﻿ Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time? from Physics System of Particles and Rotational Motion Class 11 Manipur Board

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Torques of equal magnitude are applied to a hollow cylinder and a solid sphere, both having the same mass and radius. The cylinder is free to rotate about its standard axis of symmetry, and the sphere is free to rotate about an axis passing through its centre. Which of the two will acquire a greater angular speed after a given time?

Let m and r be the respective masses of the hollow cylinder and the solid sphere.

The moment of inertia of the hollow cylinder about its standard axis, I1 = mr

The moment of inertia of the solid sphere about an axis passing through its centre, I2 = mr

We have the relation,

τ = Iα

where,

α = Angular acceleration,

τ = Torque,

I = Moment of inertia,

For the hollow cylinder, τ1 = Iα1

For the solid sphere, τn = Iαn

As an equal torque is applied to both the bodies, ττ2,

∴        =   =

α2 > α1                                      ...(i)

Now, using the relation,

ω = ω0 + α

where,

ω0 = Initial angular velocity

t = Time of rotation

ω = Final angular velocity

For equal ω0 and t, we have

ω ∝ α                                                 … (ii)

From equations (i) and (ii)

ω2 > ω

Hence, the angular velocity of the solid sphere will be greater than that of the hollow cylinder.
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What is the significance of defining the center of mass of a system?

The motion of n particle system can be reduced to one particle motion.

An equivalent single point object would enable us to discuss the overall motion of the system.
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Define centre of mass.

Centre of mass of a body or a system of bodies is a point at which the entire mass of the body or system is supposed to be concentrated.
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Is it necessary for centre of mass to lie within the body?

No, centre of mass needs not to lie within the body. It is not necessary that the total mass of the system be actually present at the centre.

The position of the centre of mass is calculated using the usual Newtonian type of equations of motion.
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Is it necessary that there should be matter at the centre of mass of system?

No, it is not necessary that there be matter at the centre of mass of the system.

For e.g., if two equal point masses are separated by certain distance, the centre of mass lies at the mid point of two point masses and there is no mass at that point.
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What is the need of centre of mass?

Newton’s second law of motion is strictly applicable to point masses only. To apply the Newton's law of motion to rigid bodies, the concept of centre of mass is introduced.

The concept of centre of mass of a system enables us to discuss overall motion of the system by replacing the system by an equivalent single point object.
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