175 Name Date Partners University of Virginia Physics Department PHYS 635, Summer 2009 LAB 11: FREE, DAMPED, AND FORCED OSCILLATIONS OBJECTIVES • To understand the free oscillations of a mass and spring. • To understand how energy is shared between potential and kinetic energy. • To understand the effects of damping on oscillatory motion. • To understand how driving forces dominate oscillatory motion. • To understand the effects of resonance in oscillatory motion. OVERVIEW You have already studied the motion of a mass moving on the end of a spring. We understand that the concept of mechanical energy applies and the energy is shared back and forth between the potential and kinetic energy. We know how to find the angular frequency of the mass motion if we know the spring constant. We will examine in this lab the mass-spring system again, but this time we will have two springs — each having one end fixed on either side of the mass. We will let the mass slide on an air track that has very little friction. We first will study the free oscillation of this system. Then we will use magnets to add some damping and study the motion as a function of the damping coefficient. Finally, we will hook up a motor that will oscillate the system at practically any frequency we choose. We will find that this motion leads to several interesting results including wild oscillations. Harmonic motions are ubiquitous in physics and engineering - we often observe them in mechanical and electrical systems. The same general principles apply for atomic, molecular, and other oscillators, so once you understand harmonic motion in one guise you have the basis for understanding an immense range of phenomena. INVESTIGATION 1: FREE OSCILLATIONS An example of a simple harmonic oscillator is a mass m which moves on the x-axis and is attached to a spring with its equilibrium position at 0x= (by definition). When the mass is moved from its equilibrium position, the restoring force of the spring tends to bring it back to the equilibrium position. The spring force is given by springF kx= − (1) where k is the spring constant. The equation of motion for m becomes 22d xm kxdt= − (2)176 Lab 11 – Free, Damped, and Forced Oscillations University of Virginia Physics Department PHYS 635, Summer 2009 This is the equation for simple harmonic motion. Its solution, as one can easily verify, is given by: ()sinF F Fx A tω δ= + (3) where Fk mω= (4) Note: The subscript “F” on Fω, etc. refers to the natural or free oscillation. FA and Fδ are constants of integration and are determined by the initial conditions. [For example, if the spring is maximally extended at 0t=, we find that FA is the displacement from equilibrium and 2δ π=.] We can calculate the velocity by differentiating with respect to time: ( )cosF F F Fdxv A tdtω ω δ= = + (5) The kinetic energy is then: ()22 21 12 2cosF F FKE mv A tkω δ= = + (6) The potential energy is given by integrating the force on the spring times the displacement: ( )22 21 12 20 sinxF F FPE kxkxdx kA tω δ= = =+∫ (7) We see that the sum of the two energies is constant: 212 FKE PEkA+ = (8) ACTIVITY 1-1: MEASURING THE SPRING CONSTANT We have already studied the free oscillations of a spring in a previous lab, but let's quickly determine the spring constants of the two springs that we have. To determine the spring constants, we shall use the method that we used in Lab 8. We can use the force probe to measure the force on the spring and the motion detector to measure the corresponding spring stretch. To perform this laboratory you will need the following equipment: • force probe • motion detector • mechanical vibrator • air track and cart glider • two springs with approximately equal spring constants • electronic balance • four ceramic magnets • masking tape • string with loops at each end, ~30 cm long 1. Turn on the air supply for the air track. Make sure the air track is level. Check it by placing the glider on the track and see if it is motionless. Some adjustments may beLab 11 – Free, Damped, and Forced Oscillations 177 University of Virginia Physics Department PHYS 635, Summer 2009 necessary on the feet, but be careful, because it may be impossible to keep the track level over its entire length. 2. Tape four ceramic magnets to the top of the glider cart and measure the mass of the glider cart on the electronic balance. glider cart mass _______________ kg(with four magnets) 3. Set up the force probe, glider, spring, motion detector, and mechanical vibrator as shown below on the air track. If not already done, tie a loop at each end of a string, so that it ends up about 30 cm long. Loop one end around the force probe hook and the other end around the metal flag on the glider cart. Note the spring you put on the apparatus as Spring 1. Make sure the mechanical vibrator-oscillator driver is in the locked position. FlagForce ProbeGliderSpringMechanical VibratorMotion Detector Open the experiment file called L11.A1-1 Spring Constant. 4. Zero the force probe with no force on it. 5. Pull the mechanical vibrator back slowly until the spring is barely extended. 6. Start the computer and begin graphing. Use your hand to slowly pull the mechanical vibrator so that the spring is extended about 30 cm. Hold the vibrator still and stop the computer. 7. The data will appear a little jagged, because your hand cannot pull back smoothly, but overall you should see a straight line. Use the mouse to highlight the region of good data. Then use the fit routine in the software to find the line that fits your data, and determine the spring constant from the fit equation (the slope). Include your best estimate of the uncertainty (the fit routine reports this). 1k _______________ N/m 8. Print out one set of graphs for your group that includes the fit and include it in your group report. Never move items on the air track unless the air is flowing! You might scratch the surfaces and create considerable friction. Figure 1178 Lab 11 – Free, Damped, and Forced Oscillations University of Virginia Physics Department PHYS 635, Summer 2009 Question 1-1: Was the force exerted by the spring proportional to the displacement of the spring? Explain. Question 1-2: What