Solutions to Quiz 7
November 10, 2009
1
Finding the launch speed of the Bowling
Ball
You throw a bowling ball (mass M , radius R) onto a lane with a backspin
of ω0 . As soon as it hits the lane, friction begins to act on the ball. As
shown in the graph, friction decreases the balls velocity, but it also increases
its angular velocity. To your dismay, you find that the ball comes to a stop
somewhere on the lane.
Figure 1: Velocity and R times angular velocity plotted over time
If the ball has moment of inertia I = βM R2 , find the initial velocity v0
that you gave the ball. Express your answer in terms of β, R and ω0 .
Hint: You can use kinematics to get equations for v(t) and ω(t). You
know the final velocities. So you then have 2 equations relating v0 , ω0 and t,
and you can eliminate t.
Remember that the ball IS slipping so you can NOT use v = Rω or
a = Rα.
1
1.1
Solution
Figure 2: Motion of the ball
As soon as the ball hits the lane, friction begins to act on it. The effect
of friction is to decrease the forward velocity but to increase the angular
velocity in the clockwise direction. Following the hint, we use kinematics to
derive a relation. We know that when the ball stops, its final velocity v and
angular velocity ω are both 0.
So we have the kinematic equations
0 = v0 − at
0 = ω0 + αt
(1)
(2)
where I’ve chosen a and α to be positive, and put the signs in by hand.
We can rewrite these as t = −ω0 /α, and t = v0 /a. So equating t gives
a
(3)
v0 = − ω0
α
So now it’s just a matter of finding a and α. We can do so using Newton’s
laws.
Ff ric = ma
Ff ric R = Iα
(4)
(5)
combing the two gives a/α = I/mR = βR. Substituting this into our result
(3) gives
v0 = −βRω0
(6)
2
2
What happens when you throw it faster?
What would happen if you had thrown the ball with more initial speed v0
(keeping ω0 fixed)? Draw a graph of v and Rω vs. t for this case.
Hint: When you finally reach v = Rω, friction ‘switches off’ (because
now the bottom part of the ball has no velocity) and the ball is locked into
whatever speed it is at.
2.1
Solution
If you throw the ball with more initial speed v0 , while keeping ω0 fixed,
then from the graph we can see that we will reach the point that v = Rω
at some non-zero velocity. Once this happens, friction ‘switches off’ (or to
throw around the jargon, friction decouples) and in the absence of any forces
or torques, our ball will keep moving at speed v and the angular velocity
ω = v/R.
Figure 3: Velocity and R times angular velocity curve for the ball
You can try this experiment out by stepping on a tennis ball and launching
it forward from under your shoe. Depending on the ratio of |v0 /ω0 |, once the
ball stops slipping you should be able to get it to move forward (|v0 /ω0 | >
βR), stop (|v0 /ω0 | = βR), or even come backwards (|v0 /ω0 | < βR)!
3