Asteroid/Angular Momentum Problem

[rwx1606]

Homework Statement

A spherical asteroid with radius r = 123m and mass M = 2.10×1010 kg rotates about an axis at four revolutions per day. A "tug" spaceship attaches itself to the asteroid's south pole (as defined by the axis of rotation) and fires its engine, applying a force F tangentially to the
asteroid's surface as shown in the figure.If F = 265N, how long will it take the tug to rotate the asteroid's axis of rotation through an angle of 10.0 degrees by this method?

The Attempt at a Solution

I solved for the angular acceleration using that net torque is equal to the moment of inertia times the angular acceleration. After that I used constant angular acceleration equations to solve for the final angular velocity and then solved for time t. However, this is not giving the right answer and I would appreciate any help. The system I considered was the asteroid itself, so the spaceship applies a external torque and I said angular momentum wasn't conserved. Thanks in advance for the help.

 

[jbsteele]

$\tau = Iα$ $I = \frac{2}{5}MR^2$ $α = \frac{\tau}{I}$ $α = \frac{F*R}{\frac{2}{5}MR^2}$ $α = \frac{265*123}{\frac{2}{5}(2.1*10^{10})(123)^2} = 2.32*10^{-7} rad/s^2$ $ω_f = ω_i + α*t$ $ω_f = 0 + 2.32*10^{-7}*t$ $Δθ = ω_ft$ $Δθ = (2.32*10^{-7})*t$ $Δθ = 10°$ $t = \frac{10°}{2.32*10^{-7}}$ $t = 4.35*10^8 s$

 

[thomate1]

Dear student,

Thank you for sharing your attempt at a solution to the asteroid/angular momentum problem. It is clear that you have approached the problem correctly by considering the net torque on the asteroid and using the equations of constant angular acceleration. However, it seems that there may be an error in your calculations or assumptions.

Firstly, it is important to note that in this problem, angular momentum is conserved. This is because the system (the asteroid and the spaceship) is isolated, meaning there are no external torques acting on it. Therefore, the initial angular momentum of the system must be equal to the final angular momentum after the spaceship applies the force.

To solve this problem, you can use the equation for angular momentum, L = Iω, where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity. The initial angular momentum of the asteroid can be calculated using its given mass and rotational speed (four revolutions per day). The final angular momentum can be calculated by taking into account the additional angular momentum added by the force applied by the spaceship.

Once you have the final angular momentum, you can use the equation L = Iω to solve for the final angular velocity. Then, using the equation ω = ω0 + αt, where ω0 is the initial angular velocity and α is the angular acceleration, you can solve for the time t it takes for the asteroid to rotate through an angle of 10.0 degrees.

I hope this helps you to solve the problem correctly. If you need any further assistance, please do not hesitate to ask. Keep up the good work in your studies of angular momentum and celestial bodies.

Best regards,
[Your name]