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Phy 121
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
angular and linear vs. radius
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Working on a problem for a ball on a string moving in a circle. I want to verify the constants as the string length is shortened. Said another way, for that ball on a string as the radius is shortened what stays the same? Angular velocity and accel?
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For the ball on string of different radii,
Is it true that angular velocity and accel are independent of radius?
Is it true that linear velocity and accel do change with radius?
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If there is no net torque, angular momentum is conserved.
Assuming that the string is shortened by pulling it toward the center of the circular path, there is no component of the tension perpendicular to the string so there is zero torque. The angular momentum will therefore remain constant.
Angular momentum is equal to angular velocity multiplied by moment of inertia.
The moment of inertia of the ball is m r^2. So it's the product
omega * m r^2
that remains constant.
Thus
omega_2 * m r_2^2 = omega_1 * m r_1^2.
A simple rearrangement (divide both sides by m, divice both sides by omega_1, divide both sides by r_2) yields
omega_2 / omega_1 = r_1^2 / r_2^2,
which can also be written
omega_2 / omega_1 = (r_1 / r_2)^2.
That is, for the given conditions the angular velocity is inversely proportional to the radius.
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