Q#1
The axis of rotation of a purely rotating body(a) must pass through the center of mass
(b) may pass through the center of mass
(c) must pass through a particle of the body
(d) may pass through a particle of the body.
Answer: (b), (d)
Explanation: It is not a necessary condition that the axis of rotation should pass through the center of mass or through a particle of the body. But it may pass through them.
Q#2
Consider the following two equations
(A) L = I ⍵ (B) dL/dt = Γ
In noninertial frames
(a) both A and B are true
(b) A is true but B is false
(c) B is true but A is false
(d) both A and B are false
Answer: (b)
dL/dt = total torque of external force, not equal to any torque. So, dL/dt = Γ is not correct.
(A) L = I ⍵ (B) dL/dt = Γ
In noninertial frames
(a) both A and B are true
(b) A is true but B is false
(c) B is true but A is false
(d) both A and B are false
Answer: (b)
dL/dt = total torque of external force, not equal to any torque. So, dL/dt = Γ is not correct.
Q#3
A particle moves on a straight line with a uniform velocity. Its angular momentum
(a) is always zero
(b) is zero about a point on the straight line
(c) is not zero about a point away from the straight line
(d) about any given point remains constant.
Answer: (b), (c), (d)
The angular momentum of a particle about a point
= mvr {Where r is the perpendicular distance of the line of velocity from the point}
A particle moves on a straight line with a uniform velocity. Its angular momentum
(a) is always zero
(b) is zero about a point on the straight line
(c) is not zero about a point away from the straight line
(d) about any given point remains constant.
Answer: (b), (c), (d)
The angular momentum of a particle about a point
= mvr {Where r is the perpendicular distance of the line of velocity from the point}
Since for a point on the line, r = 0, and for a point away from the line, r is not zero. Hence (b) and (c).
And for a given point r = constant, hence (d).
Q#4
If there is no external force acting on a nonrigid body, which of the following quantities must remain constant
(a) angular momentum
(b) linear momentum
(c) kinetic energy
(d) moment of inertia.
Answer: (a), (b)
If L be the angular momentum, then external torque = dL/dt
But given that it is zero, so dL/dt =0
L = a constant.
Similarly, it can be shown that linear momentum = a constant.
(c) is not correct because in such situation total energy is constant.
(d) is not correct because the moment of Inertia does not depend upon the external force, it depends upon the shape and the axis of rotation. In a nonrigid body, these two may not remain constant.
Q#5
Let IA and IB be moments of inertia of a body about two axes A and B respectively. The axis A passes through the center of mass of the body but B does not.
(a) IA < IB
(b) If IA < IB, the axes are parallel
(c) If the axes are parallel, IA < IB
(d) If the axes are not parallel, IA ≥ IB.
Answer: (c)
From the parallel axis theorem IB = IA + IAr² where r is the distance between the parallel axes. Hence IA < IB .
Others are not necessary conditions.
Q#6
A sphere is rotating about a diameter,
(a) The particles on the surface of the sphere do not have any linear acceleration.
(b) The particles on the diameter mentioned above do not have any linear acceleration.
(c) different particles on the surface have different angular speeds.
(d) All the particles on the surface have same linear speed.
Answer: (b)
The particles on the surface of the sphere revolve in concentric circles. So, at least they have centripetal accelerations. (a) is not true.
Different particles on the surface take equal time to complete a revolution, hence they have equal angular speeds. So, (c) is not true.
The linear speed v = ɷr. That means the particles at different r will have different linear speed. So, (d) is not true.
Since for the particles on the diameter r = 0. So, v = 0. Since v is not changing so linear acceleration is zero. (b) is correct.
Q#7
The density of a rod gradually decreases from one end to the other. It is pivoted at an end so that it can move about a vertical axis through the pivot. A horizontal force F is applied on the free end in a direction perpendicular to the rod. The quantities, that do not depend on which end of the rod is pivoted, are
(a) angular acceleration
(b) angular velocity when the rod completes one rotation
(c) angular momentum when the rod completes one rotation
(d) torque of the applied force.
Answer: (d)
The options (a), (b) and (c) involves the moment of Inertia which depends upon the pivoting end. Option (d) is unchanged because torque = Fl.
The density of a rod gradually decreases from one end to the other. It is pivoted at an end so that it can move about a vertical axis through the pivot. A horizontal force F is applied on the free end in a direction perpendicular to the rod. The quantities, that do not depend on which end of the rod is pivoted, are
(a) angular acceleration
(b) angular velocity when the rod completes one rotation
(c) angular momentum when the rod completes one rotation
(d) torque of the applied force.
Answer: (d)
The options (a), (b) and (c) involves the moment of Inertia which depends upon the pivoting end. Option (d) is unchanged because torque = Fl.
Q#8
Consider a wheel of a bicycle rolling on a level road at a linear speed v₀ (Figure 10-Q4).
(a) The speed of the particle A is zero.
(b) The speed of B, C and D are all equal to v₀
(c) The speed of C is 2v₀.
(d) The speed of B is greater than the speed of O.
Answer: (a), (c), (d)
The option (b) is not true because the wheel is not in pure rotation about O. It can be assumed that its diameter AC is instantaneously in pure rotation about A. So options (a) and (c) are correct. The point B has one more component of velocity downward, hence its total velocity > velocity of O. Hence option (d) also.
Consider a wheel of a bicycle rolling on a level road at a linear speed v₀ (Figure 10-Q4).
(a) The speed of the particle A is zero.
(b) The speed of B, C and D are all equal to v₀
(c) The speed of C is 2v₀.
(d) The speed of B is greater than the speed of O.
The option (b) is not true because the wheel is not in pure rotation about O. It can be assumed that its diameter AC is instantaneously in pure rotation about A. So options (a) and (c) are correct. The point B has one more component of velocity downward, hence its total velocity > velocity of O. Hence option (d) also.
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