How does acceleration affect the period of a pendulum in an elevator?

In summary, the period of a simple pendulum is dependent on the gravitational force and the length of the pendulum. If the pendulum is mounted in an elevator that accelerates, the gravitational force acting on the pendulum will change, affecting the period. In the case of an elevator accelerating upwards at 5 m/s/s, the amplitude will decrease but the period will remain the same. If the elevator moves at a constant speed, the period will also remain the same. Accelerating downwards at 5 m/s/s or 9.81 m/s/s will have no effect on the period. However, in the case of free fall, the period will increase to infinity.
  • #1
ThatDude
33
0

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
Justify your answers.

My answer:

(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.

I don't know if I approached these questions correctly; can someone please help me out?
Thank you.
 
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  • #2
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?
 
  • #3
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)?
 
  • #4
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.
 
  • #5
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
 
  • #6
Well, to make it swing, there must be some sort of restoring force.
 
  • #7
ThatDude said:
Well, to make it swing, there must be some sort of restoring force.
Correct. What's the usual source of that restoring force?
 
  • #8
The gravitational force.
 
  • #9
ThatDude said:
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?
 
  • #10
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.
 
  • #11
ThatDude said:
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! :)
 
  • #12
ThatDude said:
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.
 

1. What is SHM in an accelerating frame?

SHM (Simple Harmonic Motion) in an accelerating frame refers to the motion of a body that is undergoing simple harmonic oscillation while also experiencing a constant acceleration. This type of motion can occur in various systems, such as a pendulum swinging on a moving object or a mass attached to a spring on a rotating platform.

2. What causes SHM in an accelerating frame?

SHM in an accelerating frame is caused by the combined effects of a restoring force and an external acceleration. The restoring force, such as gravity or tension in a spring, tries to bring the body back to its equilibrium position, while the external acceleration continuously changes the position of the body.

3. What is the equation for SHM in an accelerating frame?

The equation for SHM in an accelerating frame is given by x(t) = A sin(ωt + φ) + (1/2)at², where x(t) is the displacement of the body at time t, A is the amplitude of the motion, ω is the angular frequency, φ is the phase constant, and a is the external acceleration.

4. How does SHM in an accelerating frame differ from SHM in a stationary frame?

SHM in an accelerating frame differs from SHM in a stationary frame in that the acceleration of the frame must be taken into account in the equations of motion. In a stationary frame, the acceleration is assumed to be zero and the equations for SHM are simplified. Additionally, the amplitude and frequency of the motion may also change in an accelerating frame due to the external acceleration.

5. What are some real-life examples of SHM in an accelerating frame?

Some examples of SHM in an accelerating frame include a pendulum on a moving train, a mass attached to a spring on a rotating merry-go-round, and a mass hanging from a swinging crane. These systems all experience SHM while also being subjected to a constant acceleration due to the motion of the frame they are attached to.

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