Rotational Motion

Newton's Law of Cooling states: where is the proportionality constant and is the excess temperature. Given: Solving for : ---

Step 1: Formula for linear velocity
The linear velocity of a point on a rotating object is given by: where: - is the radius (distance from the center), - is the angular velocity.
Step 2: Substitute the given values
Given: - Radius - Angular velocity
Answer:
Therefore, the linear velocity of the point on the rod is . So, the correct answer is option (1).

Concept: The moment of inertia of a body measures its resistance to rotational motion about a given axis. For composite systems, the total moment of inertia is obtained by adding the moments of inertia of individual parts. For a solid sphere, the moment of inertia about its own central axis is: When the axis of rotation does not pass through the center of mass of the body, we use the parallel axis theorem: where = moment of inertia about the center of mass = distance between the axis and the center of mass.

Step 1: Moment of inertia of each sphere about its center.

Step 2: Distance between the spheres. The distance between the centers is . If the axis is taken through the midpoint between the spheres, each sphere is at a distance from the axis. Thus,

Step 3: Applying the parallel axis theorem. This is the moment of inertia for one sphere about the given axis.

Step 4: Total moment of inertia of two spheres. Thus, the moment of inertia of the system is:

Concept: The moment of inertia measures how difficult it is to change the rotational motion of a body about a given axis. It depends on how the mass of the body is distributed relative to the axis of rotation. For a continuous body, where:
• is the perpendicular distance of the mass element from the axis,
• is the small mass element. For standard rigid bodies, these integrals are already evaluated and known as standard results. One such result is the moment of inertia of a uniform solid disc about its central axis.

Step 1: Identify the body and the axis of rotation. The object is a uniform solid disc of: The axis of rotation passes:
• Through the center of the disc
• Perpendicular to its plane This is the standard central axis for a disc.

Step 2: Use the standard formula for a solid disc. The moment of inertia of a uniform disc about this axis is:

Step 3: Select the correct option. Thus, Hence the correct answer is

Concept: For a uniform ring, about an axis perpendicular to its plane through the centre. Using the parallel axis theorem: where is the distance between the axes.

Step 1: Identify the required axis. The required axis is a tangent perpendicular to the plane of the ring. Distance between the centre and tangent axis:

Step 2: Apply the parallel axis theorem.

Step 3: Simplify the expression.

Concept: The moment of inertia of a body measures its resistance to rotational motion about a given axis. For a solid sphere, the moment of inertia about an axis passing through its centre (diameter) is a standard result.

Step 1: Recall the standard formula. The moment of inertia of a solid sphere about its diameter is where = mass of the sphere, = radius of the sphere.

Step 2: Interpret the result. This means the rotational inertia of a solid sphere depends on both its mass and the square of its radius.

Step 1: Understanding the Concept:
The moment of inertia measures an object's resistance to rotational acceleration about a specific axis.
For a continuous mass distribution, it is calculated using the integral .

Step 2: Key Formula or Approach:
The approach involves taking an elemental mass at a distance from the axis and integrating over the entire length of the rod.
The relevant formula is , where the limits of integration are from to .

Step 3: Detailed Explanation:
Consider a thin uniform rod of mass and length .
Let the axis of rotation pass through its center and be perpendicular to its length.
We take an elemental mass of length at a distance from the center.
The linear mass density is , so the mass of the element is .
The moment of inertia of this elemental mass is .
Integrating from to gives the total moment of inertia:





Step 4: Final Answer:
The correct option is (B) as the calculated moment of inertia is .

Concept: The moment of inertia of a body measures its resistance to rotational motion about a given axis. For a solid sphere of mass and radius , the moment of inertia about its diameter is: When two identical solid spheres are considered together about a suitable axis, the total moment of inertia becomes the sum of the individual moments of inertia according to the principle of superposition.

Step 1: Moment of inertia of one sphere. For one solid sphere:

Step 2: Moment of inertia for two spheres. If two identical spheres contribute to the rotational system and the axis arrangement leads to the combined effect, the total moment of inertia becomes:

Step 3: Simplification. Considering the geometrical configuration of the spheres and the effective contribution of mass distribution in the system, the resulting moment of inertia becomes: Thus, the moment of inertia for the given system of two spheres is