Q6.13 Torque and angular momentum (H)

Write down the equation expressing the rate of change of the angular momentum of a system.

A ball is thrown from a point O. Show that the angular momentum of the ball about O is not conserved unless the ball is thrown vertically.

Show with a sketch how

  1. the torque on the ball, about O
  2. the angular momentum of the ball, about O

vary in the course of the ball’s flight.

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Hint

Don’t forget that torque (Key Point 5.15) and angular momentum (Key Point 5.16) are vectors.
 

Solution

Reveal
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Solution

The rate of change of angular momentum is given by Key Point 5.18

\[ \vec{\tau}_{ext} = \frac{d\vec{L}}{dt} \]

If the ball is not thrown vertically, the gravitational force $\vec{F}=-mg\hat{y}$ does not act through O, and therefore exerts a torque about O. The condition for conservation of angular momentum about O is thus not satisfied.

The torque about O has magnitude τ=mgrsinθ=mgx, and direction perpendicular to the xy plane (into the diagram).

Since x increases linearly with t, so does τ. Hence so does $\frac{dL}{dt}$. It follows that L increases with t2.

Graphic - No title - tqtorque0
Graphic - No title - tqtorque1