# How high does a pendulum go after pushing at equilibrium?

• LightningBolt226
In summary, the force applied to the pendulum is constant over a small angle, and so the pendulum bob moves in a straight line. The average speed and distance traveled is given by v=at, and the energy is given by W=Fd.
LightningBolt226
Usually, the pendulum problems I encounter relate to initial velocity. What happens if, at equilibrium, I push a pendulum with a certain amount of force? (E.g. 10N) Is there a way to calculate how high the pendulum will go? I guess it's complicated considering torque done by gravity, etc

LightningBolt226 said:
Usually, the pendulum problems I encounter relate to initial velocity. What happens if, at equilibrium, I push a pendulum with a certain amount of force? (E.g. 10N) Is there a way to calculate how high the pendulum will go? I guess it's complicated considering torque done by gravity, etc
Integrate the force through the position displacement -- that will give you an energy increase, which you can translate into a higher zero velocity pause position. Makes sense?

I'm not exactly sure what you mean. What are the limits of integration for this case?

LightningBolt226 said:
I'm not exactly sure what you mean. What are the limits of integration for this case?
It's a path integral, from the start of the push to the end of the push. How is the push force versus position defined in this problem? Is the push force constant over some arc length? Where on the pendulum is the push force applied?

Sorry, I forgot to mention earlier. Suppose that it was an instantaneous force that was applied. Also, suppose that the pendulum is a point mass and is where the force is applied.

Actually, this may be a more complicated situation than just a path integral. If the force is applied for more than a very small angle near the bottom of the pendulum's travel, it may require a free body diagram of the forces on the pendulum mass during its travel. I'll flag this thread for the PF Science Advisers to have a look at...

Last edited:
Thanks. I realize it might be complicated so I was wondering first about the situation wherein the application of the force was hypothetically instantaneous at the equilibrium position.

LightningBolt226 said:
Sorry, I forgot to mention earlier. Suppose that it was an instantaneous force that was applied. Also, suppose that the pendulum is a point mass and is where the force is applied.
That simplifies the problem a lot. But there is no such think as an instantaneous force. It can be applied over a small angle of the pendulum's travel, which makes the force*distance = change in energy approach a lot easier. Can you show us that math?

LightningBolt226 said:
Sorry, I forgot to mention earlier. Suppose that it was an instantaneous force that was applied. Also, suppose that the pendulum is a point mass and is where the force is applied.
There is no such thing as an instaneous application of force. It may not be very long, but there's always some non-zero time between the moment when we start applying the force and the moment when we stop.

If that time is very short, then it's a good approximation to say that the pendulum bob moves in a straight line and then: ##F=ma## gives us the acceleration; ##v=at## gives us the final speed; ##v_{ave}=v/2## is the average speed; ##d=tv_{ave}## is the distance the force is applied; and now we can get the energy from ##W=Fd##.

Also, you will want to google for "impulse momentum" to see another standard way of handling problems in which a force is applied for a very short time.

berkeman

## 1. What is a pendulum?

A pendulum is a weight suspended from a pivot so that it can swing freely. It is an example of a simple harmonic oscillator, meaning its motion follows a predictable pattern.

## 2. What is meant by pushing at equilibrium?

Pushing at equilibrium means giving the pendulum a small initial displacement, then releasing it. This causes the pendulum to swing back and forth with a constant period and amplitude.

## 3. How high does a pendulum go after pushing at equilibrium?

The height of a pendulum after being pushed at equilibrium depends on several factors, including the length of the pendulum, the initial displacement, and the force of gravity. However, in a perfect scenario with no friction or air resistance, the pendulum will continue swinging back and forth at the same height indefinitely.

## 4. How does the length of a pendulum affect its swing height?

The length of a pendulum affects its swing height because it determines the period of the pendulum's motion. The longer the pendulum, the longer its period, meaning it will take longer for the pendulum to complete one full swing. This also means that the pendulum will reach a higher height during its swing.

## 5. Can the height of a pendulum be increased by pushing it at different times during its swing?

No, the height of a pendulum cannot be increased by pushing it at different times during its swing. The pendulum's motion is driven by gravitational force, so regardless of when it is pushed, it will follow the same pattern and reach the same height at each swing. Pushing it at different times will only change the phase of the pendulum's motion, not its overall height.

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