Mr.p: Good morning. Bo, could you please read the problem and Bobby, could you please translate?
"Flipping Physics" Intro
I apologize for not wearing safety glasses while performing this demonstration.
If this egregious oversight causes you emotional harm, I am sorry.
Seriously, I should have worn safety glasses and it was asinine of me not to have done so.
Billy: Yeah, the dart could have bounced back towards you instead of getting stuck in that thing.
Mr.p: Actually, it did bounce back a couple of times.
Billy: Oh. That is dangerous. Bo: That's not so smart. Bobby: What were you thinking?
Mr.p: I know. I am fallible. Billy: Yeah.
Mr.p: Bo please read.
Bo: As shown, a 5.3 gram dart moving horizontally at 16.9 meters per second collides with and sticks to a
stationary Rotational Inertia Demonstrator
a distance of 31.7 cm from the axis of rotation of the Rotational Inertia Demonstrator.
What is the final angular velocity of the Rotational Inertia Demonstrator?
The rotational inertia of the rotational inertia demonstrator equals
0.0237 kilograms times meters squared.
Bo: Okay. The mass of the dart is 5.3 grams,
actually, we should probably make sure everything is in base SI units, so 0.0053 kilograms.
The initial velocity of the dart is 16.9 meters per second. How fast is that?
Billy: 37.8 miles per hour or 60.8 kilometers per hour. Bo: Cool.
Bobby: And the distance the dart hits the rotational inertia demonstrator from its axis of rotation,
let's call that distance "y", equals 0.317 meters.
You can see the initial angular velocity of the rotational inertia demonstrator is zero.
The rotational inertia of the demonstrator equals 0.0237 kilograms times meters squared
and the final angular velocity of the demonstrator equals question mark.
Mr.p: Alright, we have already shown in a previous lesson
that angular momentum is conserved when a point particle, in this case the dart,
collides with a rigid object with shape, in this case the rotational inertia demonstrator.
Billy, could you please begin by just showing us the basics
of how we know angular momentum is conserved during this collision and begin solving the problem?
Billy: Okay. Well,
we know angular momentum is conserved because the net torque acting on the system equals zero.
In this case the system is the dart and the rotational inertia demonstrator,
and the axis of rotation is at the center or axle of the demonstrator.
We know this is true because net torque equals change in angular momentum over change in time,
and if that equals zero, then the change in angular momentum equals zero,
so the angular momentum is not changing. Therefore angular momentum is conserved.
Before the collision that means we have the sum of the initial angular momenta of the dart and demonstrator
and after the collision we have the sum of the final angular momenta of the dart and demonstrator.
Mr.p: And what are the two general equations we have for the angular momentum of objects?
Billy: The angular momentum of a point particle equals r,
the vector from the axis of rotation to the center of mass of the point particle,
times its mass, times its velocity, times the sine of the angle between the r vector and its velocity.
The angular momentum of a rigid object with shape equals the rotational inertia times angular velocity.
Mr.p: Thank You Billy. Bobby, what equations should we use for the angular momenta of the objects?
Bobby: Okay. The dart is a point particle, so before the collision, let's use the point particle equation.
Before the collision, the rotational inertia demonstrator is not moving, so it has zero angular momentum initial.
After the collision, the dart is still a point particle, so
Bo: Remember, we already showed in a previous lesson
that you can use either of the two equations for angular momentum when a point particle is moving in a circle,
like it is, after the collision.
Bobby: Right. Thanks, Bo. Bo: Sure.
Bobby: So let's choose rotational inertia of the dart final times angular velocity of the dart final
for the angular momentum of the dart final.
And for the final angular momentum of the demonstrator, well, it is a rigid object with shape,
so its angular momentum will be its rotational inertia times its angular velocity final.
Mr.p: Great. Now, looking specifically at the equation for
the initial angular momentum of the dart as it moves toward the demonstrator before the collision.
If you recall, we have already shown in the previous lesson that,
while the values of r and theta change as the point particle moves towards the rigid object with shape,
the value of "y", which equals r for the dart initial times sine of theta for the dart initial, does not change.
So we can substitute the value "y" in for r for the dart initial times sine of theta for the dart initial.
Now Bo, what can we do from here?
Bo: We can substitute the mass of the dart times the square of r for the dart final
in for the rotational inertia of the dart.
Actually, after the collision, because the dart moves in a circle around the axis of rotation of the demonstrator,
the r for the dart final is just "y" the distance from the axis of rotation to the dart.
And, after the collision, the dart and demonstrator are both moving with the same final angular velocity.
That means we can factor out final angular velocity
and divide the whole equation by what is parenthetically in front of the final angular velocity
and the final angular velocity equals y times the mass of the dart times the velocity of the dart initial
all divided by the quantity mass of the dart times y squared plus the rotational inertia of the demonstrator.
With numbers that is
0.317 times 0.0053 times 16.9
all divided by the quantity 0.0053 times
0.317 squared plus 0.0237
which equals 1.1717
or 1.2 radians per second with two significant digits.
Mr.p: Very nice, Bo.
Now, 1.2 radians per second is our predicted final angular velocity of the system.
We can also measure the final angular velocity of the system.
Now, you can see that, for a few moments after the collision, the rotational inertia demonstrator
acts a little bit less like a rigid object with shape and does change shape slightly.
Eventually it stops oscillating and we can then begin measuring its angular velocity.
One spoke of the demonstrator takes 69 frames to go through an angular displacement of 5.0 degrees.
We can convert the angular displacement to pi over 36 radians and,
knowing the original video was filmed at 960 frames per second,
we can convert the change in time to 0.071875 seconds.
Angular velocity equals change in theta over change in time,
so 1.21414 radians per second is our measured final angular velocity of the system.
The percentage difference between our predicted and measured values ends up being roughly 3.6 percent.
Which makes me think we can say, "the physics works!" BBB: Uh huh, uh huh.
Mr.p: The physics works. The physics works. BBB: Uh huh. Uh huh.
Mr.p: The physics works.
Thank you very much for learning with me today, I enjoyed learning with you.