Gravity in the vertical oscillator In this derivation, we

Chapter , Problem 42

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Gravity in the vertical oscillator In this derivation, we show why the gravitational force does not appear in the equation for a vertically suspended oscillator. a. Consider the figure in which a block hangs at rest from a spring. The spring is stretched an amount _y by the weight mg of the block. We denote this equilibrium position of the block y = 0. When the system is at equilibrium, explain why mg = k_y, where k is the spring constant. Unstretched spring mg Equilibrium y _ 0 y _y b. Suppose the block is pulled down a distance y from its equilibrium position. We now write Newtons second law using the end of the unstretched spring as a reference point, assuming no damping or external force and assuming the y-axis points downward. Explain why my_ = -k1y + _y2 + mg. c. Use the equilibrium relationship mg = k_y, to express the equation of motion as my_ = -ky, which is the equation derived earlier. Notice that the gravitational force does not appear.