Torque on a spherical planetoid galaxy

[Borat321]

Hi - I had a question on webassign - here is the question.

A spherical planetoid in a galaxy far, far away has spin angular momentum of magnitude L = 5.9e+35 kg m2/s directed out of its north pole. An external torque acts on it, such that the planetoid's axis of rotation, and hence its angular momentum vector, gradually changes direction, describing a cone with half-angle 23.5 degrees as shown in the figure.
Define the y-axis as straight up in the figure (the vertical arrow shown). Define the x-axis as to the right.

Suppose the angular momentum vector takes 21200 years to swing once around the cone shown. What is the magnitude of the rate of change of the planetoid's angular momentum in that direction at the instant shown? (Hint: consider the analogy between how the component of angular momentum changes with time, and how the position of a particle in circular motion changes with time.

What is the magnitude of the external torque exerted on the planetoid?

I thoiught that the rate of cahnge of the planetoid's angular momentum can just be Lsin(23.5)w, where L=5.9e+35 and w = 2pi/21200 converted into seconds.

Also, I thought torque would just be rFsin23.5, where r = radius of Earth, but where am I going wrong?

 

[Nate810]

The rate of change of the planetoid's angular momentum in that direction would be: Lsin(23.5)w, where L is the magnitude of the spin angular momentum (5.9e+35 kg m2/s) and w is the angular velocity (2π/21200). The magnitude of the external torque exerted on the planetoid is: T = Lsin(23.5)w, where L is the magnitude of the spin angular momentum (5.9e+35 kg m2/s) and w is the angular velocity (2π/21200).

 

[kyle8921]

I would like to clarify and expand on the concept of torque and angular momentum in this scenario. Torque is a measure of the force that causes an object to rotate around an axis. In the case of the planetoid, the external torque is causing the planetoid's axis of rotation to change direction, thus changing its angular momentum vector.

The rate of change of the planetoid's angular momentum in the direction shown can be calculated using the cross product of the planetoid's angular momentum vector and the external torque vector. This will give us the magnitude and direction of the change in angular momentum.

To calculate the magnitude of the external torque, we need to consider the radius of the planetoid and the force acting on it. The force can be calculated using the formula F=ma, where m is the mass of the planetoid and a is its acceleration. This acceleration is caused by the external torque, so we can use the formula T=Iα (torque= moment of inertia x angular acceleration) to find the torque.

In summary, the rate of change of the planetoid's angular momentum can be calculated using the cross product of its angular momentum vector and the external torque vector. The magnitude of the external torque can be found using the formula T=Iα, where I is the moment of inertia and α is the angular acceleration. These calculations will give us a better understanding of the dynamics of the planetoid's rotation and the effects of external forces on it.