Non-holonomic

b. Holonomic

c. a & b

d. None

253: Holonomic constraints are also known as?

a. Differentiable constraints

b. Integrable constraints

c. Partially integrable constraints

d. Non-integrable constraints

254: Gas molecules in a container, particles moving in a sphere are the examples of?

a. Holonomic constraint

b. Non-Holonomic constraint

c. Scleronomic constraint

d. Rheonomic constraint

255: The constraints which contain time explicity are called _____?

a. Holonomic constraint

b. Non-Holonomic constraint

c. Scleronomic constraint

d. Rheonomic constraint

256: A scleronomic constraint is independent of time.

a. True

b. False

257: Configuration space is ____ dimensional space?

a. 1

b. 3

c. n

d. Infinite

258: Which one of the following is the property of virtual displacement?

a. Infinitesimal

b. Occur at given instant of time

c. Consistent with constraints

d. All

258: For what type of equilibrium Principle of Virtual Work is applicable?

a. Static equilibrium

b. Dynamic equilibrium

c. Both

d. None

259: D’ Alembert Principle is suggested by and developed by ?

a. D’Alembert, D’Alembert

b. D’ Alembert , Bernoulli

c. Bernoulli , D’ Alembert

d. Bernoulli, Bernoulli

260: What is the angle between constraints and virtual displacement?

a. 0

b. 90

c. 45

d. 180

261: Which law of newton is used in D’ Alembert principle?

a. 1st

b. 2nd

c. 3rd

d. None

262: For what type of equilibrium is D’ Alembert principle applicable?

a. Static equilibrium

b. Dynamic equilibrium

c. Both

d. None

263: Sum of internal forces is equal to ____?

a. 0

b. 1

c. F

d. None

264: The Lagrange from Lagrangian equation is equal to?

a. T = L –V

b. L= T- V

c. L= V – T

d. L = T –Q

265: What is the degree of freedom of Atwood machine?

a. One

b. Two

c. Three

d. Four

267: What is the kinetic energy of simple pendulum?

a. $\mathbf{12m}\mathbf{l}^\mathbf{2}\mathbf{\theta}^\mathbf{2}$

b. $12mv^2$

c. $12mx^2$

d. None

268: Which principle is used to find the trajectory of path at any instant of time?

a. Lagrange

b. Hamilton

c. D’ Alembert

d. None

269: Hamilton’s principle is also known as ___?

a. Variational principle

b. Principle of least action

c. Action integral

d. a & b

270: Path of projectile is parabola is an example of?

a. Lagrange

b. Hamilton

c. D’ Alembert

d. None

272: What is the equation of parabola?

a. $\mathbf{y}=\mathbf{a}\mathbf{x}^\mathbf{2}+\mathbf{bx}$

b. $x =-ax^2+bx$

c. $x=-ay^2+by$

d. None

273: Which one of the following is a central force?

a. Gravitational force

b. Coulomb force

c. Magnetic force

d. a & b

274: What is the property of central force motion?

a. P=0

b. L=0

c. F=0

275: For a central force the magnitude depends only on distance from center.

a. True

b. False

276: In central motion direction of angular momentum is always _____ to plane?

a. Tangent

b. Perpendicular

c. Normal

277: The system in which one of the particle is moving while other is at rest is called _______?

a. Mass coordinate system

b. Laboratory coordinate system

c. Acceleration coordinate system

d. None

278: The direction of motion of incident particle is center of mass is _____ as the direction of motion of incident particle in laboratory frame.

a. Same

b. Different

c. Opposite

d. None

279: At which angle in laboratory coordinate system, the two particles move off to one another after collision?

a. Normal

b. Tangent

c. Right angle

d. None

280: Impact parameter is the perpendicular distance between center of force and incident velocity.

a. True

b. False

281: From which particle of atomic nuclei Rutherford scattering formula is derived?

a. Alpha

b. Beta

c. Gamma

282: The set of 2n equation is known as ______?

a. Hamilton equation

b. Canonical equation

c. Equation of motion

d. All of above

283: A coordinate that do not appear explicitly in the Lagrangian of a system is said to be _____?

a. Cyclic coordinate

b. Ignorable coordinate

c. a & b

d. None

284: The generalized momentum conjugate to a cyclic coordinate remains _____ during a motion

a. Constant

b. Conserved

c. Non-conserved

d. None

285: Homogeneity of time leads to the conservation of ______?

a. Linear momentum

b. Energy

c. Angular momentum

d. All

286: Homogeneity of space leads to the conservation of ______?

a. Energy

b. Linear momentum

c. Angular momentum

d. All

287: Isotropy of space leads to the conservation of ______?

a. Energy

b. Linear momentum

c. Angular momentum

d. All

288: For which quantity Hamiltonian is constant?

a. t

b. p

c. L

d. V

289: Which one of the following equation is true about Hamiltonian?

a. H=E

b. H=p

c. H=Q

d. H=L

290: Inverse of canonical transformation is_____?

a. Identity

b. Canonical

c. Non-canonical

d. Conserved

291: Which system constitute Poisson bracket?

a. Holonomic constraint

b. Non-Holonomic constraint

c. Scleronomic constraint

d. Rheonomic constraint

292: The concept of Poisson bracket was introduced by?

a. F.D. Poisson

b. S.D. Poisson

c. G.C. Poisson

d. None

293: Poisson first theorem states that ____?

a. $\left[\mathbf{u},\mathbf{H}\right]=\mathbf{0}$

b. $\left[u,P\right]=0$

c. $\left[H,P\right]=0$

d. All of these

294: The equation $\left[F,G\right]=constant$ belongs to?

a. Poisson first theorem

b. Poisson second theorem

c. Poisson third theorem

d. All of these

295: The canonical equations of motion are ______ in the property of Poisson bracket?

a. Constant

b. Implicit

c. Explicit

d. Conserved

296: Which one of the following is the type of canonical transformation?

a. Point transformation

b. Coordinate transformation

c. a & b

d. None

297: According to the principle of least action:

a. $\int{\left(\sum_{k}{p_k\dot{q_k}}-H\right)dt=0}$

b. $\int{\sum_{k}{p_k\dot{q_k}}dt=0}$

c. $\int{(H+L)dt=0}$

d. b & c

298: The modified Hamilton’s principle is given by:

a. $\mathbf{\delta\sum_{j}{\int_{t_1}^{t_2}{p_jdq_j-\delta}\int_{t_1}^{t_2}Hdt=0}}$

b. $\delta\sum_{j}{\int_{t_1}^{t_2}{p_jdq_j+\delta}\int_{t_1}^{t_2}Hdt=0}$

c. $\delta\sum_{j}{\int_{t_1}^{t_2}{p_jdq_j-\delta}\int_{t_1}^{t_2}{Hdq_j}=0}$

d. $\delta\sum_{j}{\int_{t_1}^{t_2}{p_jdq_j-\delta}\int_{t_1}^{t_2}{Hdq_j}=0}$

299: Hamilton’s principle function S and Hamilton’s characteristic function W for conservative system are related as:

a. S=W

b. $\mathbf{S}=\mathbf{W}-\mathbf{Et}$

c. S=W+Et

d. S is not related to

300: The action and angle variables have dimensions of:

a. Force and angle

b. Energy and angle

c. Angular momentum and angle

d. Dimensionless quantities

MCQ's with Answers

1: The number of independent ways in which a mechanical system can move without violating any constraint is called

2: The generalized momentum Pi need not always of linear momentum

3: A Single particle in a space requires three coordinates, so it has degree of freedom

4: The sixth coordinates must satisfy a single constraints equation for a · · ·

5: In degree of freedom the · · · number of coordinates required to specify the configuration of a dynamical system

6: If constrains can be expressed as a function of position then it is called

7: If constrains are stationary then it is called

8: First difficulty of constraints in the solution of mechanical problems

9: Rigid body and gas molecules within the wall of container are example of

10: The square of the period is proportional to semi major axis of

11: Further away from planet is from sun · · · it takes what period reaches orbit to sun

12: The area per unit time swept out by a radius vector from sun to planet becomes

13: Planets move in · · ·. Around sun

14: the angular momentum of rigid body is L and its K.E is halved. What happens to its angular momentum

15: If a linear momentum is increased by 50% K.E will increase by what percent?

16: The gravitational force between two masses is

17: The value of universal gravitational constant G is

18: The angular momentum is in a central force field

19: The areal velocity of the particle in a central force field is

20: At the turning point in an arbitrary potential field the radial velocity is

21: For hyperbolic orbit the values of energy E and eccentricity ϵ are

22: For parabolic orbit the values of energy E and eccentricity ϵ are

23: For elliptical orbit the values of energy E and eccentricity ϵ are

24: For circular orbit the value of eccentricity

25: All the planet moves around the Sun in orbit

26: In the formation of the cycloidal curves, the circle which rolls with a fixed point without

27: · · · is a curve generated by a point on the circumference of a circle, which rolls without

28: Type of curve is created by the intersection of a plane parallel to the side of cone?

29: Type of curve is created by the intersection of a plane with a cone which makes an angle with the axis greater than the angle between the side of the cone and the axis?

30: Curve generated by a point on the circumference of a circle, which rolls without slipping along outside of another circle is known as

31: Curved traced out by a point which moves uniformly both about the center and at the same time away or towards the center is known as

32: Poisson bracket is used to define a?

33: The Poisson bracket is an important binary operation in

34: Hamilton’s formulation of classical mechanics made use of a mathematical tool called

35: the bracket of p with any reasonably smooth function of q is

36: Lagrange’s bracket is

37: Poisson’s bracket is

38: As there are three generalized coordinates, then Hamilton’s canonical equations will be · · · in number

39: The generalized momentum is also called

40: Which of the following answer is true for the Lagrange’s bracket

41: If the Poisson bracket of a function with the Hamiltonian vanishes

42: If we make a canonical transformation from the set of variables (pk, qk) to new set of Variables (Pk, Qk) and the transformed Hamiltonian is identically zero, then

43: Hamiltonian H is defined as

44: Whenever the Lagrangian for a system does not contain a coordinate explicitly.

45: The dimensions of generalized momentum

46: If the Lagrangian does not depend on time explicitly

47: Any · · · on the freedom of movement of a system of particles in the form equation is

48: Example of the Constraints is

49: In the Constraints the particles can move

50: There are only · · · types of constraints

51: The constraints which can be expressed as a function of positions and time are called

52: Example of the Scleronomic constraints

53: If the Constraints depends on time explicitly then it is called

54: The constraints which cannot be expressed as a function of position and time are called

55: In Lagrangian approach, the flow parcels follows

56: Each parcel in the Lagrangian formulation is tagged using

57: Which of these is an acceptable tag for Lagrangian parcels?

58: The Lagrangian equation of motion is order differential equations.

59: In Lagrange’s equation virtual displacement does not involve

60: In Lagrange’s equation if there are N number of particle and so the generalized coordinated are

61: In a simple one-constraints Lagrange multiplier setup, the constraint has to be always one dimension lesser than the objective function.

62: Whenever the Lagrangian for a system does not contain a coordinate explicitly

63: Euler equation govern flows

64: In Euler form of energy equations, which of these terms is not present?

65: The gravitational force between two masses is

66: All the planet moves around the sun in orbit?

67: The combination of total and partial derivative that enters into Euler equation is called

68: One of the possible paths systems will travel through a path which is

70: The acceleration due to gravity (g) on the surface of Earth is approximately 2.6 times that on the surface of Mars. Given that the radius of Mars is about one half the radius of Earth, the ration of the escape velocity on Earth to that on Mars is approximately

71: A constant force F is applied to a relativistic particle of rest mass m. If the particle starts from rest at t = 0, its speed after time t is

74: A planet of mass m moves in the inverse square central force field of the sun of mass M. If the semi-major and semi-minor axes of the orbit are a and b, respectively, the total energy of the planet is

75: An annulus of mass M made of a material of uniform of uniform density has inner and outer radii a and b respectively. Its principle moment of inertia along the axis of symmetry perpendicular to the plane of the annulus is

76: Two events separated by a (spatial) distance 99 m, are simultaneous in one inertial frame. The time interval between these two events is a frame moving with a constant speed 0.8 c (where speed of light c = 3 × 108 m/s) is

77: If the Lagrangian of a particle moving in one dimension is given by L = x2/2 − V (x) the Hamiltonian is

78: A horizontal circular platform mutes with a constant angular velocity Ω directed vertically upward. A person seated at the center shoots a bullet of mass m horizontally with speed v. The acceleration of the bullet, in the reference frame of the shooter, is

80: What is proper time interval between the occurrence of two events i f in one inertial frame events are separated by 7.5 × 108 m and occur 6.5 s apart?

81: A solid cylinder of height H, radius R and density ρ, floats vertically on the surface of a liquid of density ρ◦. The cylinder will be set into oscillatory motion when a small instantaneous downward force is applied. The frequency of oscillation is

82: Two bodies of equal mass m are connected by a massless rigid rod of length l lying in the xy−plane with the center of the rod at the origin. If this system is rotating about the z−axis with a frequency ω, its angular momentum is

83: The muon has mass 105 MeV/c2 and mean lifetime 2.2 µs in its frame. The mean distance traversed by a muon of energy 315 MeV before decaying is approximately

42: If we make a canonical transformation from the set of variables (p_k, q_k) to new set of Variables (P_k, Q_k) and the transformed Hamiltonian is identically zero, then

76: Two events separated by a (spatial) distance 9⁹ m, are simultaneous in one inertial frame. The time interval between these two events is a frame moving with a constant speed 0.8 c (where speed of light c = 3 × 10⁸ m/s) is

77: If the Lagrangian of a particle moving in one dimension is given by L = x²/2 − V (x) the Hamiltonian is

80: What is proper time interval between the occurrence of two events i f in one inertial frame events are separated by 7.5 × 10⁸ m and occur 6.5 s apart?

83: The muon has mass 105 MeV/c² and mean lifetime 2.2 µs in its frame. The mean distance traversed by a muon of energy 315 MeV before decaying is approximately

87: A particle moves in a potential V = x² + y² + z²/2 . Which components(c) of the angular momentum is/are constant(s) of motion?

89: A canonical transformation relates the old coordinates (q, p) to the new ones (Q, P) by the relation Q = q² and P = pq/2. The corresponding time independent generating function is

116: Newton’s 1^st law of motion is applicable to only: