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.
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.
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.
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
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.
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?
2. Calculate the percentage error between the expected theoretical values of and the experimental values you measured for all cases. Use the equation .
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?
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?
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?
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.