Chapter 11, Problem 34P

To determine

To Derive: The formula for maximum speed of pendulum.

Answer to Problem 34P

The formula for maximum speed of a simple pendulum bob in relation with gravity, length and angle is $v_{\max }=\sqrt{2\times g\times L\times \left(1-\cos \theta \right)}$ .

Explanation of Solution

Given info:

Gravity = $g$

Length = $l$

Angle of swing= $\theta$

Formula used:

The expression for law of conservation of energy.

  $E_{\text{top}}=E_{\text{bottom}}$

This can be written as:

  $K{E}_{\text{top}}+U_{\text{bottom}}=K{E}_{\text{top}}+U_{\text{bottom}}$

Here, $KE$ defines the kinetic energy and $U$ defines the potential energy.

Calculation:

Relate the energy conservation principle to the potential energy at the maximum height of the pendulum to the kinetic energy at the lowest point of the swing. Relating the energy at top kinetic energy is zero and at bottom potential energy is zero is obtained.

  $K{E}_{\text{top}}+U_{\text{bottom}}=K{E}_{\text{top}}+U_{\text{bottom}}$

Substitute the values in the above expressions.

  $\begin{array}{l}0+m\times g\times h=\frac{1}{2}\times m\times v_{\max }{}^2+0\\ v_{\max }{}^2=2\times g\times h\\ v_{\max }=\sqrt{2\times g\times h}\\ \text{but }h=L-L\cos \theta \\ v_{\max }=\sqrt{2\times g\times (L-L\cos \theta )}\\ v_{\max }=\sqrt{2\times g\times L(1-\cos \theta )}\end{array}$

Conclusion:

Thus, the formula for maximum speed of a simple pendulum bob in relation with gravity, length and angle is $v_{\max }=\sqrt{2\times g\times L\times \left(1-\cos \theta \right)}$ .