Chapter 8, Problem 25P

To determine

To find: The magnitude and direction of the net torque.

Answer to Problem 25P

The magnitude of the torque is $mg\left(l_1-l_2\right)$ and the direction of the torque isclockwise.

Explanation of Solution

Given information:

Initially the rod is held in the horizontal position and then released. Calculatethe magnitude and direction of the net torque on this system when it is first released.

Formula used:

  $\begin{array}{l}{\tau }_{\text{counter-clock}}=mg{l}_1\\ {\tau }_{\text{clock}}=mg{l}_2\end{array}$

Calculation:

The free body diagram of the given scenario is shown in the below figure.

In the above figure m is the mass of each block, g is the acceleration due to gravity, l₁ and 1₂arethe lengths.

From the free body diagram, the weight of the right block provides the clockwise torque and the weight of the left block provides the counter-clockwise torque.

  $\begin{array}{l}{\tau }_{\text{counter-clock}}=mg{l}_1\\ {\tau }_{\text{clock}}=mg{l}_2\end{array}$

The torque ${\tau }_{net}$ is in general expressed as the product of force and the perpendicular distance ofthat force.

Consider the counter-clockwise force as positive and clockwise force as positive.

  ${\tau }_{net}={\tau }_{counter-clock}-{\tau }_{clock}$

Rearranging the above equation as shown below,

  $\begin{array}{l}{\tau }_{net}=mg{l}_1-mg{l}_2\\ {\tau }_{net}=mg\left(l_1-l_2\right)\end{array}$

From the figure, it is clear that the length $l_1<l_2$ . So, the term $l_1-l_2$ in the torque equation wouldbe negative. This in term results in negative net torque. As per our consideration, if the torque isnegative, the direction of net torque must be clockwise.

Therefore, the magnitude of the torque is $mg\left(l_1-l_2\right)$ and the direction of the torque isclockwise.

Note: The magnitude of the torque can also be considered as $mg\left(l_2-l_1\right)$ depending upon theconsiderations of sign conventions for the clockwise and anti-clockwise torques.

Conclusion:

The magnitude of the torque is $mg\left(l_1-l_2\right)$ and the direction of the torque isclockwise.