Max Velocity of Real-Life Pendulum

[gup08]

Homework Statement

Video was recorded with a 240 fps camera. It took 93 frames for the pendulum to go from 90 deg to the bottom. The pendulum length is 51 cm.

Homework Equations

Find the max velocity

The Attempt at a Solution

From the simple pendulum equation, I should get around 11.36 km/hr. Given that its 51 cm, with a 90 deg angle.

If it takes 0.3875 seconds, based off that each frame has 0.004 seconds.. 0.0041667 sec * 93 = 0.3875 seconds. Then velocity is 51cm(difference in height)/ 0.3875seconds = 131.613 cm/s and when converted to km/hr it results in 4.738 km/hr!

Roughly half of the theoretical. Is it the height that I am not thinking correctly?

 

[Merlin3189]

Video was recorded with a 240 fps camera. It took 93 frames for the pendulum to go from 90 deg to the bottom. The pendulum length is 51 cm.
Find the max velocity

3. The Attempt at a Solution
From the simple pendulum equation, I should get around 11.36 km/hr. Given that its 51 cm, with a 90 deg angle.

If it takes 0.3875 seconds, based off that each frame has 0.004 seconds.. 0.0041667 sec * 93 = 0.3875 seconds. Then velocity is 51cm(difference in height)/ 0.3875seconds = 131.613 cm/s and when converted to km/hr it results in 4.738 km/hr!

Roughly half of the theoretical. Is it the height that I am not thinking correctly?

In your first calculation you use an energy formula to give the speed at the bottom of the swing (I think - it is not shown.)

But in your second calculation (4.738 km/hr) you take the total distance and total time. What speed does this give you?

 

[gup08]

I used the energy formula to get an approximation of what the value should be (roughly) given that its larger than a small angle approximation. That was from v = √2g * L * (1-cos(a)) where g=9.81, L=0.51m, a= 90deg

But in the second equation, I get 4.738 km/hr for the velocity. which is half of what I should be getting. I believe the distance is what's messing me. Would the distance be the arc length of the swing instead of the height difference?

 

[Merlin3189]

I believe the distance is what's messing me. Would the distance be the arc length of the swing instead of the height difference?

I must admit I hadn't thought about that, but I don't think it is the cause of the problem. You have taken a distance that the pendulum moves and divided it by the time taken. That gives you a speed, but not the maximum speed. Your pendulum starts at rest and accelerates, so it is moving faster and faster as it falls. The speed is varying, but your calculation gives only a single speed - the speed it would need to go at if it moved at constant speed, ie the average speed. (Actually the average speed if it just fell vertically, but changing the distance to an arc would still leave you with an average speed.)

I'm not sure what the question is getting at, as I'd just use the energy equation to work out the ideal max speed. But in that case you don't need the information about the camera. So I wonder if they want you to use that information to estimate frictional or air resistance effects? I haven't done that myself, so I'm not sure off hand what you could get. You know only the total time, the geometry, the position at start & end, gravity and the initial speed. What sort of resistance function you could estimate from that, I don't know.

Your own question about the discrepancy is simply the difference between the instantaneous speed at the bottom of the arc and the average speed over the arc. (Your 4.738 would be even smaller if you calculated correctly, but that's irrelevant.)

I'm going qrt now, but I'll see what I can work out tomorrow, if you're still not sorted.

 

[NascentOxygen]

It took 93 frames for the pendulum to go from 90 deg to the bottom. The pendulum length is 51 cm.

Maybe the pendulum swings farther than 90°, and all that was filmed was its travel through the last 90°. So perhaps at 90° it already has a velocity > 0. That might be something to consider?

 

[gup08]

I must admit I hadn't thought about that, but I don't think it is the cause of the problem. You have taken a distance that the pendulum moves and divided it by the time taken. That gives you a speed, but not the maximum speed. Your pendulum starts at rest and accelerates, so it is moving faster and faster as it falls. The speed is varying, but your calculation gives only a single speed - the speed it would need to go at if it moved at constant speed, ie the average speed. (Actually the average speed if it just fell vertically, but changing the distance to an arc would still leave you with an average speed.)

I'm not sure what the question is getting at, as I'd just use the energy equation to work out the ideal max speed. But in that case you don't need the information about the camera. So I wonder if they want you to use that information to estimate frictional or air resistance effects? I haven't done that myself, so I'm not sure off hand what you could get. You know only the total time, the geometry, the position at start & end, gravity and the initial speed. What sort of resistance function you could estimate from that, I don't know.

Your own question about the discrepancy is simply the difference between the instantaneous speed at the bottom of the arc and the average speed over the arc. (Your 4.738 would be even smaller if you calculated correctly, but that's irrelevant.)

I'm going qrt now, but I'll see what I can work out tomorrow, if you're still not sorted.

You make a good point, Merlin3189. So from a video footage, there's no way of obtaining the maximum velocity... at least nothing I can think of. It would just be the average speed. I do have the footage to watch.

The simple pendulum equation of v = √2g * L * (1-cos(a)) would not be an accurate representation, as it its an angle of 90 deg?

I thought this was going to be a easier problem than it actually is.

Maybe the pendulum swings farther than 90°, and all that was filmed was its travel through the last 90°. So perhaps at 90° it already has a velocity > 0. That might be something to consider?

Good thought, but the film starts at 90 degrees with a initial velocity of 0.

 

[NascentOxygen]

So from a video footage, there's no way of obtaining the maximum velocity.

Sure there is. The distance it travels between the last two frames will tell you its speed, thereabouts.

In practice, you could determine its speed between each of the 93 frames, plot these on a graph, and join all the dots with a smooth curve. This method irons out errors and imprecision, to give you an accurate answer. You'll then be able to compare ideal behaviour vs. practical result and try to explain the difference.

 

[gneill]

I notice that no one has inquired about what physical form the pendulum takes. Is it a simple compact mass on a light string, or an extended mass on a string, or a mass on a rod, or something else?

 

[gup08]

Given that I have the footage. I have found a software called Tracker (http://physlets.org/tracker/), which is free. It allows to track the movement of an object and like NascentOxygen mentioned, it will plot the time vs position of each plot. I have not finished yet but it has been a big help in getting the maximum velocity.

Gneill, the pendulum is a mass on a rod.

I can't thank you guys enough for the brainstorming. I will be more active around the forum!

 

[gneill]

Gneill, the pendulum is a mass on a rod.

That being the case, depending upon the specifics of the geometry and mass distribution the behavior may depart more from the classic pendulum formula than you might think. Take a few minutes and research (google) "Physical Pendulum".

 

[NascentOxygen]

Would the distance be the arc length of the swing instead of the height difference?

Yes, the length of the arc. Alternatively, if you can measure the angular movement, calculate speed using the known radius.