L07.  Maximum Speed of a Pendulum Bob

About your lab report:  For your final report, include title and goal, your method, data, analysis, and conclusion in sections of the same name. 

Goal

Measure the speed of a pendulum bob at its lowest point and compare to the value predicted by using conservation of energy.

Prelab: Theory

Do the following in advance of your lab period.

  1. Read sections 8.1-3 of your text.
  2. Read this page completely.
  3. Study the Guide to Solving Conservation of Energy Problems, Part I.
  4. Complete the Chapter 8 assignments through E.8.3 and P11.

Introduction

An ideal, simple pendulum consists of a point mass suspended from a massless string.  The bob swings in a circular arc under the influence of gravity.  In the real world, the string has mass and the bob isn't a point.  However, it's easy to approach the ideal by using a lightweight string, and a small, dense bob.  In this context, small means that the size of the bob is small compared to the length of the string.  Dense means that the bob has a large mass to volume ratio similar to that of, say, a metal or even wood.  (A wad of paper wouldn't be a good choice.)  With choices such as these and a firm support for the string, the pendulum becomes a good device for studying conservation of energy, because non-conservative forces such as air friction and friction in the support do not play a large part over the course of a cycle of the motion.

The experimental situation that you'll be investigating is shown in Figure 1 below.  The string of length L is pulled to an angle of q with the vertical and released from rest at point A.  The goal is to determine the speed of the bob at point B using both theoretical and experimental methods.  The acceleration of the bob isn't constant, so one can't, in principle, use dvat equations.  Let's see why.

Figure 1 Figure 2 Figure 3

In Figure 2, we've drawn the forces acting on the bob when in position A. Tension acts toward the center of the circle and weight acts vertically downward. Unlike previous problems we've had, there are two acceleration components. There's a component toward the center of the circle, but there's also a component tangent to the circular path. This latter component causes the bob to speed up if it's headed down or causes the bob to slow down if it's headed up.

In order to write net force equations, we select axes parallel and perpendicular to the bob's velocity vector. In Figure 3, we've resolved the weight force into components along the axes. The net force equations are the following.

Applying Newton's 2nd law to the Fnet,x equation yields ax.

This shows that the acceleration of the bob tangent to the path is non-uniform. The acceleration depends on the angle q, which is changing. This is the reason we can't apply dvat equations to the solution of the problem of finding the speed of the bob at the lowest point. The dvat equations assume uniform acceleration.

This leaves conservation of energy as the method of choice for the problem. For conservative forces, of which gravity is one, we need know only the initial (A) and final (B) states of the pendulum. We don't need to know the path followed by the bob, because the work done by gravity is independent of the path, and the work done by tension is 0. We're leaving it to you to solve the conservation of energy problem. (See P11.)

Next, we'll look at ways to measure vB experimentally.

Theoretically, you need to find the instantaneous velocity at the bottom of the swing.  In practice, you'll have to settle for an approximation.  Consider the diagram to the right.  If you measured the time, Dt, for the ball to travel from P to Q (equally-spaced on either side of the vertical), then an approximation to the speed at the lowest point would be:  vave = Dx/Dt, where Dx is the straight-line distance from P to Q.  The closer together P and Q are, the better your approximation would be, in principle.  We have to add the phrase, in principle,  because in practice the uncertainty increases as the time interval decreases.  Suppose, for example, that Dt is 0.4 s with an uncertainty of 0.1 s in starting and stopping a stopwatch.  That's a 25% uncertainty.  If a smaller Dx is used so that Dt is, say, 0.2 s, then the percentage uncertainty increases to 50%.  So there are two things working against each other here.  Theoretically, vave approaches the instantaneous velocity, vB, as Dx decreases.  But experimentally, the measurement becomes highly uncertain.  This provides a challenging problem in experimental design.  Here are two approaches to dealing with it.

  1. Use a timing device that minimizes starting and stopping error.  A photogate is such a device.  If placed at the bottom of the pendulum's swing, the photogate starts a timer when the bob enters it and stops the timer when the bob leaves.  With such a device, timing uncertainties can be reduced to thousandths of a second.  Moreover, the distance Dx over which the timing occurs is limited to the width of the bob.  Very accurate results can be expected with this method.  If you have access to a photogate, then you can use it for the measurement.  Here are some tips for good results:  a)  Make sure the photogate is positioned at the lowest point of the swing, b) Make sure you know which points of the bob are triggering the photogate on and off.  You have to know this, because that gives you your distance, Dx.  If the bob is say, a cube, then Dx will depend on how the cube is turned as it passes through the gate. You'll need to come up with a reliable way to deal with that.

  2. Design the pendulum in such a way that the time for the pendulum bob to move from P to Q is as great as you can make it without having P and Q too far apart.  This is actually simple to do.  Just make the string as long as you can.  The pendulum swings slower when the string is longer.  If you can make the string extend from the ceiling to as close to the floor as possible without grazing it, then that's what you should do.  Something else is working in your favor, too.  The speed of the bob changes much less in moving from P to Q than it does in moving from, say, A to P.

Equipment and setup

If you use Method I, then we'll leave it to you to figure out how to set up your photogate.

  • about 3 meters of light string
  • small, dense object for the pendulum bob.  You have to be able to tie the string to it.
  • meter stick or tape measure
  • stopwatch or photogate
  • some means to hang the pendulum, the higher the better
  • 2-3 ring stands.  You can use these as markers.  Two of them can mark the positions P and Q (for method II) and a third can mark the position of A so that you always drop the bob from the same point.  If you don't have ring stands, a strategically placed table may work.  See the diagram to the right.  The table legs can be the markers for P and Q.  You would need a third marker for point A.

Timing for Method II

You'll need to find a way to release the bob at point A, start the stopwatch as the bob reaches point P, and stop the watch as the bob reaches point Q.  Practice this several times before recording measurements, because you'll need to learn to anticipate.  This can actually be done with good reproducibility.  Perhaps your facilitator or another student can help by releasing the bob while you time.  In the event that you're working with another student in the course, you're each expected to obtain your own set of measurements.

Method and Diagram

In your lab report, describe your set up and method in sufficient detail that the teacher, who can't see you work, could repeat your method.  Include a labeled diagram at least half a page in size.  You may leave this part of the report to write later if you have limited time to take the data.  You may also discover while taking data that you want to make alterations to your procedure.

Data

No matter what method you're using, you'll need to measure some distances.  Review the introduction as needed to remind yourself what distances need to be measured.  If you use Method II, distances measured to the nearest millimeter are sufficient.  If you use Method I, measure the width of the bob as precisely and accurately as you can.

For either method, take ten time trials and find the deviations as you did in L05.  Here's a typical data table.  You will, of course, need to add 5 rows.

Trial Time
(s)		 Deviation
(s)
1    
2    
3    
4    
5    
Means:    
 

%:

  

Analysis and Interpretation

As always, start with equations in symbols before substituting numerical values and units.

  1. Using your mean time, calculate vave.  This is your experimental value of vB.
  2. Use the theory that you developed in P11 to calculate your theoretical value of vB.  If you need to measure the length of the pendulum, measure from the point of support to the center of the bob.
  3. Find the percentage difference between your theoretical and experimental values of vB.
  4. Since timing is the primary source of error, at least in Method II, compare the percentage difference from step 9 with the percentage mean deviation in your time trials.  Can timing account for the error?
  5. Is your experimental value less than your theoretical value, as might be expected from item 6 in the Introduction?  If it's not less, what might be the reason? (Note added 11-5-07: The missing question 6 refers back to the Introduction. The question was: Give an argument why one would expect vave to be less than vB.)
  6. For this item, refer to the introduction to this lab. The statement was made that the velocity didn't change much from P to Q. This is a reason that the approximation of vB with vave is a good one. Use the physics of the situation to explain why doesn't the velocity change much between P and Q.
  7. Give an argument why one would expect vave to be less than vB.

Conclusion

Summarize what you did and what you found out.

Submitting your work

You may fax your hand-written report or send a file through the digital dropbox. If you send a file, email the teacher to alert him that you're doing so. Name the file in the usual way.