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Rev 1/01,10/05,11/06 Experiment 14: Hooke’s Law and Simple Harmonic Motion Purpose (1) To study Hooke’s Law for an elastic spring (2) To study Simple Harmonic Motion of a mass suspended from an elastic spring Apparatus Helical steel spring with supporting stand and scale, set of slotted weights with hanger, timer, laboratory balance. Theory: Hooke’s Law A spring exerts a force which is given by Hooke’s Law: 1 Fs = - kx where x is the amount of displacement from the equilibrium position. The negative sign in this equation shows that the spring’s force is opposite to x. If the spring is stretched (x is positive) then the spring pulls back. If the spring is compressed (x is negative) the spring pushes. The parameter k is the spring constant and is a property of the spring. It is different for different springs. An elastic spring subjected to a stretching force of magnitude F will be stretched from its equilibrium position by an amount x given by Hooke’s Law until the spring force, which pulls back when the spring is stretched, balances the stretching force. 67 Experiment 14 If you hang your spring on the supporting stand (Figure 1), it will be at its unstretched length. Hanging a slotted weight of mass mload on it will subject it to the force of gravity on the slotted weight F= mload g. This will cause the spring to stretch a distance x from its equilibrium position according to Hooke’s Law until Fs = mload g. Simple Harmonic Motion (SHM) If a spring with a weight hanging on it is given an additional displacement of magnitude a (Fig. 2) and released, it will undergo Simple Harmonic Motion. In simple harmonic motion a spring will oscillate up and down with an amplitude a and a period T. The minus sign in Hooke’s Law tells us why this happens. When the spring is stretched downward, it pulls upwards and then becomes compressed. When it is compressed, it pushes downward and then becomes stretched, and so on. The equation for period T is: 2 T = 2 Meff k where Meff is the “effective mass” = Mload + 1/3 Mspring. This correction follows from a more accurate theory. 68 Experiment 14 Procedure Part I - Testing Hooke’s Law: 1. Measure and record the mass mhanger of the hanger, the mass mspring of your spring. 2. Adjust the scale so that the pointer on the hanger is near the upper end of the scale when the hanger is unloaded. Record its position as So and prepare a table on your data sheet: mweights mload = mweights + mhanger Scale Reading S (cm) The units of the scale are centimeters. 3. Load the hanger by m = 50 grams. Record the scale reading S. Keep increasing the load by suitable increments (they need not be equal) to obtain 8 data points before the lower end of the scale is reached. Record all data. CAUTION: Do not stretch the spring excessively. You may damage it by deforming it permanently. 4. Reverse the process by unloading the same loads as in (3). Again, record S for each load. Procedure Part II - Study of Simple Harmonic Motion: 1. Prepare a table: mweights mload = mweights + mhanger N tN T=tN/N meff average 2. Load the hanger with m = 200 g and set the suspended mass into oscillation (as in Fig. 2(b)) with an amplitude of roughly 1 cm. Measure the time tN for N = 100 complete cycles. Remember that in a complete cycle the mass starts in the original position and goes back to the original position, i.e. from the top of the cycle down to the bottom and back to the top of the cycle. 69 k Experiment 14 3. Repeat (2) with an amplitude of about 2 cm and a value of N between 90-110. 4. Repeat (1) and (2) with m = 250g. 5. Repeat (1) and (2) with m = 300g. Lab Report Part I Theory: In equation 1 above, x represents the amount the spring stretched = (S - So). You can calculate the spring constant k from your data using equation 1 : Fs = k (S - So) Since the spring force exactly balances the weight of the mass mloadg, we can write: mload g = k (S - So) = k S - k So Solving for s we find S = (g/k) mload + So This shows a linear relationship between m and s with the slope = g/k and the y intercept of So. For a plot of S vs m slope = S/mload = g/k k = g / slope 1. Plot S vs mload for all your data points (some points may be on top of each other). Label all axes and write the units. Calculate the slope = s/m (don’t forget the units). Then calculate k using the equation k = g / slope. What are the units of k? (Remember g = 980 cm/sec2) 70 Experiment 14 Question #1 What is the purpose of measuring S both when loading and unloading. What is your conclusion of this effect, based on your data? Question #2 From looking at your graph, can you claim that you verified Hooke’s Law? Explain your reasoning. Lab Report Part II Theory: You can calculate k for all runs from this data using the equation for the period T: meff k Squaring both sides and solving for k. we find: T 2 3 k 4π 2 m eff T2 2 Find the average value of k for Part II using Equation 3. Remember to write all units. Question #3 Did you verify that the period of SHM is independent of amplitude? Explain. Conclusion 3, Compare your results from Part I to your average results from Part II by calculating the discrepancy as follows: k (Part I) – k (Part II) .5*k (Part I) + k (Part II) 100% What are some reasons for the discrepancy? 71