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speed max, an amplitude A, and an angular frequency . When the box is passing through the point where the spring is unstrained (x = 0 m), a second box of the same mass m and speed max is attached to it, as in part b of the drawing. Discuss what happens to (a) the maximum speed, (b) the amplitude, and (c) the angular frequency of the subsequent simple harmonic motion.

speed max, an amplitude A, and an angular frequency . When the box is passing through the point where the spring is unstrained (x = 0 m), a second box of the same mass m and speed max is attached to it, as in part b of the drawing. Discuss what happens to (a) the maximum speed, (b) the amplitude, and (c) the angular frequency of the subsequent simple harmonic motion.

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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SOLVE STEP BY STEP IN DIGITAL FORMAT GIVEN THE FOLLOWING PROBLEM Conceptual Example 8 takes advantage of energy conservation to illustrate what happens to the maximum speed, amplitude, and angular frequency of a simple harmonic oscillator when its mass is changed suddenly at a certain point in the motion. Conceptual Example 8 Changing the Mass of a Simple Harmonic Oscillator Figure 10.18a shows a box of mass m attached to a spring that has a force constant k. The box rests on a horizontal, frictionless surface. The spring is initially stretched to x = A and then released from rest. The box executes simple harmonic motion that is characterized by a maximum speed max, an amplitude A, and an angular frequency w. When the box is passing through the point where the spring is unstrained (x = 0 m), a second box of the same mass m and speed max is attached to it, as in part b of the drawing. Discuss what happens to (a) the maximum speed, (b) the amplitude, and (c) the angular frequency of the subsequent simple harmonic motion. Reasoning and Solution (a) The maximum speed of an object in simple harmonic motion occurs when the object is passing through the point where the spring is unstrained (x = 0 m), as in Figure 10.18b. Since the second box is attached at this point with the same speed, the maximum speed of the two-box system remains the same as that of the one-box system. (b) At the same speed, the maximum kinetic energy of the two boxes is twice that of a single box, since the mass is twice as much. Subsequently, when the two boxes move to the left and compress the spring, their kinetic energy is converted into elastic potential energy. Since the two boxes have twice as much kinetic energy as one box alone, the two will have twice as much elastic potential energy when they come to a halt at the extreme left. Here, we are using the principle of conservation of mechanical energy, which applies since friction is absent. But the elastic potential energy is proportional to the amplitude squared (4²) of the motion, so the amplitude of the two-box system is √2 times as great as that of the one-box system. (c) The angular frequency w of a simple harmonic oscillator is w = √k/m (Equation 10.11). Since the mass of the two-box system is twice the mass of the one-box system, the angular frequency of the two-box system is √2 times as small as that of the one-box system. V V₂ V max max max x = 0 m (a) x = A m %0 = 0 m/s ₁x = 0 m m m (b) Figure 10.18 (a) A box of mass m, starting from rest at x = A, undergoes simple harmonic motion about x = 0 m. (b) When x = 0 m, a second box, with the same mass and speed, is attached. Answer why the Amplitude of the mass-spring system increased to 2 times the mass is sqrt(2) greater than that of the one-mass system and the angular frequency is sqrt(2) times less than that of the one-mass system. PROVE YOUR SOLUTION USING THE APPROPRIATE PHYSICAL EQUATIONS AND DEMONSTRATIONS DO IT STEP BY STEP THANK YOU!
Transcribed Image Text:SOLVE STEP BY STEP IN DIGITAL FORMAT GIVEN THE FOLLOWING PROBLEM Conceptual Example 8 takes advantage of energy conservation to illustrate what happens to the maximum speed, amplitude, and angular frequency of a simple harmonic oscillator when its mass is changed suddenly at a certain point in the motion. Conceptual Example 8 Changing the Mass of a Simple Harmonic Oscillator Figure 10.18a shows a box of mass m attached to a spring that has a force constant k. The box rests on a horizontal, frictionless surface. The spring is initially stretched to x = A and then released from rest. The box executes simple harmonic motion that is characterized by a maximum speed max, an amplitude A, and an angular frequency w. When the box is passing through the point where the spring is unstrained (x = 0 m), a second box of the same mass m and speed max is attached to it, as in part b of the drawing. Discuss what happens to (a) the maximum speed, (b) the amplitude, and (c) the angular frequency of the subsequent simple harmonic motion. Reasoning and Solution (a) The maximum speed of an object in simple harmonic motion occurs when the object is passing through the point where the spring is unstrained (x = 0 m), as in Figure 10.18b. Since the second box is attached at this point with the same speed, the maximum speed of the two-box system remains the same as that of the one-box system. (b) At the same speed, the maximum kinetic energy of the two boxes is twice that of a single box, since the mass is twice as much. Subsequently, when the two boxes move to the left and compress the spring, their kinetic energy is converted into elastic potential energy. Since the two boxes have twice as much kinetic energy as one box alone, the two will have twice as much elastic potential energy when they come to a halt at the extreme left. Here, we are using the principle of conservation of mechanical energy, which applies since friction is absent. But the elastic potential energy is proportional to the amplitude squared (4²) of the motion, so the amplitude of the two-box system is √2 times as great as that of the one-box system. (c) The angular frequency w of a simple harmonic oscillator is w = √k/m (Equation 10.11). Since the mass of the two-box system is twice the mass of the one-box system, the angular frequency of the two-box system is √2 times as small as that of the one-box system. V V₂ V max max max x = 0 m (a) x = A m %0 = 0 m/s ₁x = 0 m m m (b) Figure 10.18 (a) A box of mass m, starting from rest at x = A, undergoes simple harmonic motion about x = 0 m. (b) When x = 0 m, a second box, with the same mass and speed, is attached. Answer why the Amplitude of the mass-spring system increased to 2 times the mass is sqrt(2) greater than that of the one-mass system and the angular frequency is sqrt(2) times less than that of the one-mass system. PROVE YOUR SOLUTION USING THE APPROPRIATE PHYSICAL EQUATIONS AND DEMONSTRATIONS DO IT STEP BY STEP THANK YOU!
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