What is the horizontal displacement L of a point mass m hung on an ideal rod of length z, when a torque of magnitude r is applied?

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To find the horizontal displacement ( L ) of the point mass ( m ) hung on an ideal rod of length ( z ) when a torque of magnitude ( r ) is applied, we can use the relationship between torque, force, and distance.

The torque ( \tau ) created by a force ( F ) acting at a distance ( z ) from the pivot is given by:

[

\tau = F \cdot z

]

In this situation, the force acting on the mass ( m ) due to gravity is ( F = mg ), where ( g ) is the acceleration due to gravity. Therefore, the torque due to the weight of the mass can be expressed as:

[

\tau = mg \cdot z

]

Now, we also have the applied torque ( r ). The equilibrium of torques gives us the relationship:

[

r = mg \cdot L

]

Here, ( L ) is the horizontal displacement from the pivot point to where the force acts. Rearranging this equation to solve for ( L ), we find:

[

L = \frac{r}{mg}

]

This indicates that the horizontal displacement depends