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Unformatted text preview: CHAPTER 10 SIMPLE HARMONIC MOTION AND ELASTICITY ANSWERS TO FOCUS ON CONCEPTS QUESTIONS 1. 0.12 m 2. (c) The restoring force is given by Equation 10.2 as F = − kx , where k is the spring constant (positive). The graph of this equation is a straight line and indicates that the restoring force has a direction that is always opposite to the direction of the displacement. Thus, when x is positive, F is negative, and vice versa. 3. (b) According to Equations 10.4 and 10.11, the period T is given by 1 2 m T k π = . Greater values for the mass m and smaller values for the spring constant k lead to greater values for the period. 4. (d) The maximum speed in simple harmonic motion is given by Equation 10.8 ( 29 max v A ϖ = . Thus, increases in both the amplitude A and the angular frequency ϖ lead to an increase in the maximum speed. 5. (e) The maximum acceleration in simple harmonic motion is given by Equation 10.10 ( 29 2 max a A ϖ = . A decrease in the amplitude A decreases the maximum acceleration, but this decrease is more than offset by the increase in the angular frequency ϖ , which is squared in Equation 10.10. 6. 1.38 m/s 7. (b) The velocity has a maximum magnitude at point A, where the object passes through the position where the spring is unstrained. The acceleration at point A is zero, because the spring is unstrained there and is not applying a force to the object. The velocity is zero at point B, where the object comes to a momentary halt and reverses the direction of its travel. The magnitude of the acceleration at point B is a maximum, because the spring is maximally stretched there and, therefore, applies a force of maximum magnitude to the object. 8. 0.061 m 9. +7.61 m/s 2 10. 0.050 m 240 SIMPLE HARMONIC MOTION AND ELASTICITY 11. (c) The principle of conservation of mechanical energy applies in the absence of nonconservative forces, so that KE + PE = constant. Thus, the total energy is the same at all points of the oscillation cycle. At the equilibrium position, where the spring is unstrained, the potential energy is zero, and the kinetic energy is KE max ; thus, the total energy is KE max . At the extreme ends of the cycle, where the object comes to a momentary halt, the kinetic energy is zero, and the potential energy is PE max ; thus, the total energy is also PE max . Since both KE max and PE max equal the total energy, it must be true that KE max = PE max . 12. (e) In simple harmonic motion the speed and, hence, KE has a maximum value as the object passes through its equilibrium position, which is position 2. EPE has a maximum value when the spring is maximally stretched at position 3. GPE has a maximum value when the object is at its highest point above the ground, that is, at position 1....
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