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phys22-lab1-shm - Experiment SIMPLE HARMONIC MOTION Part I...

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Experiment: SIMPLE HARMONIC MOTION Part I: Mass Hanging from Spring PURPOSE The object of this experiment is to study the periodic motion of simple system, the oscillations of a mass hung on a vertical spring. EQUIPMENT A spiral spring, mounted on a vertical holder with a scale, set of weights, stopwatch. BACKGROUND Periodic motion (also called simple harmonic motion) is characterized by a repetitive sequence of events for a moving body. In general, the force on the body is directly proportional to its displacement, but in the opposite direction. This means that when the body moves in a certain direction, a force acts so as to pull the body back to its original (equilibrium) position. This type of force is called a " restoring " force, since it tends to "restore" the body back to its equilibrium position. When the body is displaced from equilibrium and released, this force will accelerate the body back toward its equilibrium position — but on the way back, the body's own inertia will carry it past the equilibrium point, causing a new restoring force from the opposite direction to pull it back again. In this manner, the body undergoes periodic motion, and we say that the body oscillates about its equilibrium point. This periodic motion will continue until friction or other force acts to stop it. Consider a mass m hanging from a spring, as shown in Fig. 6. If the mass is displaced by an amount x , then the spring exerts a restoring force F in the opposite direction: F = kx (1) where the minus sign indicates that the force is in the opposite direction to the displacement. This is known as Hooke's Law, and the constant of proportionality k is known as the spring constant , which essentially is a measure of the "stiffness" of the spring. Since the dimensions of the spring constant are Force/Displacement, we can see that the units of k must be N/m.
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2 m w x F Figure 6: Elongation of a spring at vertical distance x If we hang a mass m from the spring, then the spring will be stretched some amount x from its initial position. The downward force on the mass is due to gravity, mg . The upward force on the mass is due to the spring, kx . Since the mass is at rest, the magnitudes of these two forces must be equal: F grav = mg = F spring = kx . (2) Therefore, if various weights are hung from the spring, and the corresponding elongations of the spring are measured, then a plot of the weight mg versus the displacement x should yield a linear relationship with a slope equal to the experimental value of the spring constant k . In the above discussion, we treated the static case, in which the mass hanging from the spring was at rest. What if we pull the mass down from its equilibrium position and then let go? The mass will oscillate vertically, bobbing up and down as the spring stretches and compresses. For one-dimensional periodic motion, Newton's Second Law can give the displacement of the body as a function of time, x(t), with some period T, such that
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phys22-lab1-shm - Experiment SIMPLE HARMONIC MOTION Part I...

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