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For undamped simple harmonic motion, the position of the mass as a function of time is: x(t) = A cos(wt + $0) + Xeq where xeq is the equilibrium position, the amplitude A is the maximum deviation from equilibrium, w is the angular frequency, and is the phase constant. The graph below is the position of a 50-gram mass on a spring as a function of time. a. Determine the following: Xeq (equilibrium position): A (amplitude): T (period): f (frequency): (angular frequency): AA x (cm) -2 0.4 time (s) b. Determining Phase Constant: Consider the form x(t) = A cos(wt+) + Xeq- If = 0, the first peak in the curve 0.2 0.6 0.8

For undamped simple harmonic motion, the position of the mass as a function of time is: x(t) = A cos(wt + $0) + Xeq where xeq is the equilibrium position, the amplitude A is the maximum deviation from equilibrium, w is the angular frequency, and is the phase constant. The graph below is the position of a 50-gram mass on a spring as a function of time. a. Determine the following: Xeq (equilibrium position): A (amplitude): T (period): f (frequency): (angular frequency): AA x (cm) -2 0.4 time (s) b. Determining Phase Constant: Consider the form x(t) = A cos(wt+) + Xeq- If = 0, the first peak in the curve 0.2 0.6 0.8

College Physics
College Physics
10th Edition
ISBN:9781285737027
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter13: Vibrations And Waves
Section: Chapter Questions
Problem 14P: An object-spring system moving with simple harmonic motion has an amplitude A. (a) What is the total...
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d. Write an expression for the x-position as a function of time. Use your results from (a) to include numerical values with units for all constant coefficients. e. Write an expression for the x-velocity as a function of time. Include numerical values with units for all constant coefficients. f. Write an expression for the x-acceleration as a function of time. Include numerical values with units for all constant coefficients. g. What is true of the x-velocity and x-acceleration when the x position is most positive? The x-velocity is: • most positive ◆ zero The x-acceleration is: • most positive ◆ zero h. What is true of the x-velocity and x-acceleration when x is increasing as it goes through x = xq? The x-velocity is: • most positive The x-acceleration is: • most positive ◆ zero most negative • most negative ◆ zero • most negative • most negative
Transcribed Image Text:d. Write an expression for the x-position as a function of time. Use your results from (a) to include numerical values with units for all constant coefficients. e. Write an expression for the x-velocity as a function of time. Include numerical values with units for all constant coefficients. f. Write an expression for the x-acceleration as a function of time. Include numerical values with units for all constant coefficients. g. What is true of the x-velocity and x-acceleration when the x position is most positive? The x-velocity is: • most positive ◆ zero The x-acceleration is: • most positive ◆ zero h. What is true of the x-velocity and x-acceleration when x is increasing as it goes through x = xq? The x-velocity is: • most positive The x-acceleration is: • most positive ◆ zero most negative • most negative ◆ zero • most negative • most negative
Activity 4.16 - Simple Harmonic Motion Graph For undamped simple harmonic motion, the position of the mass as a function of time is: x(t) = A cos(wt + o) + Xeq where xeq is the equilibrium position, the amplitude A is the maximum deviation from equilibrium, is the angular frequency, and is the phase constant. The graph below is the position of a 50-gram mass on a spring as a function of time. a. Determine the following: Xeq (equilibrium position): A (amplitude): C. T (period): f (frequency): w (angular frequency): AA и x (cm) -2 0 0.2 Flag =-2 T 0.4 time (s) 0.6 b. Determining Phase Constant: Consider the form x (t) = A cos(wt + n) + Xeq- If = 0, the first peak in the curve will occur at t = 0. So is related to how far past t=0 the first peak occurs. Define ting as the time at which the first peak occurs (at or after t = 0). The phase constant is given by 0.8 Note that this will give a negative phase constant. We can always get a positive phase constant by adding 2π since shifting the angle of a cosine function by 2π has no effect. Determine ting and the phase constant for the motion shown in the graph. tlag = On the previous activity, we found that the period (and therefore the frequency and angular frequency) only depends on the spring constant & and the mass m. Based on the graph, what is the spring constant?
Transcribed Image Text:Activity 4.16 - Simple Harmonic Motion Graph For undamped simple harmonic motion, the position of the mass as a function of time is: x(t) = A cos(wt + o) + Xeq where xeq is the equilibrium position, the amplitude A is the maximum deviation from equilibrium, is the angular frequency, and is the phase constant. The graph below is the position of a 50-gram mass on a spring as a function of time. a. Determine the following: Xeq (equilibrium position): A (amplitude): C. T (period): f (frequency): w (angular frequency): AA и x (cm) -2 0 0.2 Flag =-2 T 0.4 time (s) 0.6 b. Determining Phase Constant: Consider the form x (t) = A cos(wt + n) + Xeq- If = 0, the first peak in the curve will occur at t = 0. So is related to how far past t=0 the first peak occurs. Define ting as the time at which the first peak occurs (at or after t = 0). The phase constant is given by 0.8 Note that this will give a negative phase constant. We can always get a positive phase constant by adding 2π since shifting the angle of a cosine function by 2π has no effect. Determine ting and the phase constant for the motion shown in the graph. tlag = On the previous activity, we found that the period (and therefore the frequency and angular frequency) only depends on the spring constant & and the mass m. Based on the graph, what is the spring constant?
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