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  • Observe collisions of extended bodies in two dimensions.
  • Examine collision at the point of percussion.

Bowling pins are sent flying and spinning when hit by a bowling ball—angular momentum as well as linear momentum and energy have been imparted to the pins. (See [link] ). Many collisions involve angular momentum. Cars, for example, may spin and collide on ice or a wet surface. Baseball pitchers throw curves by putting spin on the baseball. A tennis player can put a lot of top spin on the tennis ball which causes it to dive down onto the court once it crosses the net. We now take a brief look at what happens when objects that can rotate collide.

Consider the relatively simple collision shown in [link] , in which a disk strikes and adheres to an initially motionless stick nailed at one end to a frictionless surface. After the collision, the two rotate about the nail. There is an unbalanced external force on the system at the nail. This force exerts no torque because its lever arm r size 12{r} {} is zero. Angular momentum is therefore conserved in the collision. Kinetic energy is not conserved, because the collision is inelastic. It is possible that momentum is not conserved either because the force at the nail may have a component in the direction of the disk’s initial velocity. Let us examine a case of rotation in a collision in [link] .

A bowling ball, just as it is striking the pins.
The bowling ball causes the pins to fly, some of them spinning violently. (credit: Tinou Bao, Flickr)
Figure a shows a disc m sliding toward a motionless stick M of length r pivoted about a nail, on a frictionless surface. In figure b, a disk hits the stick at one end and adheres to it, and the stick rotates, pivoting around the nail in a direction shown by the arrow in the clockwise direction and angular velocity omega.
(a) A disk slides toward a motionless stick on a frictionless surface. (b) The disk hits the stick at one end and adheres to it, and they rotate together, pivoting around the nail. Angular momentum is conserved for this inelastic collision because the surface is frictionless and the unbalanced external force at the nail exerts no torque.

Rotation in a collision

Suppose the disk in [link] has a mass of 50.0 g and an initial velocity of 30.0 m/s when it strikes the stick that is 1.20 m long and 2.00 kg.

(a) What is the angular velocity of the two after the collision?

(b) What is the kinetic energy before and after the collision?

(c) What is the total linear momentum before and after the collision?

Strategy for (a)

We can answer the first question using conservation of angular momentum as noted. Because angular momentum is size 12{Iω} {} , we can solve for angular velocity.

Solution for (a)

Conservation of angular momentum states

L = L , size 12{L=L'} {}

where primed quantities stand for conditions after the collision and both momenta are calculated relative to the pivot point. The initial angular momentum of the system of stick-disk is that of the disk just before it strikes the stick. That is,

L = , size 12{L=Iω} {}

where I size 12{I} {} is the moment of inertia of the disk and ω size 12{ω} {} is its angular velocity around the pivot point. Now, I = mr 2 size 12{I= ital "mr" rSup { size 8{2} } } {} (taking the disk to be approximately a point mass) and ω = v / r size 12{ω=v/r} {} , so that

L = mr 2 v r = mvr . size 12{L= ital "mr" rSup { size 8{2} } { {v} over {r} } = ital "mvr"} {}

After the collision,

L = I ω . size 12{L'=I'ω'} {}

It is ω size 12{ω rSup { size 8{'} } } {} that we wish to find. Conservation of angular momentum gives

I ω = mvr . size 12{I'ω'= ital "mvr"} {}

Rearranging the equation yields

ω = mvr I , size 12{ω'= { { ital "mvr"} over {I'} } } {}

where I size 12{I'} {} is the moment of inertia of the stick and disk stuck together, which is the sum of their individual moments of inertia about the nail. [link] gives the formula for a rod rotating around one end to be I = Mr 2 / 3 size 12{I= ital "Mr" rSup { size 8{2} } /3} {} . Thus,

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s. Reply
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Source:  OpenStax, Mechanics. OpenStax CNX. Apr 15, 2013 Download for free at http://legacy.cnx.org/content/col11506/1.2
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