Two identical tops spin with angular velocities 45π rad/s up and 19π rad/s down, respectively, about vertical axes on a table. The tops bump into one another and separate. After the collision, one of the tops has an angular velocity of 35π rad/s in its original direction.
What is the angular velocity of the other top if the 45π rad/s top ends up with the angular velocity of 35π rad/s? (Answer: -28.3rad/s)
What is the angular velocity of the other top if the 19π rad/s top ends up with the angular velocity of 35π rad/s? (Answer: 192rad/s)
Answers are provided above. Please show all work on how to get the answers.
45π - 19π - 35π = -28.3rad/s
45π - 19π + 35π = 192radrad/s
To solve both problems, we can apply the conservation of angular momentum. The total angular momentum before the collision is equal to the total angular momentum after the collision.
Let's consider the angular momentum of each top before the collision:
1. For the first top spinning up with an angular velocity of 45π rad/s:
Angular momentum 1 = Moment of inertia * Angular velocity 1
= (I1 * 45π)
2. For the second top spinning down with an angular velocity of 19π rad/s:
Angular momentum 2 = Moment of inertia * Angular velocity 2
= (I2 * -19π) (negative sign due to opposite direction)
Now, let's calculate the angular momentum of each top after the collision:
1. For the first top spinning with an angular velocity of 35π rad/s:
Angular momentum 1' = Moment of inertia * Angular velocity 1'
= (I1 * 35π)
2. For the second top spinning with an unknown angular velocity:
Angular momentum 2' = Moment of inertia * Angular velocity 2'
= (I2 * Angular velocity 2')
Since the total angular momentum before and after the collision remains the same, we can equate the two values:
Angular momentum 1 + Angular momentum 2 = Angular momentum 1' + Angular momentum 2'
(I1 * 45π) + (I2 * -19π) = (I1 * 35π) + (I2 * Angular velocity 2')
Now, let's solve for the unknown angular velocity in each case:
1. When the 45π rad/s top ends up with an angular velocity of 35π rad/s:
(I1 * 45π) + (I2 * -19π) = (I1 * 35π) + (I2 * Angular velocity 2')
Rearrange the equation to solve for Angular velocity 2':
(I2 * Angular velocity 2') = (I1 * 45π) + (I2 * -19π) - (I1 * 35π)
(I2 * Angular velocity 2') = (I1 * 10π) - (I2 * 19π)
Divide both sides by I2:
Angular velocity 2' = (I1 * 10π - I2 * 19π) / I2
Since the tops are identical, I1 = I2:
Angular velocity 2' = (I2 * 10π - I2 * 19π) / I2
Angular velocity 2' = (10π - 19π)
Angular velocity 2' = -9π
Therefore, the angular velocity of the other top is -9π rad/s, or approximately -28.27 rad/s.
2. When the 19π rad/s top ends up with an angular velocity of 35π rad/s:
(I1 * 45π) + (I2 * -19π) = (I1 * 35π) + (I2 * Angular velocity 2')
Rearrange the equation to solve for Angular velocity 2':
(I2 * Angular velocity 2') = (I1 * 45π) + (I2 * -19π) - (I1 * 35π)
(I2 * Angular velocity 2') = (I1 * 10π) - (I2 * 19π)
Divide both sides by I2:
Angular velocity 2' = (I1 * 10π - I2 * 19π) / I2
Since the tops are identical, I1 = I2:
Angular velocity 2' = (I2 * 10π - I2 * 19π) / I2
Angular velocity 2' = (10π - 19π)
Angular velocity 2' = -9π
Therefore, the angular velocity of the other top is -9π rad/s, or approximately -28.27 rad/s.
In both cases, the angular velocity of the other top after the collision is approximately -28.27 rad/s.