Simple Harmonic Motion

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1 Simple Harmonic Motion Objective: In this exercise you will investigate the simple harmonic motion of mass suspended from a helical (coiled) spring. Apparatus: Spring 1 Table Post 1 Short Rod 1 Right-angled Clamp 1 Ring with Hook 1 Mass Set 1 Stopwatch 1 Introduction: A mass attached to an ideal coiled spring is said to experience a restoring force proportional to the amount of extension or compression of the spring from its rest, or equilibrium, position. This restoring force can be written as: F = k(x x 0 ) (1) where x is the stretched or compressed length and x 0 is the equilibrium length (in metres), and k is the spring constant (in N/m). The negative sign indicates that the force exerted by the spring is opposite to the direction of the stretch or compression so that the force always acts to restore the spring to its equilibrium position. In the circular motion lab, you studied the static properties of a spring by attaching a mass and measuring how far the spring stretched. Here you will study the dynamic properties by attaching a mass and setting it in motion. A useful way to analyse this, and many other systems where the force can be described, is to recognize that Newton s second law (F = ma) can be applied. Since F is known (i.e., the spring force), the unknown is the acceleration. But the acceleration is the rate of change of the rate of change of position, so the unknown is really the position of the mass on the end of the spring. And that can, in principle, be measured. If the mass is allowed to oscillate vertically, it will do so about the position x 0, with some frequency, ω (omega), and will take a time T (the period) to make one full oscillation. Without loss of generality, we can set x 0 = 0, since it is simply a reference point. Newton s second law (usually written F=ma) states that a force applied to a mass, m, will cause the mass to experience an acceleration, a, proportional to the applied force, inversely proportional to the mass, and acting in the same direction as the applied force.

2 2 Newton s second law applied to this system gives: ma = kx (2) d 2 x dt 2 or = " k % $ 'x (3) # m & The term on the left is the acceleration of the mass written as the second derivative (with respect to time) of the displacement of the mass from equilibrium. This is an example of a second order differential equation. It is important to recognise that: the acceleration is directed opposite the extension of the spring (i.e., in the same direction as the restoring force), and the acceleration is proportional to the displacement These are characteristics of simple harmonic motion. Equation 3 does not help directly with the experiment, because the acceleration is not easy to measure, and depending on how fast the mass oscillates, the position may not be easy to measure either. What you need is to find something measurable. Instead of focusing on the position of the mass, perhaps it is better to think about time instead, as in the period, or how long it takes for the mass to complete one oscillation. Equation 3 does not contain time explicitly, so you need to find a relationship between position and period. One solution (but not the most general one) to equation 3 is where A is the amplitude of the oscillation (in metres), ω is the frequency of oscillation (in radians/second), and t is time (in seconds). x = Acos( ωt) (4) Recall that the derivative of cosine is -sine, and the derivative of -sine is -cosine. If you substitute equation 4 into 3, and solve the derivatives, you get: ω 2 = k m (5)

3 3 The relationship between period and frequency is defined to be: T = 2π ω (6) This is the time required for one complete oscillation (since the cosine function repeats after 2π radians). Combining equations 5 and 6 gives the result:! T = 2π m $ # & " k % 1 2 (7) So the above analysis predicts that the period of oscillation of an applied mass suspended from an ideal, massless spring is proportional to the square root of the applied mass. The purpose of the lab is to investigate the behaviour of a real spring, however. Generally, the mass of a real spring causes the spring to oscillate, even when no additional mass is added to the spring. Equation 7 must be modified to account for this:! T = 2π m + m $ 0 # & " k % 1 2 (8) m is the mass applied to the spring, m 0 is the contribution the mass of the spring itself makes to the oscillation of the spring. It is not the entire mass of the spring, but rather a fraction of the spring mass (sometimes quoted as 1 3 m spring ). The term m+m 0 is called the effective mass. You will test equation 8. Procedure: To test equation 8, you will systematically add masses to a vertically hanging spring and measure the time required for multiple oscillations of the masses to occur. Use seven masses in the range from 200 to 500 grams in steps of 50 grams. Select a mass and hang it from the spring. Allow the mass to come to rest. Pull the mass down a short (5 mm or less) distance and release it. Time how long it takes for 10 oscillations to occur. Do three time trials for each mass. Performing the runs in succession from smaller to larger mass is not necessarily the best method. The main drawback is the risk of inadvertently recording the expected outcome instead of the actual outcome (as in That doesn t look right. It should be this. ) Try randomising the order of the runs.

4 4 Weigh the masses in the combinations you use. Do not weigh the masses singly only, then add the numbers in combination, because this will increase the measurement uncertainty. Also weigh the spring. You should have 21 trials altogether: 7 different masses with three time trials each. Remember to estimate uncertainties in timing. These are primarily reaction times (try the drop test method to estimate reaction time, or look it up from the Projectile lab experiment where you tested your reaction time). If you can time some oscillations better than others, then make a note of this. Measurement uncertainties are not necessarily equal for all measurements. Note the behaviour of the mass-spring system: Are the oscillations vertical only? Does the spring oscillate when no mass is attached? Try to measure the period. Analysis: From your data of three trials of 10 oscillations, determine the average period of oscillation of each mass (Don t forget: a period of oscillation means for one oscillation, not for ten!). Also calculate the standard deviations (σ T ) in the period data for three runs: the smallest, largest, and one middle value. Plot a graph of T 2 (y-axis: average period squared) vs. applied mass. You can plot this by hand or use a plotting program. Calculate the slope of the line, and the y-intercept, and include the correct units and error estimates (if you use a plotting program, use it to find a linear least squares fit, or trendline ). Error bars: Calculate error bars for the T 2 data (just the three runs for which you have calculated the standard deviations): δ ( T 2 ) = 2TδT (9) where δt is the error in the average period. This error is the bigger of: σ T 3 or 2 ( δt ) 3 (10) where δt is the reaction time error for each period measurement. The 2 is there because there are two reaction times for each trial, one for starting, and one for stopping. Error bars for the x-axis masses are just reading errors from the digital balance scale. Note that these are too small to draw on your graph.

5 5 Generate a maximum and minimum line using the error bars for the lowest and highest points. Find the slopes and intercepts. Relating the graph to equation 8: First, note that you plotted period squared, not period, so a good first step would be to square both sides of equation 8: which can be rewritten as:! T 2 = 4π 2 m + m 0 $ # & (11) " k % T 2 = 4π 2 k m + 4π 2 k m 0 (Recall that m is the mass you hung on the spring, and m 0 is a correction factor (which, at the moment, is an unknown)). Next, note that the equation of a line is: y = ax + b (12) Make a direct correspondence between equations 11 and 12 so that you know how the slope of the line and the y-intercept relate to k and m 0. (Note: k is not the slope!) Once you have correctly determined how these terms relate to the graph and the best fit line, calculate the spring constant, k, and correction term m 0, and error estimates. Compare your value of m 0 with the mass of the spring. Testing your result: According to your graph, what is the expected period for a mass of 150 grams? Hang the required mass on the spring, set it oscillating, and measure the period. Compare the measured period against the expected period. Use your error estimates from above to determine if the predicted and measured masses agree within error. PHYS120 students: write out the full error term for the period T based on equation 8. Additional activity Resonance Frequency is the inverse of the period. Predict what mass you should hang from the spring so that the natural frequency is 1.0 Hertz. Hang this mass (or as close as you can manage to this mass). Try driving the spring (with very low amplitude!) at this frequency, and at frequencies different from this. What do you notice about the amplitude of the oscillations?

6 6 Some sources of error to think about: The analysis above assumes that the motion of the mass is in one dimension only. Was the mass moving only in one dimension? Was there any evidence of damping behavior (i.e., did the oscillations get smaller as the ten periods were measured)? Equation 1 (and hence equation 8) is valid if the spring compressions and extensions are small compared with the length of the spring. Was this condition observed?

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