Question

A pendulum is made from a thin cylindrical rod pivoted at the end. It has a length of 0.30 m and radius 0.001 m. It swings up to a max angle. Measured from vertical. The angular speed at the bottom of its swing is 2.9 rad/s. What is the max angle in radians?

Answer 1

Answer:

Maximum angle is $23.91^{\circ}$ rad

Explanation:

As per the question:

length of the pendulum, L = 0.30 m

Radius of the pendulum, R = 0.001 m

Angular speed at the bottom, $\omega = 2.9\ rad/s$

Now,

To calculate the maximum angle, $\theta_{m}$:

For the pendulum, the moment of inertia, I = $\frac{ML^{2}}{3}$

Now, using the principle of the conservation of energy:

Kinetic energy = Potential energy

$\frac{1}{2}\times I\omega^{2} = mgh$

where

h = $\frac{L}{2}(1 - cos\theta_{m})$

Thus

$\frac{1}{2}\times \frac{mL^{2}}{3}\times \omega^{2} = m\times 9.8\times \frac{L}{2}(1 - cos\theta_{m})$

$1 - cos\theta_{m} = \frac{0.3\times 2.9^{2}}{3\times 9.8}$

$theta_{m} = cos^{- 1}(0.914) = 23.91^{\circ}$