Question

Given the frequency of a pendulum and its maximum angle, how do you calculate the speed of the ball in the pendulum?

a) Use the formula v = √(2 * g * L * (1 - cos(θ)))
b) Use the formula v = g * L * sin(θ)
c) Use the formula v = g * L * cos(θ)
d) Use the formula v = √(2 * g * L * cos(θ))

Answer 1

To calculate the speed of a pendulum's bob, use the formula v = √(2 * g * L * (1 - cos(θ))), where g is the acceleration due to gravity, L is the length of the pendulum, and θ is the angle from the vertical.

Step-by-step explanation:

To calculate the speed of the ball in a pendulum when it is released from a certain angle, we use the principle of conservation of mechanical energy. The potential energy at the maximum angle is converted into kinetic energy at the lowest point. The correct formula to use in this case would be:

v = √(2 * g * L * (1 - cos(θ)))

Here, v is the speed of the pendulum's bob, g is the acceleration due to gravity, L is the length of the pendulum, and θ is the maximum angle with respect to the vertical.

Option (a) gives the correct formula to find the velocity of the pendulum's bob at different positions in its arc:

1. Vertically down: θ = 0, so the speed is the maximum due to the entire potential energy being converted to kinetic energy.
2. Angle of 20° with the vertical: Plug in θ = 20° into the formula to find the speed at that angle.
3. Angle of 10° with the vertical: Use θ = 10° in the formula for the corresponding speed.