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Unformatted text preview: COLLEGE PHYSICS, Part I Chapter 10: Simple Harmonic Motion and Elasticity The Ideal Spring and Simple Harmonic Motion Simple Harmonic Motion and Reference Circle Energy and Simple Harmonic Motion The Pendulum Damped Harmonic Motion Driven Harmonic Motion and Resonance Elastic Deformation Stress, Strain, and Hooke’s Law The Ideal Spring and Simple Harmonic Motion kx F applied = You can stretch a spring by applying force. For relatively small displacements, the displacement, x , is proportional to the force: Here, k is the spring constant, and its unit is N/m. A spring that obeys this law is an ideal spring . Periodic motion or oscillation is simply periodically repeating motion. Let’s analyze periodic motion using an example of a ball attached to one end of a spring, as shown in the picture at the right. The equilibrium position corresponds to x = 0. When the x coordinate of the ball is changed, the spring either stretches or compresses; in both cases the spring exerts a force on the ball that tends to bring the ball back to the equilibrium position. This restoring force is simply given by Hooke’s law : Periodic motion under the action of a restoring force that is directly proportional to the displacement from the equilibrium is called simple harmonic motion (SHM). If you attach a pen to the ball and let it trace the position of the ball on a strip of paper moving with a constant speed, you will get a graph that shows the change of the ball’s position with time. This is a sinusoidal graph, and the maximum deviation of the ball from the equilibrium is the amplitude , A , of the simple harmonic motion. The acceleration of the ball at any position is This indicates that the acceleration is a function of the position, x , and therefore is not constant: The change of position with time is the velocity. A sinusoidal dependence of the position on time indicates that the velocity of an object that undergoes SHM is not constant. const v x ≠ This graph shows the change in the x component of velocity and the acceleration of an object attached to the spring. The object speeds up when it moves toward the equilibrium position ( x = 0) and slows down when it moves away from the equilibrium position. The acceleration is maximum at extreme positions ( x = A and x = A ) and is zero at the equilibrium position, and so is the force, according to Hooke’s law: speed slow speed slow t t x = A a 0 aA a 0 A v x a x The amplitude , A , of a simple harmonic motion is the maximum displacement from the equilibrium position, i.e., the maximum value of , and is always positive. The range is 2 . The period , T , is the time per one cycle of oscillation. It is always a positive quantity....
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 Force, Simple Harmonic Motion

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