Notes

• Words in blue are links to additional reading or videos.

• Text in gray boxes are hints and things to take note of.

• Text in red boxes are important instructions or prompts that guide you to DISCUSS some of the key RESULTS and CONCEPTS learned in the lab. These prompts may not be the only items that need to be included in your report.

1    Objectives

The goals of this lab are as follows:

• to explore the relationship between the period T and length L of a swinging pendulum

• to determine g, the acceleration due to gravity at the Earth’s surface

• to determine if and how the moment of inertia of an object affects how it will swing as the mass in a pendulum

2    Introduction

A simple pendulum consists of a compact object of mass m suspended from a fixed point by a string of length L, as shown in Figure 1.  You may have explored the motion of a pendulum in previous courses, but for this lab we are going to be talking about the motion in terms of angular motion.

The Torque (r=r × F) on the pivot point of the pendulum is

|r | = L * (-mg) * sin(9 )

We also know that r is equal to the moment of inertia of the object (I) times the angular acceleration (a)

|r| = I|a|

Which direction does angular the angular acceleration vector point in Figure 1 if the ball is moving up and to the right?

Figure 1: Two-dimensional (plane) trajectory of the simple pendulum

Together this gets us the equation

I|a| = -Lmgsin(9)

If we only move the mass by a few degrees (<15О ) we can use the small angle approxima- tion sin(9) = 9 to simplify our equation.

a = 9

While it might not be obvious now, there is a solution to this equation - but it involves some extra calculus - for now you will have to believe that the solution to this equation gives us an equation for 9 in terms of time where

9 = 90 cos(ot)                                                 (1)

where 9 is the initial angle of the pendulum before it is set in motion, 90 is the displaced

angle on the opposite side, and                              

o = ′

For this experiment, the string is assumed to be massless, and does not contribute to the moment of inertia of the pendulum. What we have is essentially a mass (m) rotating about a distance L from the center of rotation. The moment of inertia of our simple pendulum is then I = mL2 and equation 2becomes

o = ′                                                      (3)

[In this experiment, we don’t record 9 as we’re using a rangefinder, so we record the

distance displaced by the small angle down at the level of the rangefinder.]

Lets think about how one might change the period of oscillation from equation 3.

2T

=

1. How would the period T of the pendulum change if the length L were doubled and everything else remained the same?

2. How would T change if the mass m were doubled and everything else remained the same?

3. How would T change if the gravitational acceleration g were less; for example, at the surface of the Moon instead of the Earth?

This would be a good time to double check that your calculator is in radians for this lab!

3    Procedure

The pendulum apparatus consists of:

• a vertical post and fixed arm from which the pendulum bob, an aluminum ball of diameter d, is suspended by a light nylon string of negligible mass (m ≈ 0);

• a sliding arm used to adjust the pendulum swing length. If the arm is properly cal- ibrated to the scale on the pendulum post, so that the scale reads ‘0’ when the arm touches the top of the ball, then the scale can be used to directly measure, or set, this length;

• a scale that, when calibrated to the sliding arm, displays the length of the string s from the bottom of the sliding arm, the pivot point of the pendulum, to the top of

the ball, to a precision of one millimetre (mm).

This length s is not exactly equal to the pendulum length L; the pendulum length is mea- sured from the top of the string (at the fixed point) to the centre of the ball. Therefore, L is the sum of the length of the string s and half the ball’s diameter d:

L = s ＋d ●             (4)

Set the scale to measure the pendulum string length:

Figure 2: A graphical description of the pendulum calibration steps

1. Loosen the clamping nut to release the string and lower the ball to the table.

2. Align the bottom of the sliding arm, labelled Index, with the zero mark on the scale.

3. Adjust the string length so that the top of the ball just contacts the bottom of the arm, ensuring that the string is not stretched.

4. Gently tighten the string under the clamping nut. Do not just wrap the string around the nut; it will slip and result in length measurements that are incorrect.

4    The experimental method.

As shown in this short video, you are going to use a computer-controlled range-finder and the or PhysTks software to monitor the change in distance over time of the pendulum bob from the device.

• mount the larger ball m1  and calibrate the pendulum.

• your TA will assign your group a length (s) between 30 and 90 cm.

• adjust the sliding arm so that the string length s is approximately the length given to your group. Record the actual length to a precision of 1 mm (0.001 m).

Acquire distance/time data of the pendulum motion

1. Set the pendulum swinging in a straight line, keeping 9 small (less than approxi- mately 15О ). Wait several seconds to allow for any stray oscillations present in the bob to dissipate before beginning to collect data.

2. Shift focus to the PhysTks software. Check the Dig1 box and choose to collect 50 points at 0.1 s/point. Click Get data to acquire a set of data points (x’y) of distance y as a function of time x.

3. Click Draw to plot your data.  Your points should look like a smooth sine wave, without spikes, stray points or flat spots. If any of these are noted, adjust the position of the range-finder and acquire a new data set. Flat spots at the bottom of the graph occur when the ball is too close to the range-finder.

If the pendulum was not swinging in-line with but at a significant angle β to the range-finder, how would the appearance of your graph change? Try to explain what is going on mathematically. How would this affect your results?

Fit the pendulum distance/time data to a sine wave

From the introduction we see that our data should be in the form

9 = 90 * cos(ot)

When we deal with real-world data we need to use curve fitting software to find what the expected values of the theoretical parameters are for our experiment. We will do this by fitting to a sign curve.

1. Select fit to: y= and enter A*sin(B*x+C)+D in the fitting equation box.

If you have done curve fitting before in labs you might know that for most poly- nomial fits (lines, quadratics) you do not need to give good guesses to get a good fit. When it comes to sine curve fitting this is not the case - below details HOW one learns good guesses for the sine curve fit for this experiment.

Comparing this to our experimental data, we can see that x in the fitting equation is actually time.  The amplitude A should be equal to 90  which is approximately the distance we pulled the mass back before letting it go. The value of omega in our fit is parameter B. Lastly, fit parameters C and D don’t seem to exist in the theoretical equation BUT they are required in our fit. C is a phase shift, this is required because we let the pendulum swing for a bit before we started collecting data. This means that our data likely does not start at a maximum or a minimum of our sine function. Similarly, fitting parameter D comes from the distance of the range finder to the center of motion. When we talk theoretically about motion, we never need to think about where our detector is, but in the physical world this often leaves us with an vertical shift in our motion.

The fit parameter B (in radians/s) is the rate of change in angle with time, which is also called the angular frequency o, so that B*x is an angle in radians.  After one period of oscillation, where T represents the period in seconds, x increases by T and the angle B*x increases by 2T radians.  Therefore BT = 2T, and this allows us to relate the fitting parameter B to the period T of the pendulum’s oscillation:

B =        which is equivalent to       T =

2. Click Draw. If you get a Fit timed out message on the bottom left of the screen, the initial guesses for the fitting parameters may be too distant from the required values for the fitting program to properly converge.

Look at your graph and enter some reasonable approximate values for the fitting parameters. You can get an initial guess for B by estimating the time x between two adjacent minima, or one period, of the sine wave.

3. Label the axes and give your graph a descriptive title that includes the length s of the string. Click Send to: to email yourself a copy of the graph for later inclusion in your lab report.

Include an image of your sine curve fit in your lab report.  Be sure to also include somewhere in the lab report the fitting equation you used and the values of the fit parameters with their uncertainties.

Don’t forget all fitting parameters have units. If you are stuck, remember that ev- ery term on the right hand side of the equation must have the same units overall as the left hand side of the equation. It is also helpful to remember that the argument of sine functions need to be in radians (this will help you determine the units for B and C) but the value that sine functions output have no units.

Determine Gravity with Uncertainty

We know that the mass of the large ball m1 is 0.0225kg and it has a diameter d=0.02540m. Using this information along with your fit values for B and equations 3 and 4 determine gravity and its uncertainty.

State the value you obtain for gravity in your report along with its calculated uncer- tainty and the equation that you used to calculate the uncertainty.

If you calculated the uncertainty in multiple steps you can write each step as a single error propagation equation instead of one large equation.

Is the value you obtained close to the value of gravity?

How does it compare to all of the other groups? Using PhysTKS calculate the value of gravity and the standard deviation (uncertainty) of all of the groups - were you able to accurately determine the force of gravity?

Moment of inertia

We are now going to conduct the pendulum experiment again. This time we are going to assume that the moment of inertia for the object is not that of a point object. This means that the equation for o for this trial will be from equation 2.

We will use random objects from around the room and see if we can determine the differ-ence in their moments of inertia.