System of Particles & Rotational Motion Important Questions

Very Short Answer Questions [1 Mark Questions]

Ques: Is there any possibility of coinciding with the geometrical centre and C.M. of a body? Cite example.

Ans: C.M. and geometrical centre may coincide when the body in consideration has a uniform mass density. So, yes, They may coincide. For instance, C.M. and geometrical centre are the same in the case of a cube or a sphere or cylinder etc.

Ques: Under what condition, the torque due applied can be considered to be zero?

Ans: As we know, τ = rFsinθ

Now, if θ = 0 or 180,

Or, r = 0, then τ = 0

Now, r = 0 refers to the applied force passing through the axis of rotation.

Ques: Which parameter is conserved when a planet is revolving around a star?

Ans: The parameter that is conserved when a planet revolves around a star is angular momentum.

Ques: When a cat can land on its feet after a fall, which property of Physics is being used by her?

Ans: If the cat can land on its feet after a fall, she is using the principle of conservation of angular momentum.

Ques: What are the factors on which the M.I of a body depends?

Ans: The M.I. of a body depends on the following factors-

1. Position of the axis of the rotation
2. Mass of the body.
3. Way of distribution of mass about the axis of rotation.

Ques: Does the C.M of a body necessarily have to lie inside a body?

Ans: No, the C.M of the body in consideration does not necessarily be inside the body itself. I also may lie outside the body. In the case of the semicircular ring, it is at the center which is outside the ring.

Ques: Why do we prefer to use a wrench, having a long arm?

Ans: We know, the turning effect of a force is 

τ = r × F

 Now when the arm of the wrench is long, r is larger. Therefore, a smaller force will be required to produce the same turning effect.

Short Answer Questions [2 Marks Questions]

Ques: What is the difference between the center of gravity and C.M?

Ans: The difference between the center of gravity and C.M. is as follows-

Ques: What will be the effect on the day-length if the polar ice caps of the planet melt?

Ans: Melting of the polar ice caps of the Earth will produce a huge amount of water that would spread around the planet going farther away from the axis of rotation. This will increase the radius of gyration and hence M.I. To conserve angular momentum, the angular velocity ⍵( omega) will decrease. So the length of the day( T= 2π/⍵) will increase.

Ques: Suppose you have been given two spheres of the same mass and radius, one is solid and the other is hollow. Which of them has a larger moment of inertia about its diameter?

Ans: The hollow sphere will have a greater M.I. value as its entire mass is concentrated at the boundary which is the maximum distance from its axis.

Ques: What is a rigid body?

Ans: A rigid body refers to an object in which the distance between all the constituting particles remains fixed under any influence of external force. Therefore, a rigid body conserves its shape during motion.

Ques: Distinguish between internal and external forces.

Ans: The mutual forces between the particles of a particular system are called internal forces.

The forces exerted by some kind of external source on the particles of the system are to be named external forces.

Ques: Why are spokes fitted in the cycle wheel?

Ans: The cycle wheel is generally constructed in a way that the M.I of the wheel can be increased with minimum possible mass. This can be achieved by using spokes and the M.I is increased to ensure the uniform speed.

Ques: Why cannot a single force balance the torque?

Ans: The effect of torque can be seen to produce angular acceleration and its effect is entirely different from that of the force that causes linear acceleration. Therefore, a single force cannot balance the torque.

Long Answer Questions [3 Marks Questions]

Ques: What are the important features of angular momentum?

Ans: The following are the important features of angular momentum-

• The angular momentum of a particle with respect to a point provides an idea of the strength of its rotational tendency about the point.
• The magnitude of the angular momentum can be defined in terms of mass and velocity of the particle and its distance from the reference point i.e. L = mvr.
• The vector concept of angular momentum is important and useful. Its direction is the axial direction portrayed by the right-handed rule.

Ques: What is the physical significance of M.I?

Ans: The inability of a body to change its state of uniform rotation about an axis is called rotational inertia or M.I of that concerned body.

It plays a similar part in a rotatory motion as is being played by the mass in translational motion i.e. it is a type of rotational analogue of mass. 

Ques: The moment of inertia of a solid sphere about a tangent is 7/5 MR2. What will be the moment of inertia about its diameter?

Ans:

From the diagram we can say that a tangent is drawn at the point of C of the given solid sphere of mass M and radius of R.

Now, a diameter AOB∥KCl

Then according to the Theorem of Parallel Axis, it can be inferred

I₁= I – M(OC)²

M.I about the tangent can be written as I₁= 7/5 MR²

I = I₁ – M(OC)²

⇒ I = 7/5 MR² – MR²

⇒ I = 2/5 MR²

Therefore, 2/5 MR² is the moment of inertia about the diameter.

Ques: A circular ring having a diameter of 40cm and mass of 1 kg is rotating about an axis normal to its plane and passing through the center with a frequency of 10 rotations pers second. Now, calculate the angular momentum about the axis of rotation.

Ans: From the question we can obtain,

R = 40/2 cm = 20 cm = 0.2 m

M = 1 kg

v = 10 rotations / sec

Now, we can calculate the moment of inertia as follows:

M.I = MR²  = 1 × (0.2)² = 0.04 kg sq.m

⍵ = 2π × 10 = 20π rad / sec

Thus, angular momentum can be calculated as,

⇒ L= 0.04 × 20π

⇒ L = 2.51 kg sq m / sec

Very Long Answer Questions [5 Marks Questions]

Ques: Using the expression of power and K.E. of rotational motion, derive the relation τ = lɑ.

Ans: As we know that power is given by

P = τ⍵………………(i)

Also, we know that,

K.E. = 1/2 Iω²………………(ii)

Now, the power of rotational motion is equal to time rate of work done during the rotational motion. Since the work done is stored in the form of kinetic energy,

p = d/dt (K.E. of rotational motion)

Using the equations (i) and (ii), we have,

\(\tau(\omega)=\frac{d}{d t}\left(\frac{1}{2} I(\omega)^{2}\right)=\frac{1}{2} I \frac{d}{d t}\left(\omega^{2}\right)\)

\(\begin{aligned} &=\frac{1}{2} \times I \times 2 \omega \frac{d \omega}{d t} \\ &=I \omega \alpha \quad\left(\because \frac{d \omega}{d t}=\alpha\right) \end{aligned}\)                    … (iii)

From (i) and (iii), we get:

\(\begin{aligned} \tau \omega &=I \omega \alpha \\ \tau &=I \alpha . \end{aligned}\)

Ques: Three homogeneous rigid bodies namely a solid sphere, a solid cylinder and a hollow cylinder are placed at the top of an inclined plane. If they’re released from the rest at the same elevation and roll without slipping, which one will be reaching the bottom first and which one will be reaching the last?

Ans: Let us assume that a₁, a₂, a₃ are respective linear accelerations respectively,

Therefore the solid sphere will reach first and the hollow cylinder will reach last.

Ques: Three separate particles of masses of 1g, 2g, and 3g have their center of mass at the precise point (2,2,2). What should be the position of the fourth particle of mass 4g so that the combined center of mass may be at the point (0,0,0).

Ans: Let’s assume that r₁, r₂, r₃ are the position vectors of the three mass particles in consideration w.r.t origin. 

Then their center of mass is given by

If r’ be the position vector of the new C.M. then

r’= 0

And r₄ be the position vector of the fourth particle from the origin, then

Thus the new particle should be at (-3,-3,-3).

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