Question

A child swings on a 14 m rope. The highest point of her swing is 3 m above the ground. When the child is at the lowest point, 1 m above the ground, what is her speed?

Answer 1

This is the case of a simple pendulum.

A pendulum is a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position.

For small swings the pendulum approximates a harmonic oscillator in that the equation for the angular displacement is

For real pendulums, corrections to the period may be needed to take into account the buoyancy and viscous resistance of the air, the mass of the string or rod, the size and shape of the bob and how it is attached to the string, flexibility and stretching of the string, and motion of the support

in this case according to the figure the point A is the highest point of child swing (3m above the ground)

the point C is the lowest (1 m above the ground)

the max angle is find by

So the equation is

the graphic of this angular displacement is

to know the angular velocity in t=0 (when the angle is zero; the lowest point) we have to derivate the function

and evaluate in t=0

w=0.452 rad/seg.

the speed is the tangential velocity and it is calculated

solution 6.328 m/s