1.Two masses of 4gm and 6gm respectively are attached to the ends of a light rod of negligible mass and the rod rotates anticlockwise at the rate of 2rev/sec about an axis passing through the center of mass and perpendicular to its length.

Calculate
(a)the angular momentum of each mass about the center of mass.
(b) the total momentum of the system about the center of mass.
(c) the angular momentum of the system about the axis of rotation.
How will things change if the axis of rotation were inclind at 30 degrees to the length of the rod and why?

To calculate the angular momentum of each mass about the center of mass, we can use the formula:

Angular momentum = mass x velocity x radius

For mass 1 (4gm):
(a) The mass of 4gm is given. The velocity of the mass can be calculated by multiplying the angular velocity (2rev/sec) by the distance of the mass from the center of mass. Since it is attached to the end of the rod of negligible mass, the distance from the center of mass is half the length of the rod.

Velocity (mass 1) = angular velocity x radius (mass 1)

For mass 2 (6gm):
Similarly, you can calculate the velocity of mass 2 using the same formula, considering its distance from the center of mass.

Velocity (mass 2) = angular velocity x radius (mass 2)

(b) To find the total momentum of the system about the center of mass, you need to add the individual angular momenta of each mass.

Total momentum = angular momentum (mass 1) + angular momentum (mass 2)

(c) The angular momentum of the system about the axis of rotation is given by the formula:

Angular momentum (system) = angular momentum (mass 1) + angular momentum (mass 2) + (mass 1 + mass 2) x velocity (center of mass)

The last term in the equation takes into account the linear momentum of the center of mass, which is given by the total mass of the system multiplied by the velocity of the center of mass.

If the axis of rotation were inclined at 30 degrees to the length of the rod, the distances of the masses from the axis would change. This would affect the calculation of the velocities and the angular momentum of each mass. The change in the distance would alter the radius term in the formula.

To find the new velocities and angular momenta, you would need to use the new distances from the axis of rotation and apply the same formulas as described above. The inclination of the axis of rotation affects the distribution of masses and how they contribute to the total angular momentum of the system.

(a) To calculate the angular momentum of each mass about the center of mass, we can use the formula:

Angular momentum = mass * velocity * perpendicular distance from the axis of rotation

Given:
Mass1 (m1) = 4 gm = 0.004 kg
Mass2 (m2) = 6 gm = 0.006 kg
Angular velocity (ω) = 2 rev/sec = 2 * 2π rad/sec = 4π rad/sec (since 1 revolution = 2π radians)

Let's assume the distance from the center of mass to the center of mass is 'd'. Since the rod is of negligible mass, the center of mass will be at the midpoint of the rod.

For Mass1:
Distance from the axis of rotation = d/2
Velocity = ω * distance from the axis of rotation = 4π * (d/2) = 2πd

Angular momentum of Mass1 = m1 * velocity * distance from the axis of rotation = 0.004 * (2πd) * (d/2)

For Mass2:
Distance from the axis of rotation = -d/2 (since it's on the other side of the center of mass)
Velocity = ω * distance from the axis of rotation = 4π * (-d/2) = -2πd

Angular momentum of Mass2 = m2 * velocity * distance from the axis of rotation = 0.006 * (-2πd) * (-d/2)

(b) The total momentum of the system about the center of mass is simply the sum of the individual angular momenta of the masses:
Total angular momentum about the center of mass = Angular momentum of Mass1 + Angular momentum of Mass2

(c) The angular momentum of the system about the axis of rotation is given by:
Angular momentum about the axis of rotation = Angular momentum about the center of mass + (mass of the system * velocity of the center of mass * perpendicular distance between the two axes)

If the axis of rotation were inclined at 30 degrees to the length of the rod, the perpendicular distance between the two axes would change, affecting the angular momentum calculations. The distances from the masses to the new axis of rotation would also change, leading to different values for angular momentum.