JEE Exam:

1. For the solid sphere, we can use the formula:

$$ I = \frac{2}{5}MR^2 $$ $$ \alpha = \frac{\Delta \omega}{\Delta t} $$

where:

• I is the moment of inertia of the sphere
• M is the mass of the sphere
• R is the radius of the sphere

Then, the torque required to stop the sphere can be calculated using:

$$ \tau = I \alpha $$

2. The angular momentum of the disc is given by:

$$ L = I\omega $$

We can calculate the moment of inertia of the disc using the formula:

$$I = \frac{1}{2}MR^2$$

where:

• M is the mass of the disc
• R is the radius of the disc

Thus, we have :

$$L= \frac {1} {2}MR^2 \times \omega $$

3. The angular momentum of the particle is given by:

$$ L = mvr $$

where:

• m is the mass of the particle
• v is the speed of the particle
• r is the radius of the circular path

4. Treating the two particles as point masses, the total angular momentum of the system is:

$$ L = I\omega $$

where:

I = the total moment of inertia of the system I can be calculated as:

$$I = m_1r_1^2 + m_2r_2^2 $$ $$ I= (4)(0.25)^2 + (6)(0.25)^2 $$ $$ I = 2.5 \ kg \ m^2$$

where:

• m1 and m2 are the masses of the two particles
• r1 and r2 are the distances of the particles from the center of mass

5. The kinetic energy of the flywheel is given by:

$$ K = \frac{1}{2}I\omega^2$$

where:

• I is the moment of inertia of the flywheel
• ω is the angular velocity of the flywheel

To calculate the moment of inertia of the flywheel, we can use the formula:

$$ I = \frac{1}{2}MR^2$$

where:

• M is the mass of the flywheel
• R is the radius of the flywheel

Hence, the kinetic energy becomes:

$$ K=\frac {1} {2}MR^2 \omega^2 $$

CBSE Board Exam:

1. For the pivoted rod, the angular acceleration can be calculated using the formula:

$$ \alpha = \frac{\tau}{I}$$ where:

• τ is the torque acting on the rod
• I is the moment of inertia of the rod

We can calculate the moment of inertia of the rod using the formula:

$$ I = \frac{1}{3}ML^2 $$ $$I = \frac {1} {3}(2)(1)^2$$ $$ I=\frac {2} {3}\ kg \ m^2$$ where:

• M is the mass of the rod
• L is the length of the rod

2. The kinetic energy of the rolling wheel is given by:

$$ K = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2 $$

where:

• M is the mass of the wheel
• v is the velocity of the wheel
• I is the moment of inertia of the wheel
• ω is the angular velocity of the wheel

We can calculate the moment of inertia of the wheel using the formula:

$$ I = \frac{1}{2}MR^2$$

The angular velocity of the wheel can be calculated as:

$$ \omega = \frac{v}{R} $$

3. The torque acting on the pulley is given by:

$$ \tau = Fr$$

where:

• F is the force applied to the rope
• r is the radius of the pulley

4. The angular momentum of the cylinder is given by:

$$ L = I\omega $$

We can calculate the moment of inertia of the cylinder using the formula:

$$I = \frac{1}{2}MR^2$$

where:

• M is the mass of the cylinder
• R is the radius of the cylinder

5. The centripetal force acting on the particle is given by:

$$ F_c = \frac{mv^2}{r}$$

where:

• m is the mass of the particle
• v is the speed of the particle
• r is the radius of the circular path