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

Maximum angle is
`23.91^(\circ)` rad

Step-by-step 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 =
`(ML^(2))/(3)`

Now, using the principle of the conservation of energy:

Kinetic energy = Potential energy

`(1)/(2)* I\omega^(2) = mgh`

where

h =
`(L)/(2)(1 - cos\theta_(m))`

Thus

`(1)/(2)* (mL^(2))/(3)* \omega^(2) = m* 9.8* (L)/(2)(1 - cos\theta_(m))`

`1 - cos\theta_(m) = (0.3* 2.9^(2))/(3* 9.8)`

`theta_(m) = cos^(- 1)(0.914) = 23.91^(\circ)`