NCERT Class 11 Physics Solutions: Chapter 7 – System of Particles and Rotational Motion Part 6

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Question 7.17:

A metre stick is balanced on a knife-edge at its centre. When two coins, each of mass are put one on top of the other at the mark, the stick is found to be balanced at . What is the mass of the metre stick?

Answer:

Let and be the respective weights of the metre stick and the coin.

W and W' be the respective weights of the metre stick and th …

Figure Shown the Weights of the Metre Stick and the Coin

The mass of the metre stick is concentrated at its mid-point, i.e., at the mark.

Mass of the meter stick

Mass of each coin,

When the coins are placed away from the end , the centre of mass gets shifted by from point toward the end . The centre of mass is located at a distance of from point .

The net torque will be conserved for rotational equilibrium about point

Hence, the mass of the metre stick is

Question 7.18:

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?

Answer:

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

Explanation:

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, we can write:

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

Hence, equation becomes:

But we have the relation,

Hence, the velocity of the sphere at the bottom depends only on height and acceleration due to gravity Both these values are constants. Therefore, the velocity at the bottom remains the same from whichever inclined plane the sphere is rolled.

(b) and (c)

Explanation:

Consider two inclined planes with inclinations and , related as:

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

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

The various forces acting on the sphere are shown in the fol …

Various Forces Acting on the Sphere. LHS

is the normal reaction to the sphere.

Similarly, the acceleration produced in the sphere when it rolls down the plane inclined at is:

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

The various forces acting on the sphere are shown in the fol …

Various Forces Acting on the Sphere. RHS

is the normal reaction to the sphere.

Initial velocity,

Final velocity,Constant

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

For inclination

For inclination

From equations, we get:

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