# How Do You Correctly Cancel Angular Momentum in Physics Problems?

• pbnj
pbnj
Homework Statement
A ball with mass ##m## and diameter ##D## is thrown with speed ##v## at an angle ##\theta## with the horizontal from a height ##h_i##. How much spin (in rad/s) must the thrower impart on the ball so that at its maximum height, it has no angular momentum with respect to a point on the ground directly beneath the ball?
Relevant Equations
##v_f^2 = v_i^2 + 2a\Delta h##
##L = I\omega##
##L = \vec r \times \vec p##
First, we calculate the maximum height using the first equation, noting that at maximum height, the velocity is purely horizontal with speed ##v\cos\theta##, and with initial vertical speed ##v\sin\theta##:

\begin{align} v_f^2 &= v_i^2 - 2g(h_f - h_i) \\ 0 &= (v\sin\theta)^2 - 2g(h_f - h_i) \\ h_f &= \frac{(v\sin\theta)^2}{2g} + h_i \end{align}

The angular momentum from translation is given by ##L_t = \vec r \times \vec p = h_f\hat j \times mv\cos\theta\hat i = -h_fmv\cos\theta\hat k##. The moment of inertia of a sphere is ##I = \frac 2 5 mR^2## where ##R = \frac D 2##, and its angular momentum due to spin is ##L_s = I\omega##.

We need these to cancel, so

\begin{align} L_t + L_s &= 0 \\ -h_fmv\cos\theta + \frac 2 5 mR^2\omega &= 0 \\ \omega &= \frac{h_fmv\cos\theta}{\frac 2 5 mR^2} \\ &= \frac{5h_fv\cos\theta}{2R^2} \end{align}

The problem uses ##m = 625g##, ##D = 22.9cm##, ##\theta = 45^\circ##, ##v=5m/s## and ##h_i = 1.5m##. Using these values I get a spin of about ##1440.82##, in rad/s.

pbnj said:
Using these values I get a spin of about 1440.82, in rad/s.
Have you a reason to doubt it?

haruspex said:
Have you a reason to doubt it?
It's for a Coursera course, and it tells me it's incorrect. I've tried incrementing/decrementing my answer by 5 a few times, I tried using 2 sigfigs for everything, I tried assuming "diameter" meant "radius," but no luck. On the discussion forum there are no questions about this particular answer (the course is 3 years old, and it doesn't seem like new posts get replies). If my approach is correct and my calculations are correct, I can only assume something in the backend changed in the 3 years between then and now.

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