# SHM in an accelerating frame

## Homework Statement

The question is:

A simple pendulum is mounted in an elevator. What happens
to the period of the pendulum (does it increase, decrease, or
remain the same) if the elevator
(a) accelerates upward at 5 m/s/s
(b) moves upward at a steady 5 m/s
(c) accelerates downward at 5 m/s/s
(d) accelerates downward at 9.81 m/s/s

(a) If the elevator accelerates upward, there is a greater upward force, therefore the amplitude will decrease. However, the amplitude and period are independent of each other, therefore, the period shall remain the same.

(b) At constant speed, the period is the same

(c) If it accelerated downward, lesser upward force, therefore the amplitude will decrease --> period is still the same.

(d) No effect on the period for the same reasons as mentioned above.

Thank you.

gneill
Mentor
Can you write the equation for the period of a simple pendulum? What variables does it depend upon? Can any of them be affected during an elevator ride?

Nathanael
Homework Helper
Situation (d) where you are accelerating downwards at 9.81 m/s/s, this is just free fall, right?

So if I dropped a pendulum and let it freely fall, are you saying it would oscillate like normal (while it's falling)?

Ok. For some reason I was thinking this was a vertical spring-block system in an elevator!

@gneill , the equation for the period of a simple pendulum is T = 2pi (L/g). The gravitational force is constant throughout as well as the length of the spring, so I don't think that amplitude would be affected.

@Nathanael , no, it would just fall the same way an apple would fall from a tree.

Well what if you moved the pendulum to deep outer space? Would it swing? Can this environment be replicated?

Be careful to trust equation merely on their variables and learn to understand how they are formed/derived

Well, to make it swing, there must be some sort of restoring force.

gneill
Mentor
Well, to make it swing, there must be some sort of restoring force.
Correct. What's the usual source of that restoring force?

The gravitational force.

gneill
Mentor
The gravitational force.
Right. So some a force Mg acts on the bob, and some component of that resolves into the restoring force. What happens if the pendulum is being accelerated upwards or downwards?

That force mg changes. If it is being accelerated upward, from the frame of the pendulum, the acceleration is greater. If it is being accelerated downward, then that force would be less. The greater the acceleration, the less the period.

gneill
Mentor
That force mg changes. If it is being accelerated upward, from the frame of the pendulum, the acceleration is greater. If it is being accelerated downward, then that force would be less. The greater the acceleration, the less the period.
Bingo! :)

Nathanael
Homework Helper
The greater the acceleration, the less the period.
Just to drill the point a bit more; it depends on what direction it is being accelerated in.
In the free fall case (acceleration = 9.81 m/s/s downwards) the period increases to "infinity." But if the acceleration were 9.81 m/s/s upwards the period would be decreased.