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

1. Dec 10, 2016

### 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

2. Dec 10, 2016

### Staff: Mentor

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?

3. Dec 10, 2016

### LightningBolt226

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

4. Dec 10, 2016

### Staff: Mentor

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?

5. Dec 10, 2016

### LightningBolt226

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.

6. Dec 10, 2016

### Staff: Mentor

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: Dec 10, 2016
7. Dec 10, 2016

### LightningBolt226

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.

8. Dec 10, 2016

### Staff: Mentor

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?

9. Dec 10, 2016

### Staff: Mentor

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.