Rolling Motion Questions

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Very Short Answers Questions [1 Mark Questions]

Ques. If the radius of the wheel is R, then what will be the displacement of the center of the wheel in one rotation?

1. 3πR
2. 4R
3. 0
4. 2πR

Ans. The correct answer is d. 2πR

Explanation: The angular speed of the wheel is given by

ω = 2π/T

Where T is the time period of the rotation of the wheel.

At the center, the velocity is given by: ωR = (2π/T) x R

Therefore, the distance moved by the wheel

d = (2π/T) x R x T = 2πR

Ques. A flywheel is attached to an engine to _____.

1. Increase its energy
2. Decrease its energy
3. Help in overcoming the dead point
4. None of the above

Ans. The correct answer is c. Help in overcoming the dead point

Explanation: Due to the large inertia of motion, a flywheel helps the engine to overcome the dead points and this is why they are attached to an engine.

Ques. A body rolls down an inclined plane. If the kinetic energy of rotation is 40% of its kinetic energy of translation motion, then the body is

1. Hollow cylinder
2. Solid sphere
3. Disc
4. Ring

Ans. The correct answer is b. Solid sphere

Explanation: Given the kinetic energy of rotation is 40% of the kinetic energy of translation motion i.e.

Erotational = 40% E_{Translational}

⇒ E_rotational / E_{Translational} = 0.4

We have

E_rotational / E_{Translational} = (1/2 mv²)(K²/R²) / (1/2 mv²) = K²/R²

Where K is the radius of the gyration of the body.

⇒ K²/R² = 0.4

⇒ K² = 0.4R² = 2/5 R²

The moment of inertia of the body about the axis or rotation through its center is given by

I = mK²

⇒ I = m x 2/5 R² = 2/5 mR²

This represents the moment of inertia of a solid sphere.

Ques. A ring rolls down an inclined plane. The ratio of the rotational kinetic energy to translational kinetic energy is

1. 1 ∶ 1
2. 2 ∶ 1
3. 3 ∶ 2
4. 1 ∶ 2

Ans. The correct answer is a. 1 ∶ 1

Explanation: Rotation kinetic energy is given by

E_R = 1/2 Iω²

For a rolling ring, I = mR²

Also, ω = V/R

⇒ E_R = 1/2 mR²(V/R)² = 1/2 mV²

Also, the kinetic energy of translational motion is given by

E_T = 1/2 mV²

The ratio of the rotational kinetic energy to translational kinetic energy

E_R/E_T = 1/2 mV²/1/2 mV² = 1 ∶ 1

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Short Answers Questions [2 Marks Questions]

Ques. What are the applications of rolling motion?

Ans. The applications of rolling motion are

• The use of the wheels for displacement by the vehicles.
• In rotating devices, the use of rolling-element bearings, such as ball bearings.
• Motors used in ceiling cars, fans, and drills, rely on ball bearings.
• The use of rolling objects for transportation such as placing an object on a series of rollers or wheels.

Ques. What is the difference between pure and impure rolling?

Ans. Pure rolling in the rolling motion of a rigid body is a condition when there is no sliding or slipping during rolling. The point of contact of the body with the ground is always at rest. While in impure rolling, the body slips as it rolls.

Ques. Define Translational motion.

Ans. If any line drawn on the rigid body remains parallel to itself throughout the motion, then the motion is said to be pure translational motion.

In pure translational motion, all the particles of the body have equal velocity and acceleration at all instants and they cover equal distance and displacement in equal time.

Ques. Define rotational motion.

Ans. If any line drawn on the rigid body does not remain parallel to itself throughout its motion, then the body is said to be in rotational motion.

In rotational motion, every particle of the body moves in a circle with its center on the axis of rotation.

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Long Answers Questions [3 Marks Questions]

Ques. What is the rolling motion of rigid bodies?

Ans. The rolling motion of a body is a combination of both translational and rotational motion of a round-shaped body placed on a surface.

In rolling motion, every particle of the body has two velocities, one due to its rotational motion and the other due to its translational motion, and the resulting effect is the vector sum of both velocities of all particles.

Ques. Define pure rolling motion.

Ans. Pure rolling is a case of rolling motion when there is no relative motion at the point of contact between the rolling body and the surface, and the body is considered to be rotating about this point of contact frame.

Rolling motion can be assumed as a pure rotation about its point of contact, which is also called Instantaneous center of rotation (I).

Ques. What is the condition for pure rolling?

Ans. Pure rolling is a condition when there is no sliding of the body during rolling. This can happen when the point of contact of the rolling object with the surface has zero velocity. 

It happens when the velocity due to the rotational motion cancels out the velocity due to the translation motion. 

This will happen if 

V = rω 

Where 

• V denotes the linear velocity and 
• ω denotes the angular velocity.

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Very Long Answers Questions [5 Marks Questions]

Ques. If two cars are moving at such speeds that each of them makes a complete circle at the same time t, then what will be the ratio of angular speeds of the first car to the second car, if the mass of the first car is m₁ and mass of the second car is m₂, and both cars are moving in circles of radii r₁ and r₂, respectively?

Ans. The angular speed of an object moving in a circular path is given by

ω = 2π/T

Where

• ω = angular speed
• T = Time period of the revolution

Given that the time period of revolution of both the cars is the same i.e. t. Therefore

The angular speed of the first car, ω₁ = 2π/t

The angular speed of the second car, ω₂ = 2π/t

The ratio of angular speeds of the first car to the second car

ω₁/ω₂ = (2π/t) / (2π/t) = 1

Ques. An ice skater or ballet dancer can increase her angular velocity by folding her arms and bringing stretched legs close to other. why?

Ans. When the hand of an ice skater or ballet dancer stretched outward, then her moment of inertia increases. Therefore to conserve angular momentum angular speed will be decreased. Now she folds her arms and brings stretched legs close then her moment of inertia decreases and to conserve angular momentum, the angular velocity increases.

Ques. A solid sphere is in a rolling motion. In rolling motion, a body possesses translational kinetic energy K_t as well as rotational kinetic energy K_r simultaneously. What is the ratio of K_t : (K_t + K_r)?

Ans. Let the sphere has mass m and linear velocity v. Then the translational kinetic of the sphere will be

K_t = 1/2 mv²

If the moment of inertia of the sphere is I and ω be its angular velocity, then its rotational kinetic energy is given by

K_r = 1/2 Iω²

But I = 2/5 mr² and ω = v/r

⇒ K_r = 1/2 x (2/5 mr²) x (v/r)²

⇒ K_r = 1/5 mv²

Therefore, K_t + K_r = (1/2 mv²) + (1/5 mv²) = 7/10 mv²

⇒ K_t / (K_t + K_r) = (1/2 mv²) / (7/10 mv²) = 5/7

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