Question

a 1.0 kg ball and a 2.0 kg ball are connected by a rigid massless rod. The rod and balls are rotating clockwise about their center of gravity at 20 rpm. What torque will bring the balls to a halt in 5 seconds?

The correct answer is -0.28 N.m. I can't figure out how to get this answer. Please do not answer this question with an incorrect answer as it will be a waste of both our times.

Answer 1

First you need to find the center of mass. To do this use the equation:
MR = m1r1 + m2r2

Let's just say that baseball 1 is at a position r = 0, then baseball 2 would be at r = 1m
So, MR = 1kg * 0m + 2kg * 1m
MR = 2 kg*m
Then to find the position of the center of mass simply divide by the total mass (M, which is 1kg + 2kg = 3kg)
R = (2kg*m) / 3kg
R = 2/3 m = 'center of mass'

this makes sense too because we would expect the center of mass to be a little bit closer to the heavier ball.

Now, if it is rotating at 20rpm (rotations per minute) that means it is rotating at 20 * 2(pi) = 40pi radians per minute
OR 2/3(pi) radians per second which is equal to:
2.09 rads/s

That is the angular velocity, if we want it to stop in 5 seconds use this equation to find the acceleration:
acceleration = (Vf - Vi) / time
acceleration = (0rads/s - 2.09rads/s) / 5s
acceleration = -0.418 rads/s^2
NOTE: this is angular acceleration, usually denoted with the character "alpha", and velocity is actually 'angular velocity' with the character "omega". I used V to make it easier, as I cannot type an "omega" sign.


Then to find the torque we use the equation:
torque = (moment of intertia) * (angular acceleration)
but we don't know what the moment of inertia is, so have to calculate it, this is done with the equation: (moment of inertia is denoted with the capital letter i)
I = m1*r1^2 + m2*r2^2 + m3*r3^2 + .......
taking the center of mass (2/3m measured from the 1kg mass) to be r = 0:
I = 1kg * (-2/3m)^2 + 2kg * (1/3m)^2
I = 2/3 kg*m^2

Then using the equation state above (torque = I * a)

torque = 2/3kg*m^2 * -0.418rads/s^2

torque = -0.2787 N*m
The negative sign tells us that the torque opposes the direction of motion, which makes sense because it is supposed to stop the rotation.