Spool Critical Angle
Unit: Torque and Rotational Dynamics
Prerequisites
Later Topics
Multi-Step Problem Preview
Part 1 of 5 — sign up to solve the full problem!
Part 1
A spool of mass has an inner hub of radius and an outer rim of radius . It rests on a rough horizontal surface where the coefficient of static friction is . A string is wrapped around the inner hub (underneath) and pulled with a force at some angle with respect to the horizontal.
When the string is pulled horizontally (), the tension exerts a counter-clockwise torque about the center of the spool. However, the spool is observed to roll to the right (clockwise rotation).
Which force is responsible for producing the clockwise torque about the center that causes this motion?
Correct!
The spool experiences gravity acting downward at the center, the applied tension pulling horizontally to the right at radius (under the hub), the normal force from the ground, and static friction at the contact point.
Compute torques about the center of the spool.
The tension acts at a perpendicular lever arm , producing a counter-clockwise torque of magnitude .
Gravity acts through the center, so its torque about the center is .
The normal force acts at the contact point directly below the center, so its line of action passes through the center and its torque about the center is also .
Static friction acts at the contact point, a distance from the center. If the spool rolls to the right with clockwise rotation, the friction force must act to the left at the bottom. A leftward force applied at the bottom produces a clockwise torque of magnitude about the center.
Therefore, the force responsible for producing the clockwise torque about the center is static friction.
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