System of Particles and Rotational Motion

Physics Part I

Physics

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A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination.

(a) Will it reach the bottom with the same speed in each case?

(b) Will it take longer to roll down one plane than the other?

(c) If so, which one and why?

(a) Will it reach the bottom with the same speed in each case?

(b) Will it take longer to roll down one plane than the other?

(c) If so, which one and why?

(a) Mass of the sphere =

Height of the plane =

Velocity of the sphere at the bottom of the plane =

At the top of the plane, the total energy of the sphere = Potential energy =

At the bottom of the plane, the sphere has both translational and rotational kinetic energies.

Hence,

Total energy =

Using the law of conservation of energy,

For a solid sphere, the moment of inertia about its centre,

Hence, equation (

∴

Hence, the velocity of the sphere at the bottom depends only on height (

Both these values are constants.

Therefore, the velocity at the bottom remains the same from whichever inclined plane the sphere is rolled.

(b) Consider two inclined planes with inclinations

θ

The acceleration produced in the sphere when it rolls down the plane inclined at

a

The various forces acting on the sphere are shown in the following figure.

a

θ

∴

Initial velocity,

Final velocity,

Using the first equation of motion, we can obtain the time of roll as

∴

For inclination θ

For inclination θ

From above equations, we get,

Hence, the sphere will take a longer time to reach the bottom of the inclined plane having the smaller inclination.

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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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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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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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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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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.

The position of the centre of mass is calculated using the usual Newtonian type of equations of motion.

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