Team Quanta gladly presents all possible short questions of BS Physics book Mechanics – II’s Chapter#02: Angular Momentum for the students.
Q.1 Is it possible for the angular momentum of an object to be zero if the angular velocity is non-zero? Is it possible for the angular velocity of an object to be zero if the angular momentum is non-zero?
Answer: Angular momentum may be zero when angular velocity is non-zero for example when axis of rotation is on line of motion that is moment arm is zero. But if angular velocity is zero then angular momentum must be zero.
Q.2 Does an object moving in a straight line always have non-zero angular momentum?
Answer: No, an object moving in a straight line may or may not have angular momentum depending on location of axis of rotation.
Q.3 Can angular momentum of an object increases without changing its linear momentum? If yes, give an example. If not prove it.
Answer: Yes, consider turning of ceiling fan. In this case, angular momentum of fan increases without changing its linear momentum.
Q.4 Two moons of identical masses are in stable circular orbits with different radii about same planet. Which of two moons has larger angular momentum, the closer one or the one farther from planet?
Answer: By law of conservation of angular momentum, both have same angular momentum but angular velocities will be different.
Q.5 Two forces produce same torque. Does it follow that they have same magnitude?
Answer: No, because torque depends upon force as well as its moment arm. That is a smaller force can produce same torque as a large force depending on moment arm.
Q.6 A stone tied to one end of string is revolved round a rod in such a way that string winds over the rod and gets shortened. What happens to its angular velocity, moment of inertia and angular momentum?
Answer: As the string is shortened, so moment of inertia decreases. By law of conservation of angular momentum, angular velocity will increase. Angular momentum remains constant.
Q.7 A skater pulls in his arm, decreasing moment of inertia by a factor of two and doubling his angular speed. Is his final kinetic energy equal to his initial kinetic energy?
Answer: Kinetic energy of rotation is,
$$K_{rot}=\frac{1}{2}\ I\ \omega^2$$
When I decrease by factor of two and angular speed is doubled, then
$$K_{rot}=\frac{1}{2}\left(\frac{1}{2}\right)\ \ \left(2\omega\right)^{2\ \ }=2\ K_{rot}$$
Hence final kinetic energy increases by a factor of two.
Q.8 A pet mouse sleeps near the eastern edge of a stationary, horizontal turntable that is supported by a frictionless, vertical axle through its center. The mouse wakes up and starts to walk north on the turntable. As it takes its final steps, what is the direction of the mouse’s displacement relative to the stationary ground below?
Answer: Initial angular momentum of the mouse turntable system is zero because both are at rest. In order to conserve angular momentum, mouse must move in counter clockwise direction or towards north.
Q.9 A man is on a turntable rotating with angular speed . He has lowered his arms. If he stretches out his arms, then what will be effect on his angular velocity?
Answer: When man stretches his arms , his moment of inertia will increase due to increase in r. From law of conservation of angular momentum constant , the angular speed should decrease.
Q.10 The earth rotates about sun in elliptical orbit. At which point will angular velocity be maximum?
Answer: By law of conservation of angular momentum,
$$I\omega= constant \ \ \ \ \ \ \ \ \ \ \rightarrow\ \ I\propto\frac{I}{\omega}\ \ \ \rightarrow\ {mr}^2\propto\frac{1}{\omega}$$
This shows that angular velocity will be maximum where r is minimum that is at end points of major axis.
Q.11 A pet mouse sleeps near the eastern edge of a stationary, horizontal turntable that is supported by frictionless, vertical axle through its center. The mouse wakes up and starts to walk north on the turntable. Is the angular momentum of the system constant?
Answer: Angular momentum is conserved since both its initial and final values are zero.
Q.12 Can torque be zero when force is non-zero? Give an example of system in which net torque is zero but net force is not zero.
Answer: Yes, for example when a non-zero force is applied rapidly to a wheel it produces zero torque.
Q.13 A loaf of bread is lying on rotating plate. The crow takes the loaf of bread and the plate is rotating fast. Why?
Answer: By law of conservation of angular momentum, the angular momentum of plate is conserved i.e.,
$$I_1\omega_1=I_2\omega_{2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \rightarrow\ }{\ \ \ \ \ \ \ \ \ \ \ \ m}_1r^2\omega_1=m_2r^2\omega_2$$
In order to conserve angular momentum, as m2 decreases must increase. Hence angular velocity of plate should increase.
Q.14 Why does a long pole help a tightrope walker stay balanced?
Answer: Moment of inertia of long pole about an axis along the rope has large value. An unbalanced torque can produce only a small angular acceleration. To keep the center of mass above the rope, the tightrope walker can shift the pole left or right due to which the pole sags down at the ends to lower the center of gravity of system.
Q.15 What is angular momentum for a body which has orbital motion as well as spin motion?
Answer: When a body has spin motion along with orbital motion as motion of earth around sun or motion of electron around nucleus, the angular momentum is sum of orbital angular momentum and a spin angular momentum.
Q.16 Under what conditions angular momentum has following values:
(a) zero (b) maximum (c) minimum ( negative)
Answer: Angular momentum is defined as,
$$\vec{L}=\vec{r}\times\ \vec{p}\ \ \ \ \ \ \ \ \ \ \ \ \ \rightarrow\left|\vec{L}\right|=\left|\vec{r}\times\ \vec{p}\ \ \right|=rpsin\theta$$
Angular momentum has value zero when θ=0°,180°
Angular momentum has maximum value when θ=90°
Angular momentum has minimum value when θ=270°
Q.17 Why does a driver change his body position before and after diving in the pool?
Answer: The driver changes his position to change moment of inertia and in turn rotational velocity. By law of conservation of angular momentum, diver can spin faster when moment of inertia becomes smaller. This enables diver to take extra somersaults.
Q.18 Two boys are playing with a roll of paper towels. One boy hold the roll between the index fingers of his hands so that it is free to rotate, and the second boy pulls at constant speed on the free end of the paper towels. As the boy pulls the paper towels the radius of the roll of remaining towel decreases. How does the angular speed of the roll change in time?
Answer: Angular speed is related to radius by relation.
$$\omega=\ \frac{V}{R}$$
This equation shows that as R increase, angular speed decreases for constant value of v.
Q.19 Stars form when a large rotating cloud of gas collapses. What happens to angular speed of gas cloud as it collapses?
Answer: In this case to conserve angular momentum, angular speed should increase.
Q.20 If no torque acts on an object, will its angular velocity remain conserved?
Answer: Torque is related to angular velocity by relation,
$$\vec{\tau}=I\frac{d\vec{\omega}}{dt}$$
When torque is zero, then
$I\frac{d\vec{\omega}}{dt}=0\ \ \ \ \ \ \ \ \rightarrow\ \ \ \ \vec{\omega}= Constant$
Q.21 A particle of mass m moving in a circular orbit of radius r has angular momentum L about its center. Find its kinetic energy in terms of L, m and r.
Answer: Kinetic energy is given by relation,
$$K=\frac{1}{2}mv^{2\ }=\frac{\left(mvr\right)^2}{2mr^2}=\ \frac{L^2}{2mr^2}$$
Q.22 State the direction of following vectors in simple situations:
(a)-angular momentum (b)-angular velocity
Answer: The direction of angular momentum and angular velocity can be determined by right hand rule and is along the axis of rotation.
Q.23 Both torque and works are products of force and displacement. How are they different? Do they have same units?
Answer: Work done by a force through a displacement is given by,
$$W=\ \vec{F}.\ \vec{d}$$
This work results as a change in translational kinetic energy.
Work done by a torque through an angular displacement is given by,
$$W=\ \vec{\tau}.\ \vec{d\theta}$$
This work results as a change in rotational kinetic energy.
In either case work has same units.
Q.24 If the torque acting on a particle about an axle through a certain origin is zero, what can you say about its angular momentum about that axis?
Answer: Angular momentum and torque are related by equation,
$$\ \vec{\tau}=\vec{\frac{dl}{dt}}$$
For $\vec{\tau}=0,\ \ \ \ \ \ \ \ \ \vec{\frac{dl}{dt}}=0\ \ \ \ \ \ \rightarrow\ \ \vec{L}=$ constant
Q.25 Two spheres of equal mass and radius are rolling across the floor with same speed. Sphere A is solid and sphere B is hollow. Is work required to stop sphere A equal to work required to stop sphere B?
Answer: Work required to stop sphere B is greater than work required stopping sphere A sphere B has larger moment of inertia than sphere A due to hardness.
Q.26 If the acceleration of satellite close to earth is not 9.8m/s2, in which direction it will fly?
Answer: If the acceleration of satellite close to earth is not 9.8m/s2, then it will fly off tangent to the earth.
Q.27 Do all points on a rigid body rotating about a fixed axis have different angular velocities?
Answer: No, all points on a rigid body rotating about a fixed axis have same angular velocity.
Q.28 Two ponies of equal masses are initially at diametrically opposite points on the rim of a large horizontal turntable that is turning freely on a frictionless, vertical axle through its center. The points simultaneously start walking toward each other across the turntable. Is the angular momentum of the system conserved?
Answer: Angular momentum is constant with a non-zero value since no outside torque can influence rotation about the vertical axle.
Q.29 An ice skater starts a spin with his arms stretched out to the sides. He balances on the tip of one skate to turn without friction. He then pulls his arms in so that this moment of inertia decreases by a factor of 2. In the process of his doing so, what happens to his kinetic energy?
Answer: Rotational kinetic energy is given by,
$$K_{rot}=\frac{1}{2}\ I\ \omega^2$$
In the give process, when moment of inertia decreases by a factor of 2, angular velocity is doubled.
$$K_{rot}=\frac{1}{2}\left(\frac{1}{2}\right)\ \ \left(2\omega\right)^{2\ \ }=2\left(\frac{1}{2}I\omega^2\right)=2\ K_{rot}$$
Thus kinetic energy increases by 2.
Q.30 If global warming continues over the next on hundred years, it is likely that some polar ice will melt and the water will be distributed closer to the equator. How would that change the moment of inertia of the earth?
Answer: In order to conserve angular momentum, as mass moves away from axis of rotation , its angular speed decreases due to which moment of inertia increases.
Q.31 Give some examples in which conservation of angular momentum plays the vital role.
Answer: Conservation of angular momentum plays the vital role in sports particularly in diving, gymnastic and ice skating etc.
Q.32 If global warming continues over the next on hundred years, it is likely that some polar ice will melt and the water will be distributed closer to the equator. Would the duration of the day(one revolution) increase or decrease.
Answer: For earth being an isolated system, its angular momentum is conserved. When its mass moves away from axis of rotation, its moment of inertia increases due to which angular speed decreases.
Angular speed is related to time period by relation,
$$\omega=\frac{2\pi}{T}$$
This relation shows that as angular speed decreases, time period increases, hence duration of day increases by a small amount perhaps in nanoseconds.
Q.33 Two boys are playing with a roll of paper towels. One boy hold the roll between the index fingers of his hands so that it is free to rotate, and the second boy pulls at constant speed on the free end of the paper towels. As the boy pulls the paper towels the radius of the roll of remaining towel decreases. How does the torque on the roll change with time?
Answer: As the roll unravels, the weight of the roll decreases, due to which frictional force decreases and therefore torque is decreased.
Q.34 Is normal force exerted by ground the same for all four tyres on a vehicle?
Answer: No, normal force exerted by ground for all four tyres on a vehicle is not same.
Q.35 A student rotates on a frictionless piano stool with his arm out stretched, a heavy weight in each hand. Suddenly he lets go of weights and they fall to floor. What happens to the angular speed of student?
Answer: The angular speed of the student is unaffected.
Q.36 Two ponies of equal masses are initially at diametrically opposite points on the rim of a large horizontal turntable that is turning freely on a frictionless, vertical axle through its center. The points simultaneously start walking toward each other across the turntable. As they walk, what happens to the angular speed of the turntable?
Answer: In order to conserve angular momentum, the moment of inertia decreases, so the angular speed must increase.
Q.37 A pet mouse sleeps near the eastern edge of a stationary, horizontal turntable that is supported by a frictionless, vertical axle through its center. The mouse wakes up and starts to walk north on the turntable. In this process , is the mechanical energy and momentum of the mouse turntable system constants?
Answer: Neither mechanical energy nor linear momentum are conserved. For linear momentum, initial linear momentum is zero where as final linear momentum is not conserved.
Q.38 Consider an isolated system moving through empty space. The system consists of objects that interact with each other and can change location with respect to one another. Which of the quantities angular momentum, linear momentum or both the angular momentum and linear momentum of system can change it time?
Answer: As long as no net external force (torque) acts on the system, the linear and angular momentum of the system are constant.
Q.39 Stars originates as large bodies of slowly rotating gas. Because of gravity, these clumps of gas slowly decrease in size. What happens to the angular speed of a star as it shrinks? Explain.
Answer: In order to conserve angular momentum, as the radius of the star decreases, its moment of inertia decreases, and hence its angular speed increases.
Q.40 A ball is thrown in such a way that it does not spin about its own axis. Does this statement mean the angular momentum is zero about an arbitrary axis?
Answer: The angular momentum about any axis that does not lie along the line of motion of the ball is non-zero
