# Angular momentum (left or right movement)

## Homework Statement

A thin uniform rectangular sign hangs vertically above the
door of a shop. The sign is hinged to a stationary horizontal
rod along its top edge. The mass of the sign is 2.40 kg
and its vertical dimension is 50.0 cm. The sign is swinging
without friction, becoming a tempting target for children
armed with snowballs. The maximum angular displacement
of the sign is 25.0° on both sides of the vertical. At a
moment when the sign is vertical and moving to the left, a
snowball of mass 400 g, traveling horizontally with a velocity
of 160 cm/s to the right, strikes perpendicularly the
lower edge of the sign and sticks there.

Is the system (sign-snowball) going to move left or right?

## Homework Equations

Conservation of angular momentum.

## The Attempt at a Solution

Lost

Doc Al
Mentor
What's the angular momentum of the sign when it passes the vertical? Hint: Start by finding its moment of inertia.

The moment of inertia is $I=\frac{1}{3}ML^2$, therefore the angular momentum of the sign when it passes the vertical is $L_s=Iω=(\frac{1}{3}ML^2)ω$.

Doc Al
Mentor
The moment of inertia is $I=\frac{1}{3}ML^2$, therefore the angular momentum of the sign when it passes the vertical is $L_s=Iω=(\frac{1}{3}ML^2)ω$.
Excellent. Now find ω.

Once you've done that, figure out the initial angular momentum of the snowball.

I found the angular velocity ω from the conservation of energy:

$\frac{1}{2}Iω^2=Mgh$
$ω=\sqrt{\frac{3g(1-cosθ)}{L}}=2.35 rad/s$

The angular momentum of the sign is $L_s=\frac{1}{3}ML^2=0.47 Nms$ while the angular momentum of the snowball is $L_b=mvL=0.32 Nms$

Comparing these two, the angular momentum of the sign is greater than the angular momentum of the ball, therefore the system is going to move to the left.

Doc Al
Mentor
Good work!

Thank you for your help and for checking my work. I really appreciate it.