Uniform Circular Motion

1. Setup for Equilibrium:
We are given a system where a loop is suspended between two points and , and the system is under the influence of a magnetic field . The magnetic torque acting on the loop is balanced by the weight of the object. For equilibrium, we consider the torque and the forces involved:

2. Equation for Magnetic Torque:
The torque due to the magnetic field is given by:

3. Equating the Torque and Solving for :
For equilibrium, the torques due to the magnetic field and gravitational force must balance:

Simplifying the equation:

4. Final Expression for :
Dividing both sides by and simplifying:

5. Conclusion:
Thus, the angle at which the system is in equilibrium is given by the expression:

Conservation of Angular Momentum

By the conservation of angular momentum about the center of the large disc:

Initially, the system is at rest, so the initial angular momentum is zero. When the motor is switched on, the clockwise angular momentum of the smaller disc is balanced by the counterclockwise angular momentum of the large disc.

Angular Momentum Calculation

The angular momentum of the smaller disc is:

The angular momentum of the large disc is:

Equating and :

Simplifying to find :

Finding the Value of

Since the angular speed of the large disc is given by :

Thus, the value of is:

n = 12

Final Answer:

n = 12