Circular Motion

The shadow of the ball will move back and forth along a straight line because the child is swinging the ball in a vertical plane, causing the shadow to follow a linear motion when viewed from above. This happens as the angle between the sun and the ball changes periodically.

First, we calculate the speed of the stone using the formula for circular motion.
The linear velocity is related to the angular velocity by the relation: where and is the angular speed.
The number of revolutions per minute is given as 40 rev/min, which we convert to radians per second: So, the speed of the stone is: Now, the tension in the string is related to the centripetal force by: Substituting the known values: Next, we find the maximum speed at which the stone can be whirled around, given the maximum tension of 200 N.
Using the formula for tension: Solving for :
Thus, the correct answer is (a).

When a ring rolls on a rough surface, the frictional force acts to reduce its angular velocity.
The equation for the angular deceleration due to friction is given by: where is the mass of the ring and is the acceleration due to gravity. The torque due to the friction is: The angular acceleration is given by: where is the moment of inertia of the ring.
For a ring, , so: Simplifying: Now, to reduce the angular velocity to half, we use the equation: Substituting , we get: Solving for :
Thus, the correct answer is (b).

The rotational kinetic energy of a solid cylinder is given by: where is the moment of inertia of the solid cylinder, and is the angular velocity.
The moment of inertia for a solid cylinder rotating about its axis is: Substituting the given values: The angular momentum is given by:
Thus, the correct answer is (a).

The frictional force, the applied force, and the weight of the object all depend on the conditions of the system. If the object is in motion, the frictional force could be smaller or larger than the applied force depending on the situation (e.g., if the applied force is enough to overcome friction). Similarly, the weight of the object could be larger or smaller compared to the applied force based on the forces acting on the system. Hence, all the statements could be correct depending on the context.