Good morning.
Bo, could you please read the problem, and Bobby, could you please translate?
♫ Flipping Physics ♫
Bo: A record player is plugged in, uniformly accelerates to 45 revolutions per minute, and then is unplugged.
Bobby: Stop. Angular velocity is 45 revolutions per minute.
Billy: We need to convert 45 revolutions per minute to radians per second...
uhh...so multiply by 1 minute over 60 seconds to convert to seconds,
and 2π radians over one revolution to convert to radians, which is 1.5π radians per second.
Bobby: Don't we need to multiply through by π to get 4.7124 radians per second?
Mr. P: Actually, you know what, sometimes it is easier in the middle of the problem not to multiply through by π;
1.5π is both easier to write and more precise than 4.7124.
We will multipIy through by π when we determine our answers. Bo?
Bo: The record player (a) takes 0.85 seconds to get up to speed,
(b) spends 3.37 seconds at 45 RPMs,
and then (c) takes 2.32 seconds to slow down to a stop.
Bobby: OK, um...
the 1.5π radians per second is actually the angular velocity for part (b)...
...the initial angular velocity for part (a) is 0,
and the final angular velocity for part (a) is 1.5π radians per second.
The initial angular velocity for part (c) is also 1.5π radians per second,
and the final angular velocity for part (c) is 0.
Billy: Don't forget the changes in time - change in time for part (a) is 0.85 seconds,
part (b) is 3.37 seconds,
and part (c) is 2.32 seconds.
Mr. P: Bo, please continue reading.
Bo: What is the average angular acceleration of the record player during all three parts?
Bobby: Average angular acceleration for all three parts equals question mark.
Billy: It does not actually say in the problem that the 45 revolutions per minute is an angular velocity.
How do we know that is an angular velocity and not a linear velocity?
Bobby: The units.
Revolutions per minute are the units for angular velocity, not linear velocity.
Billy: Thanks, Bobby.
Mr. P: Billy, please solve part (a).
Billy: Okay, umm...
Average angular velocity equals change in angular velocity over change in time,
which equals angular velocity final minus angular velocity initial, all divided by change in time...
...uh, for part (a) that works out to be 1.5π minus 0, all divided by 0.85 seconds,
or 5.54399, or 5.5 radians per second squared with two significant digits.
Mr P: Thank you!
Bo, could you please do part (b)?
Bo: Sure, uhm...
We can use the same equation for angular acceleration Billy just did...
...however, we know the angular velocity, not the initial or final angular velocity.
Bobby: Actually, the record player is moving at a constant angular velocity,
so the initial and final angular velocities are both 1.5π radians per second.
Bo: Right, the record player is moving at a constant angular velocity during part (b),
so the change in angular velocity is 0, so the angular acceleration is 0. Duh.
Billy: Yeah, uh...When the angular velocity of the object is not changing, the angular acceleration is 0!
Mr. P: Correct!
And part (c) please, Bobby.
Bobby: Again, we can use the equation Billy used -
angular acceleration for part (c) equals angular velocity (c) final minus angular velocity (c) initial, divided by change in time for part (c),
which is 0 minus 1.5π divided by 2.32,
which is -2.03120, or -2.0 radians per second squared, with two significant digits.
Mr. P: Let's just take a moment to recognize that in this basic example during part (a),
when the record player is speeding up in a positive direction,
the angular acceleration of the record player is positive.
During part (b), when the record player is moving at a constant angular velocity,
the angular acceleration of the record player is zero.
And during part (c), when the record player is slowing down in a positive direction,
the angular acceleration of the record player is negative.
The same thing is true for the linear accelerations when an object has a positive linear velocity.
Thank you very much for learning with me today; I enjoyed learning with you.