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Describe the force in an elastic spring Determine the energy stored in an elastic spring Compare simple harmonic motion and the motion of a pendulum

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How does a spring react to a force that is applied to it Figure 14-1a shows a spring hanging from a support with nothing attached to it The spring does not stretch because no external force is exerted on it Figure 14-1b shows the same spring 0m with an object of weight mg hanging from it The spring has stretched by distance x so that the xm upward force it exerts balances the downward force mg of gravity acting on the object Figure 14-1c shows the same spring stretched twice as far, 2x, to support 2x m twice the weight, 2mg, hanging from it Hooke s 2mg law states that the force exerted by a spring is directly proportional to the amount that the spring Figure 14-1 The force exerted by a spring is directly proportional is stretched A spring that acts in this way is said to obey Hooke s law, to the distance the spring is which can be expressed by the following equation

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The force exerted by a spring is equal to the spring constant times the distance the spring is compressed or stretched from its equilibrium position

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In this equation, k is the spring constant, which depends on the stiffness and other properties of the spring, and x is the distance that the spring is stretched from its equilibrium position Not all springs obey Hooke s law, but many do Those that do are called elastic Potential energy When a force is applied to stretch a spring, such as by hanging an object on its end, there is a direct linear relationship between the exerted force and the displacement, as shown by the graph in Figure 14-2 The slope of the graph is equal to the spring constant, given in units of newtons per meter The area under the curve represents the work done to stretch the spring, and therefore equals the elastic potential energy that is stored in the spring as a result of that work The base of the triangle is x, and the height is the force, which, according to the equation for Hooke s law, is equal to kx, so the potential energy in the spring is given by the following equation

240 241 242 243 246

Figure 14-2 The spring constant of a spring can be determined from the graph of force versus displacement of the spring

The potential energy in a spring is equal to one-half times the product of the spring constant and the square of the displacement

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The units of the area, and thus, of the potential energy, are newton meters, or joules How does the net force depend upon position When an object hangs on a spring, the spring stretches until its upward force, Fsp, balances the object s weight, Fg, as shown in Figure 14-3a The block is then in its equilibrium position If you pull the object down, as in Figure 14-3b, the spring force increases, producing a net force upward When you let go of the object, it accelerates in the upward direction, as in Figure 14-3c However, as the stretch of the spring is reduced, the upward force decreases

In Figure 14-3d, the upward force of the spring and the object s weight are equal there is no acceleration Because there is no net force, the object continues its upward velocity, moving above the equilibrium position In Figure 14-3e, the net force is in the direction opposite the displacement of the object and is directly proportional to the displacement, so the object moves with a simple harmonic motion The object returns to the equilibrium position, as in Figure 14-3f

Figure 14-3 Simple harmonic motion is demonstrated by the vibrations of an object hanging on a spring

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