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Lab 7 - Simple Harmonic Motion

Lab 7 - Simple Harmonic Motion - PHYSICS 133 EXPERIMENT 7...

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PHYSICS 133 EXPERIMENT 7 SIMPLE HARMONIC MOTION Introduction In this lab, we study the phenomenon of simple harmonic motion for a mass-and-spring system and for a variety of pendulums. In a linear mass-spring system, the physical basis for this kind of motion is that the restoring force F exerted on a mass m that has been displaced a distance x from equilibrium must be proportional to – x . This relationship may be written F = - kx (1) where k is a constant that characterizes the stiffness of the spring. A large value of k would indicate that the spring is difficult to stretch or compress. In the case of a simple pendulum, there is no spring, and k is replaced by the quantity ( mg/L ) , where m is the mass of the pendulum bob, g is the acceleration due to gravity, L is the length of the pendulum, and x represents the (small) lateral displacement of the bob. Eq. (1) can be generalized to represent other physical situations. For example, the displacement might be given in terms of an angle, in which case the restoring variable would be a torque. See your textbook for examples. Using Newton’s second law, F = md 2 x / dt 2 , we can write Eq. (1) as a differential equation, d 2 x / dt 2 = - ( k / m ) x (2) This equation has as a possible solution the sinusoidal oscillation x = A cos ωt, which you can verify by direct substitution in Eq. (2). Here A is the amplitude and ω = √ ( k/m ) is the circular frequency. The frequency depends on physical characteristics of the system. For example, ω = √ ( k/m ) for a linear mass-spring system and ω = √ ( g/L ) for small oscillations of a simple pendulum. Since the period T = 2 π / ω , we have T = 2 π √ ( m/k ) for the mass-spring system and T = 2 π √ ( L/g ) for the simple pendulum, respectively. Equipment 1 air track with glider/spring oscillator, Low-friction pulley and thread, small masses, photogate, magnets, 1 simple pendulum, 1 meter stick pendulum, ring and disk pendulums 1 computer and interface box. Method
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Throughout this experiment, a system (glider between springs, assorted pendulums) will be displaced from equilibrium and the period, T , of oscillation will be measured. For the various oscillators, the dependence of the period on amplitude will be assessed. For true, simple harmonic motion, there should be no amplitude dependence. Procedure I. Measurement of the Spring Constant for the Mass-Spring Oscillator Record the equilibrium position of the glider. Attach a piece of string to the glider and pass it over the low-friction pulley with a mass suspended from the free end of the string. Measure the displacement of the glider. Be sure that the string moves freely and does not
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Lab 7 - Simple Harmonic Motion - PHYSICS 133 EXPERIMENT 7...

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