This chapter is all about harmonic motion, or motion that repeats itself with some associated frequency or period. This lab focuses on measuring the period of two such systems. The pendulum portion aims to determine if the swing angle affects the period of the pendulum, and the spring & mass section aims to determine the effect that mass has on angular frequency as well as what the effect of putting two springs in series and parallel has on the overall stiffness of a spring system.[br][br]I want to offer a practical suggestion for measuring the period which you will do by hand with a stop watch or a phone timer. Any time we use stop watches our measurements are muddied by our reflexes. If our pushing of the start and stop buttons adds an error of 0.1s to our measurements, for instance, it might make it impossible to discern small differences.[br][br]To minimize the error associated with button pressing, a common thing to do is to time not a single period of the motion, but perhaps five or ten periods. Press start at the beginning of the first swing, let the pendulum swing for 10 full swings and then press stop. Obviously the period is the total time divided by 10, but another consequence is that your button pressing error has a ten-time-smaller influence on the period since you divided the error over ten swings.

[list=1][*]Set up your ring stand with your largest mass hanging from it by a string that allows the mass to swing just above the table's surface, and clamp the ring stand to the table so it doesn't move. If you don't have a clamp, then just secure it with your hand during the activity.[br][/*][*]Measure the length from the attachment point of your string to the center of mass of the mass hanging from it. Consider this the "length of your string". Q1: Based on the length of your string, what do you expect the angular frequency of your pendulum to be? Look back at the math in the chapter if you don't know.[/*][*]Q2: Do you think the angular frequency should depend on mass?[/*][*]Q3: Do you think the angular frequency should depend on initial swing angle?[/*][*]Measure the period for small angle motion with a stopwatch(there is no angle too small so long as you can see it).[/*][*]Measure the period by recording the motion in MATLAB with the ultrasonic sensor and fitting the motion with a sinusoidal function. The easiest way to do this is to use the curve fitting app and use the "Fourier" option with a single term as your functional form of the curve fit.[br][/*][*]Now change to a different mass and perform parts 4 through 6 again. Q4: Does the mass affect the period.[br][/*][*]Now shorten your string (rather than cutting, tie a knot at a new spot) to around half its original length, and be sure to record this new string length. Q5: What effect do you expect to measure? Now measure the new period with only one (your choice, but record it) of the former masses. [br][/*][*]Measure the new periods for each swing angle, and after analyzing the results, find the mathematical expression that should relate the new periods, the old periods and the two lengths. This expression should have the two lengths of string in it. Q6: Please show this expression in your lab report.[br][/*][/list]

[list=1][*]Use three different masses, and measure the displacement from equilibrium that they product when hung from a spring, so that you can find the spring constant of your spring.  Do this for each single spring, the two in a series arrangement, and the two in a parallel arrangement.[/*][*]In MATLAB, plot the force F versus displacement for the four cases and use (and display) a linear curve fit to find a best value for the spring constant for each case.[/*][*]Q7: If you hang a known mass from the spring and allow it to vibrate, will the angular frequency depend on amplitude?[/*][*]Push the mass gently upward such that it still makes contact with the spring, and measure the period to get the angular frequency.  Do this using the ultrasonic sensor.[/*][*]Find the period for two different masses but only with one of the spring of your choice.[/*][/list]

To find the elastic constant for a spring, you need data relating how far the spring stretches from equilibrium versus force applied to it. You acquired this data. To find the elastic constant, just plot a linear fit to the force versus displacement data and extract the correct term. As a hint, recall that the relationship between force and displacement is [math]F_x=-k\Delta x.[/math][br][br]Please do these linear fits in MATLAB and display the equations on the graphs as shown below. Then make sure to include them in your lab submission[code].[/code][code][/code]

1. What is the benefit of allowing the pendulum to swing multiple times rather than just once if your aim is to measure the period of the motion?[br][br]2. Calculate the percentage error between the expected theoretical values of [math]\omega[/math] and the experimental values you measured for all cases. Use the equation [math]\%error=|\frac{\omega_{theo}-\omega_{exp}}{\omega_{theo}}|\times100\%[/math].[br][br]3. How does the stiffness of two springs in parallel compare with the single spring (according to your data)? What should the value have been had everything gone ideally?[br][br]4. How does the stiffness of two springs in series compare with the single spring (according to your data)? What should the value have been had everything gone ideally?[br][br]5. In which system (mass/spring or pendulum) does the period depend on the amplitude, and when should you have to worry about this dependence?[br][br]6. Some people like the look of lowered sports cars. While companies sell lowering springs to put on cars for this purpose, a cheap alternative is to simply cut the existing springs. While not recommended, it works. Suppose a spring is cut to 3/4 of its original length. Does its stiffness change? If so, by how much? Base your answer on the results of question 4 above by realizing that two springs in series are the same as a single spring with twice the length.