Team Quanta gladly presents all possible short questions of BS Physics book Mechanics – II’s Chapter#01: Rotational Dynamics for the students.
Q.1 Have all point on an object rotating about a fixed axis same angular speed?
Answer: Yes, all point on object rotating about a fixed axis have same angular speed, however all points do not have same linear speed according to relation because it changes with r.
Q.2 Why only for small values of angular displacement is a vector quantity? How can we determine the direction of angular displacement?
Answer: In case of angular motion, law of vector addition holds only in case of small angular displacements, that is why angular displacement will be vector quantity only for small angular displacements. Direction of angular displacement is determined by right hand rule.
Q.3 A disc and hoop start moving down from top of an inclined plane at same time. Which one will be moving faster on reaching bottom?
Answer: Velocity of hoop at bottom of inclined plane is,
$$v_{hoop\ \ \ \ \ \ \ \ =\ \ \ \ \ \ \sqrt{gh}}$$
Velocity of disc at the inclined plane is,
$$v_{disc\ \ \ \ \ =\ \sqrt{\frac{4}{3}gh}\ \ =\sqrt{\frac{4}{3}}\ v_{hoop}}$$
Hence velocity of disc is greater than velocity of hoop on inclined plane.
Q.4 Two cylinders are of same masses but are different radii. Which one of them has greater inertia?
Answer: The cylinder with greater radius has large moment of inertia than the cylinder of small radius of same mass.
Q.5 Can you think of a case in which tangential acceleration is non-zero while at same time centripetal acceleration is zero?
Answer: No, it is not possible.
Q.6 If an object rotates through an angle from rest in time ‘t’, in time t/2 through what angle it rotate?
Answer: Angular acceleration is defined as,
$$\alpha=\frac{w}{t}\ \ \ \ \ \ \ \ \ \rightarrow\alpha=\frac{\varphi}{t^2}\ \ \ \ \ \ \rightarrow\ \ \ \varphi=\ \alpha t^2$$
From above relation, it is clear that in time t/2, object rotates through an angle $\frac{\varphi}{4}$
Q.7 A golfer swings a golf club, making a long drive from the tee. Do all points on the club have the same angular velocity at any instant while the club is in motion?
Answer: Yes, all points on club have same angular velocity because all points on rotating body have same angular velocity at any instant. Angular variables describes the motion of entire body in simple way.
Q.8 A basketball rolls across a classroom floor without slipping, with its center of mass moving at certain speed. A block of ice of the same mass is set sliding across the floor with the same speed along a parallel line. The two objects encounter a ramp sloping upward. Which object will travel farther up the ramp?
Answer: The basketball-Earth system has more kinetic energy than the ice-Earth system due to the rotational kinetic energy of the basketball, so basketball will travel farther up the ramp.
Q.9 Give an example of case in which a body has zero rotational kinetic energy but non-zero translational kinetic energy.
Answer: In everyday life there are many examples in which an object having no rotational kinetic energy has translational kinetic energy. For example a basketball thrown has zero rotational kinetic energy but non-zero translational kinetic energy provided it does not spin.
Q.10 A large number of small dust and rain particles come from space onto earth every day. Can you say that moment of inertia of earth increases?
Answer: By definition,
Moment of inertia =
In this case M and r are both increasing, so moment of inertia of earth increases.
Q.11 What does the term rotational invariance (isotropy) of space imply?
Answer: Isotropy means rotational invariance of properties of free space. Rotational invariance of space leads to law of conservation of angular momentum.
Q.12 A solid aluminum sphere of radius R has moment of inertia I about an axis through its center. What will be change in the moment of inertia about a central axis of a solid aluminum sphere of radius 2R?
Answer: Moment of inertia of a solid sphere about its distance is,
$$I=\frac{2}{5}\ MR^2$$
For a sphere of radius 2R, moment of inertia will be,
$$I^{\prime\ }=\ \frac{2}{5}\ \left(8M\right){(2R)}^{2\ }=32\left(\frac{2}{5}MR^2\right)=32\ I$$
Q.13 Is the relationship valid if ∆φ1+ ∆φ2=∆φ2+∆φ1 valid if ∆φ1& ∆φ2 referring to different axis of rotation? Is it valid if they refer to different relations about the same axis?
Answer: In situation if order of two relations is reversed, then the given relation does not hold. Moreover this relation does not hold if refer to different axis. However this is valid if they refer to different relations about the same axis.
Q.14 A gymnast sitting on a rotating plate form with his arms outside stretched suddenly folds his out stretched arms. What happens to angular velocity of plan form?
Answer: When a gymnast sitting on a rotating plate form with his arms outside stretched suddenly folds his out stretched arms, the radius of gyration of this body decreases and as a result moment of inertia decreases. Hence in order to conserve angular momentum, angular velocity should increase.
Q.15 Suppose a bicycle wheel is rotated about an axis through its rim and parallel to its axle? What happens to moment of inertia of wheel when it rotates about its axel?
Answer: By parallel axis theorem, moment of inertia of wheel increases.
Q.16 When we say that a point on the equator of the earth has an angular speed of $\frac{2\pi rad}{day,}$ what reference frame do we have in mind?
Answer: Here w =$\frac{2\pi rad}{day} = \frac{2\pi rad}{86400s}$ = 7.27 × 10-5 As acceleration of an inertial frame of reference is zero, so we have an inertial frame of references in mind.
Q.17 Suppose just two external forces act on a stationary, rigid object and the two forces are equal in magnitude and opposite in direction. Under what condition does the object start to rotate?
Answer: Object will rotate when two forces are acting along different lines of action. Under this condition both torques will not be equal in magnitude and opposite in direction.
Q.18 Suppose you just ride on Ferris wheel and wheel rotates with constant angular velocity. Is your linear velocity constant? Is direction of centripetal acceleration constant?
Answer: No, linear velocity is not constant because direction is changing at every instant. Direction of centripetal acceleration changes as you rotate to different positions.
Q.19 Suppose you climb up on roof of a very tall building. While standing on roof, is your angular speed due to rotation of earth is same as on earth?
Answer: Yes, there is no effect on your angular speed because earth’s rotation is same for building and you.
Q.20 Explain why changing the axis of rotation of an object changes its moment of inertia.
Answer: Moment of inertia is defined by relation,
$$I={Mr}^2$$
This equation shows that moment of inertia depends upon distribution of mass and distance from given axis. By changing axis of rotation, distance of mass particles will change from axis and as a result, moment of inertia will change.
Q.21 A wheel is rotating about its axle. Consider a point on the rim. When the wheel rotates with constant angular velocity, does the point have a radial acceleration? A tangential acceleration? When the rotates with constant angular acceleration, does the point have a radial acceleration? A tangential acceleration? Do the magnitude of these acceleration change with time?
Answer: When a wheel rotates with constant angular velocity then it has radial acceleration but does not tangential acceleration. When it rotates with constant angular acceleration then it has tangential acceleration but not radial acceleration. Yes magnitude of these accelerations change with time.
Q.22 If density of the earth becomes uniform, then how it will effect the rotation of the earth?
Answer: If density of earth becomes uniform then it will have larger moment of inertia than that at present. In order to conserve angular momentum, angular speed of earth should increase.
Q.23 If we have two spheres of equal masses of copper and lead, then which sphere will be rotating fast?
Answer: Lead is more denser than cooper, so lead sphere will have small diameter than cooper sphere and lead sphere will have smaller moment of inertia, due to which it will have greater angular speed.
Q.24 Suppose that you were asked to determine the distance traveled by a needle in playing a vinyl phonograph record. What information do you need? Discuss from the point of view of reference frames (a) fixed in the room (b) fixed on rotating record. And (c) fixed on the arm of the record player.
Answer: We need a frame of reference about which body is rotating and angular displacement being a ratio of two lengths has no dimensions and point is fixed, then we need perpendicular distance center and needle. If it is fixed on rotating record then point moves with record and we should know angle through which it is rotating. If it is fixed on arm of record player then we should know perpendicular distance and angle.
Q.25 Suppose an object has non-zero rotational kinetic energy? Can it have non-zero translational kinetic energy? If yes give an example. If not, prove it.
Answer: Yes, this is possible. For Example, a bicycle wheel on a moving bicycle has both non-zero translational and rotational kinetic energies.
Q.26 Two athletes A & B ride on same merry-go-round. Athlete A is at a distance d from axis of rotation and athlete B is at a distance 2d from axis of rotation. Is the rotational period of athlete A less than that of athlete B?
Answer: No, both athlete have same period of rotation because they are on same merry-go-round which completes its one rotation in a given period of time.
Q.27 If an object is rotating, can we say that there is necessarily net torque acting on it?
Answer: This is not necessary always, however when angular velocity is changing then there is necessarily net torque acting on it.
Q.28 Which will be rotating for longer time, a raw egg or a hard boiled egg?
Answer: Raw egg will be rotating for longer period of time because it has larger moment of inertia than a solid boiled egg.
Q.29 Suppose an object has non-zero rotational kinetic energy. Can it have zero translational kinetic energy? If yes, give an example. If not prove it.
Answer: Yes, an object having zero translational kinetic energy may have non-zero rotational kinetic energy. For example, a spinning aero plane propeller on a plane that is at rest.
Q.30 What is the relation between the angular velocities of a pair of coupled gears of different radii?
Answer: In coupled gears, there are two gears- one is called driver and second is called driven. If driver rotates clockwise, then driven rotates anticlockwise and vice versa. If both gears have same radii, then both have same angular velocity. But if they have different radii, then both have different velocities. It depends upon the ratio of teeth of both gears. If number of teeth of driver are more than driven, then angular velocity of driven will be greater.
Q.31 Why the pilot experiences the negative weight while flying a jet plane?
Answer: This happens when the plane is diving, so its downward acceleration becomes greater than g and it experiences negative weight i.e. apparent weight is decreasing.
Q.32 Explain the difference between tangential and angular velocities. If one of these is given for a wheel of known radius, how you will find other?
Answer: The velocity of a body along tangent to the curved path is called tangential velocity and denoted by v. Its units are m/s. The rate of change of angular displacement is called angular velocity and represented by . Its units are . From relation , if one velocity is given then other can be calculated.
Q.33 Does the kinetic energy of a body moving in a vertical circular path remain constant?
Answer: No, because speed is different at different points of circle.
Q.34 Why is fly wheel of an engine made heavy in rim?
Answer: By making the rim of fly wheel heavier, the mass is concentrated away from axis of rotation, due to which moment of inertia of fly wheel increases. A fly wheel with larger moment of inertia helps the engine to overcome the dead points of piston and guarantees its continuous motion.
Q.35 A section of hollow pipe and a solid cylinder have the same radius, mass, and length. They both rotate about their long central axes with the same angular speed. Which object has the higher rotational kinetic energy?
Answer: Almost all of the mass of the pipe is at the same distance from the rotation axis, so hollow has a greater moment of inertia than the solid cylinder.
Q.36 A basketball rolls across a classroom floor without slipping, with its center of mass moving at a constant speed. A block of ice of the same mass is set sliding across the floor with the same speed along a parallel line. Which object has more kinetic energy?
Answer: The basketball has rotational as well as translational kinetic energy, so it has more kinetic energy.
Q.37 A person balances a meter stick in a horizontal position on the extended index fingers of her right and left hands. He slowly brings the two fingers together. The stick remains balanced, and the two fingers always meet at the central mark regardless of their original positions. Explain.
Answer: When one finger slides towards the central mark, the normal force exerted by the sliding finger on the meter stick will increase so that static friction between the sliding finger and the meter stick will stop their relative motion. At this moment the other finger starts sliding along the ruler towards the center. This process repeats until the fingers meet at the center of the ruler.
Q.38 Consider an object on a rotating disk a distance r from its center, held in place on the disk by the static friction. If the angular speed is constant , the object is accelerated or not.
Answer: Since an object moving in a circular path changes it velocity at every point on the circle due to change in direction, so motion is accelerated due to centripetal acceleration.
Q.39 Can an object be rotating to have a non-zero moment of inertia?
Answer: This is not true for an object having no mass.
Q.40 A constant net torque is exerted on an object. Which of the quantities angular position, angular velocity, angular acceleration, moment of inertia and kinetic energy for the object cannot be constant?
Answer: Angular position , angular velocity , and kinetic energy cannot be constant for the object. Moment of inertia will be constant unless there was a rearrangement of mass within object.
Q.41 A basketball rolls across a classroom floor without slipping, with its center of mass, moving at a certain speed. A block of ice of the same mass is set sliding across the floor with the same speed along a parallel line. Which object has more momentum?
Answer: The motion of their centers of mass are identical so, both have same momentum.
Q.42 Is it possible to change the translational kinetic energy of an object without changing its rotational energy?
Answer: Yes, for any object on which no net torque is acting but a net force is acting , the translational kinetic energy will change but the rotational kinetic energy will remain same.
