Angular Acceleration of a Wheel on a Turntable

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Homework Statement

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The axle of a wheel is mounted on supports that rest on a rotating turntable. the wheel has angular velocity $\omega_1 = 44.0\; \frac{\textrm{rad}}{\textrm{s}}$ about its axle, and the turntable has angular velocity $\omega_2 = 35.0\; \frac{\textrm{rad}}{\textrm{s}}$ about a vertical axis. What is the magnitude and direction of the angular acceleration of the wheel at the instant shown? Take the $z$ axis vertically upward and the direction of the axle at the moment shown to be the $x$ axis pointing to the right.

Homework Equations

$\vec{\alpha} = \frac{\textrm{d}\vec{\omega}}{\textrm{d}t}$

The Attempt at a Solution

This problem stumped me a bit, but I think I got it now and just want to make sure. $\vec{\omega}_2$, which is the angular velocity of the turntable, is constant because using the right hand rule it is always pointing up. $\vec{\omega}_1$ varies because the turntable causes the direction of it to change, although its magnitude is the same. Since $\vec{\alpha} = \frac{\textrm{d}\vec{\omega}}{\textrm{d}t}$, and $\vec{\omega}_2$ is constant, the acceleration is then $\frac{\textrm{d}\vec{\omega}_1}{\textrm{d}t}$.

The direction of $\vec{\omega}_1$ follows the circular path of $\vec{\omega}_2$, so it can be parameterized by:

$\vec{\omega}_1 = \omega_1 \cos (\omega_2 t)\hat{i} + \omega_1 \sin(\omega_2 t)\hat{j}$

If we take $t = 0$ to be shown by the picture above, then at $t=0$ $\vec{\omega}_1$ is pointing to the left, so it has to be negative. Therefore, the parametrization should be:

$\vec{\omega}_1 = -\omega_1 \cos (\omega_2 t)\hat{i} - \omega_1 \sin(\omega_2 t)\hat{j}$

To get the acceleration, I take the derivative with respect ot time and find that:

$\vec{\alpha} = \omega_1\omega_2\sin(\omega_2 t)\hat{i} - \omega_1\omega_2\sin(\omega_2 t)\hat{j}$

I then simply have to find the magnitude and direction of this to get the answers. Was my process correct?

 

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Check the j term in your final equation.

 

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Check the j term in your final equation.

Whoops! should be cosine.