Simple harmonic motion

  Objectives In this experiment we study the physical system composed by a spring and a mass attached to it. The main goals are to check the validity of Hooke’s law, which we believe is a good model for this system, and to understand the oscillatory movement that springs undergo when taken initially away from their equilibrium position. Theory A spring is a mechanical device that can compress or expand when a force is applied to it, but it returns to its original position when the force is removed. In particular, if the relation between the force applied (Fappl) and the expansion/compression (x) is linear, as in Fappl = kx (1) the system is especially simple. By using Newton’s third law, if there is an external force Fappl acting on the spring, the spring is acting with an equal and opposite force Fs = −Fappl. Then, the force exerted by the spring is given by Fs = −kx. (2) This is called Hooke’s law, and it appropriately describes the behavior of real springs if the forces applied are small. The behavior of a spring under this model depends only on the value of k, which is the spring constant. Equilibrium The equilibrium position for a spring happens when the force is equal to zero. This implies that x = 0 is the equilibrium position, which is why x is usually called displacement from equilibrium. In a situation where there are 1 several forces acting on the spring, the equilibrium position depends on all of them. For example, if we attach a mass to the end of a spring and hang it vertically, the equilibrium position corresponds to the point where the force of the spring compensates the force of gravity. In this case, the total force must cancel: Fs + Fg = 0 Fs = −Fg, −kx = −mg, x = mg k . (3) Keep in mind that in this manual, x is positive if it points down, and negative if it points up, with respect to the equilibrium position. Fg Fs equilibrium Figure 1: Mass hanging from a spring. Oscillations The nature of Hooke’s law makes springs show a peculiar behavior when placed away from their equilibrium position. When the object in the picture is slightly pulled down, away from its equilibrium position, the spring force will pull upwards. This force will overcome gravity and produce a net displacement upwards. As the mass moves up, it will pass the equilibrium position to lay above it. In this situation, the spring will be compressed, and its force will push down. At some point, this force will make the spring expand, moving the mass down, past the equilibrium position, and going back to the starting place, where the object is below the equilibrium position. This process will automatically repeat periodically, and the movement produced by it is what we call simple harmonic motion. 2 The oscillations in the spring will repeat periodically. This means that the position and velocity of the object will be the same at some time t and at a later time t + T, a later time t + 2T, a later time t + 3T, etc. The quantity T is called the period of the oscillations, and for a spring driven by Hooke’s law is given by T = 2π r m k , (4) which can be otherwise stated as T 2 = 4π 2 k × m. (5) Online experiment The online simulation for this experiment can be accessed at https://phet.colorado.edu/sims/html/masses-and-springs/latest/massesand-springs en.html Once there, enter the Lab option: • on the left, you will see an energy graph which will show how much kinetic, gravitational, elastic and thermal energy the spring has at a given moment; • on top, you will see the spring, and two selectors: one for the value of a mass, and another for the value of the spring constant (k); • at the bottom, you will see three masses. The orange has its mass (m) selected by the slider described above. The other two have, in principle, unknown values; • on the right, you will see several boxes that enable showing some reference lines in the picture (displacement, equilibrium position, ...), a slider for the value of gravity (which you don’t need to use), and a slider to select damping. You can also see a ruler and a stopwatch, which you will need to use to perform measurements. Procedure Part A: determining k using Hooke’s law 1. Choose a value for the spring constant k at random using the slider at the top. Although not impossible, part B will be harder to do if k is 3 too high. 2. Move the slider for damping all the way to the right (Lots). 3. Check the first and second boxes on the right panel to see the natural length of the spring, and the position of the masses. 4. Choose a mass for the orange object of m = 50 g, and hang it from the spring. 5. Using the ruler, measure the distance between the blue and the black lines. 6. Record the current value of the mass and the distance the first and third columns of the table. 7. Repeat steps 4 to 6 for a total of six values of m between 50 g and 300 g. m (g) m (kg) Fg (N) x (cm) x (m) Table for part A. Part B: determining k using the period of oscillations 1. Keep the value of k from part A. 2. Move the slider for damping all the way to the left (None). 3. Un-check all boxes on the right panel and put the ruler away, as none of those are needed in this part. 4. Chose a mass for the orange object of m = 50 g, and hang it from the spring. 4 5. Slightly pull the mass away from its equilibrium position. 6. Start the stopwatch when the mass is exactly at the bottom of its movement. 7. Count the number of times it reaches the bottom after that (not including the first one), and stop the stopwatch exactly as it reaches the bottom for the tenth time. 8. Record the current value of the mass and the total time indicated by the stopwatch (ts) on the first and third columns of the table. 9. Repeat steps 4 to 8 for a total of six values of m between 50 g and 300 g. m (g) m (kg) ts (s) T (s) T 2 (s2 ) Table for part B. Calculations and analysis Part A 1. Convert the masses (m) from g to kg, and the positions(x) from cm to m, filling the second and fifth columns of the table. 2. Calculate the gravitational force (third column) as Fg = mg. 3. On a spreadsheet software, make a graph with the force Fg in the vertical axis and the distance x in the horizontal axis. 5 4. Find the slope and the intercept of the best fit straight line. A general straight line is y = ax + b, where a is the slope and b the intercept. Comparing to the equilibrium equation Fg = kx, the slope of the line should be the constant of the spring: k = a. Give the value of k in the appropriate units. 5. What value of the intercept do you obtain? How does this compare with the value you expected to obtain? Part B 1. Convert the masses (m) from g to kg, filling the second and fifth columns of the table. 2. Calculate the period (fourth column) from the stopwatch time (third column) as T = ts 10 . Calculate the square of the period, to fill the fifth column. 3. On a spreadsheet software, make a graph with the squared period T 2 in the vertical axis and the mass m in the horizontal axis. 4. Find the slope and the intercept of the best fit straight line. Comparing a general straight line with equation (5) the slope (a) is related to the spring constant as a = 4π 2 k → k = 4π 2 a . Give the value of k in the appropriate units. 5. Compare the values of the spring constant obtained in part A (kA) and part B (kB). To do so, calculate their percent different with respect to their average value: % diff = 100% × kA − kB kA+kB 2 . 6