Rotational Motion

Given:
Mass of the flywheel,
Radius,
Angular velocity,
Time to stop,
Coefficient of friction,
Step 1: Calculate the angular deceleration ( ) The flywheel comes to rest in 3.14 s, so the angular deceleration is: The magnitude of angular deceleration is .
Step 2: Calculate the torque ( ) The torque required to stop the flywheel is: where is the moment of inertia of the flywheel. For a solid disk: Thus, the torque is: Step 3: Calculate the frictional force ( ) The torque is also given by: Solving for : However, considering the coefficient of friction , the normal force required to produce this frictional force is: Thus, the force to be applied against the flywheel is 200 N.
Final Answer: 200 N

Step 1: Apply conservation of angular momentum.
Before the cockroach stops, the system's angular momentum is: The negative sign arises because the cockroach’s velocity direction (anticlockwise) opposes the disc's rotation direction (clockwise).
Step 2: After the cockroach stops.
The cockroach's angular momentum becomes zero, and the disc's angular momentum must now equal the initial total angular momentum: Step 3: Solve for the final angular velocity .
Using the conservation of angular momentum:

When a metal rod of length is clamped at its midpoint, the fundamental mode of vibration has a node at the center and antinodes at the ends. This means that the length of the rod corresponds to half a wavelength, so . The speed of a transverse wave in a solid is given by where is Young's modulus and is the density. The frequency , wavelength , and speed of a wave are related by Hence, We are given that , , and , so Final Answer: 2 kHz.

Let the width of the river be .
Case 1: Shortest Path To cross the river along the shortest path (perpendicular to the river flow), the boat must counteract the river's flow. The effective velocity of the boat perpendicular to the river flow is: The time taken to cross the river along the shortest path is: Case 2: Shortest Time To cross the river in the shortest time, the boat should head directly across the river (perpendicular to the flow) without counteracting the flow. The effective velocity of the boat is simply its velocity in still water: The time taken to cross the river in the shortest time is: Ratio of Times The ratio of the times taken is: Thus, the ratio is 13:12.
Final Answer: 13:12

The velocity of a rolling object at the bottom of an inclined plane is given by:

where:

• g is the acceleration due to gravity.
• h is the height of the inclined plane.
• I is the moment of inertia.
• m is the mass.
• r is the radius.

For different objects:

• Ring: →
• Solid sphere: →
• Solid cylinder: →

Comparison:

Since , the solid sphere has the smallest value of and thus the largest velocity.

Conclusion: The solid sphere reaches the bottom with maximum velocity.