Rotational Motion

The radius of gyration of the plate about an axis perpendicular to the plane is given by: Where:
is the moment of inertia about the x-axis,
is the moment of inertia about the y-axis,
is the mass of the plate. Substituting the values:

The correct answer is (A) : 50 cm.

We need to identify the object undergoing pure rolling motion, given the relationship between its total energy ( ) and the square of the speed of its center of mass ( ), represented by a graph. The mass of the object is 1 kg.

1. Understanding Total Kinetic Energy Equations:

Total Kinetic Energy (E) in pure rolling = Translational Kinetic Energy + Rotational Kinetic Energy: $ V_{cm} = R\omega R \omega = \frac{V_{cm}}{R} \omega E I = kmR^2 k m E V_{cm}^2 V_{cm}^2 = 4 E = 3 m = 1 \, \text{kg} k k = \frac{1}{2} I = \frac{1}{2}MR^2 I = \frac{2}{5}MR^2 I = MR^2 I = \frac{1}{2}MR^2 I = MR^2 k $ value is the Disc.

Final Answer: The correct answer is (C) disc.

1. Understanding the Centripetal Formula
The centripetal force required to keep the coin moving in a circle is provided by the force of friction. At the point of slipping, the centripetal force is equal to the maximum static friction force:

,
.

Where:

: Mass of the coin,
: Distance from the center,
: Angular velocity,
: Coefficient of static friction,
: Normal force,
: Acceleration due to gravity.

At the point of slipping: $ \mu g \mu \cdot g = C r_1 \cdot \omega_1^2 = C r_2 \cdot \omega_2^2 = C C r_1 = 4 \, \text{cm} \omega_2 = 2 \cdot \omega_1 r_2 $

Final Answer: The correct answer is (A) 1 cm.

Concept:

The moment of inertia of a rigid body about a given axis is given by:

where:

• is the mass of each particle in the body
• is the perpendicular distance of the particle from the axis of rotation

Step 1: Factors Affecting Moment of Inertia

• It depends on the mass distribution relative to the axis of rotation.
• It depends on the position of the axis of rotation.
• It depends on the shape and size of the object.
• It does not depend only on the total mass but also on how the mass is distributed.

Step 2: Correct Answer

The moment of inertia depends on the position of the axis of rotation, as changing the axis changes the distances in the formula.

Answer: The correct option is B.