How to Calculate Equivalent Torque Distance on a Pivoting Rod?

[PrideofPhilly]

Homework Statement

A rod can pivot at one end and is free to rotate without friction about a vertical axis. A force F is applied at the other end, at an angle theta to the rod. If F were to be applied perpendicular to the rod, at what distance d from the axis of rotation should it be applied in order to produce the same torque t?

Homework Equations

t = rF

The Attempt at a Solution

My answer: d = Lsin(theta).

Is that correct or am I missing something?

 

[LowlyPion]

If the force is applied horizontally this would be true.

When the angle is 90 degrees the force would be fully engaged in developing torque at that displacement.

 

[nlightner]

Your answer is correct. Torque is defined as the product of the force and the perpendicular distance from the axis of rotation to the point where the force is applied. In this scenario, the perpendicular distance is given by d = Lsin(theta), where L is the length of the rod. This means that applying the force at this distance will produce the same torque as if the force were applied perpendicular to the rod. Great job!