In this problem we have a set of washers on a string spinning around in a circle with radius of .25 meters and tangential velocity of 2 meters per second the washer have a mass of .06 kilograms. We pull the string through the end of a pen which reduces the spin radius to one half or .125 meters. What will the velocity be at the second condition? Will it increase decrease or stay the same?
To solve this problem we will need to know a few equations. The first is angular velocity equals tangential velocity divided by the radius of the path. This is simply a measure of the angular speed in radians per second. Or how fast the object is moving around the center in angular units of radians.
The second equation is that the moment of inertia is equal to the radius squared multiplied times the mass. The moment of inertia is simply the resistance that must be overcome to get the object to change in angular velocity.
This is derived from the kinetic energy formula. Linear kinetic energy equation and rotational kinetic energy are similar in that it is simply one half mass times velocity squared. In the case of the rotation kinetic energy we substitute radius times angular velocity where velocity is this is due to the angular velocity being equal to velocity over r. So if we multiply the angular velocity times radius we are left with the tangential velocity
After taking the square of that we are left with ½ mass times radius squared times angular velocity squared equals kinetic energy.
Now in both equations there is something that is the resistance for the object to change velocity. In the case of the linear example this is the mass of the object. In the case of the rotational example it is the moment of inertia which is the mass times radius squared.
So now we will also need to know the angular momentum formula. The angular momentum is equal to the angular velocity times the moment of inertia.
So like linear momentum which is mass or resistance to change velocity times velocity. angular momentum is angular velocity or how fast the object is rotating in radians per second times the resistance to rotating which is the moment of inertia.
Simplifying this we are left with velocity times radius to center of rotation path times mass being equal to the angular momentum.
So we assume that at condition 1 and condition 2 the momentum is conserved. This means we can set the equations equal to one another.
Looking at the 2 equations set equal to one another. We notice that inorder to keep the equations equal to one another if the radius is decreased like in the condition 2 scenario the velocity must increase to remain equal.
You can experience this phenomenon for youself if you have a desk chair that spins. Take some weight and put it as far away from you center of rotation as possible. Start spinning the chair with your weight spread out. Now once you get going bring the weights closer to your body. We have reduced the average radius of the weight distribution which resulted in an increase in velocity. Warning use proper safety precautions.
Now reranging the formula to get velocity 2 to one side
Then simplifying the problem we are left with velocity 1 time radius 1 over radius 2 being equal to velocity 2.
Plugging in our givens we get that the velocity at point 2 is 4 meters per second. So if the radius of the rotation of the washers is reduced reduced by half the velocity is doubled.
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