Question

How do you find the maximum speed of a pendulum?

Answer 1

We shall assume the pendulum is a basic pendulum with extremely modest oscillations in order to determine its speed at any location in space.

This approximation aids in the development of the formula for the force acting on the pendulum bob as a function of acceleration at any given time, which can then be used to calculate the pendulum's speed.

Step 1: Given that

Let's assume that the pendulum's bob has a mass of $m$. The gravitational constant is $g$ while the length of the string, or the pendulum, is $\mathrm{L}$. Consequently, for extremely tiny oscillation, we can state that:

$\Rightarrow \sin \left(\theta \right)\approx \theta$

Let's examine the various forces that the pendulum bob experiences after being started in motion using the following diagram:

As may be seen in this example, the perpendicular component of $mg$, that, $\mathrm{mgsin}\left(\mathrm{\theta }\right)$ is the restoring force on the pendulum.

Step 2: Use Newton's second law of motion

The pendulum's linear acceleration can be expressed as

$\Rightarrow \mathrm{a}=-\mathrm{gsin}\left(\mathrm{\theta }\right)\left(\mathrm{equation}1\right)$.

Additionally, because the bob is travelling along the circle's arc, its angular acceleration is determined by,

$\Rightarrow \mathrm{a}=\frac{d^2\mathrm{\theta }}{d{\mathrm{t}}^2}\left(\mathrm{equation}2\right)$

Which can also be written as,

$\Rightarrow \mathrm{a}=\frac{g}{L}\left(\mathrm{equation}3\right)$

Step 3: Calculate the net restoring force

After that, using $equations\left(1\right),\left(2\right),and\left(3\right)$we have

$\Rightarrow \frac{d^2\mathrm{\theta }}{d{\mathrm{t}}^2}=-\frac{\mathrm{g}}{\mathrm{L}}\sin \left(\mathrm{\theta }\right)$

Using the earlier estimate, we have

$$\begin{gathered} \Rightarrow \frac{d^2\mathrm{\theta }}{d{\mathrm{t}}^2}=-\frac{\mathrm{g}}{\mathrm{L}}\mathrm{\theta } \\ \Rightarrow \mathrm{m}\frac{d^2\mathrm{\theta }}{d{\mathrm{t}}^2}=-\mathrm{m}\frac{\mathrm{g}}{\mathrm{L}}\mathrm{\theta }\left(\mathrm{Multiply}\mathrm{both}\mathrm{side}\mathrm{by}\mathrm{m}\right) \\ \Rightarrow \mathrm{F}=-\mathrm{kx} \end{gathered}$$

Where$F$ is the net restoring force ad $K$ is a constant equal to $\frac{\mathrm{mg}}{L}$

Step 4: Calculate the speed of the pendulum

The pendulum is therefore showing a straightforward harmonic motion. As a result, the equation for the speed of a particle showing S.H.M. can be used to calculate the pendulum's speed. The procedure is as follows,

$\Rightarrow \mathrm{v}=\mathrm{A\omega cos}\left(\mathrm{\omega t}+\mathrm{\varphi }\right)$

where,$\mathrm{\omega }$ is the angular frequency of the motion, and $\mathrm{\varphi }$ is the initial phase of the particle.

Hence, by which we find the maximum speed of a pendulum.