How to find the velocity of a wave in simple harmonic motion given time

In summary, the conversation discusses the problem of finding the speed of a wave at a specific time using different formulas. The first formula used, v=w*xmax*cos(wt), does not give the correct result because it predicts that the velocity is maximum at t=0, while the correct formula, w*sqrt(xmax^2-x^2), takes into account the displacement at t=0. It is also noted that cos should be used for displacement when it is maximum at t=0, and sin should be used when displacement is 0 at t=0.
  • #1
MattDutra123
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Homework Statement
Find the speed of this longitudinal wave at t=0.12s.
Relevant Equations
v=w*xmax*cos(wt)
The graph provided is below. The problem asks for the speed of the wave at 0.12s. I used the formula v=w*xmax*cos(wt), provided in our textbook where xmax is the amplitude of 2 cm, w (omega) is 2pi divided by the period of 0.2. However, for some reason this formula doesn't give me the correct result. Instead, the solution to the problem involves the formula w*sqrt(xmax^2-x^2), which requires us to first find the displacement at t = 0.12.

My question is: why does the formula I used not work? Why must we use the other formula to solve this problem? I apologise for the bad formatting of the formulas.
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  • #2
It doesn't work because it's the wrong formula to use. It predicts that the velocity is maximum at t = 0. If you look at the graph, the position is maximum at t = 0 which means that the speed is zero at t = 0.

Also, is this a wave or a simple harmonic oscillator? I think the latter.
 
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  • #3
kuruman said:
It doesn't work because it's the wrong formula. It predicts that the velocity is maximum at t = 0. If you look at the graph, the position is maximum at t = 0 which means that the speed is zero at t = 0.

Also, is this a wave or a simple harmonic oscillator? I think the latter.
Why does the formula I used predict that velocity is maximum at t=0?
If I use the same formula but replacing cos with sin, I get the correct answer. Why is that? Is it related to what you said? We were taught to use cos when displacement is maximum at t = 0, and sine when displacement is 0 at t = 0. Is this correct?
 
  • #4
You wrote
##v(t)=\omega~ x_{max}\cos(\omega~t)##
At ##t = 0##, ##v(0)=\omega~ x_{max}\cos(0)=\omega~ x_{max}(1)\neq 0.##
Try this
1. Find an expression for ##x(t)## consistent with the graph. Hint: As indicated above, ##x_{max}\cos(0)=x_{max}.##
2. Take the derivative with respect to time to find ##v(t)##. The result should answer your second question.
3. Evaluate ##v(t=0.12~s).## This method is neater and there is no need to do what the solution suggests.

MattDutra123 said:
We were taught to use cos when displacement is maximum at t = 0, and sine when displacement is 0 at t = 0. Is this correct?
That is correct. Your mistake was that you used cos for the velocity; you were taught to use cos for the displacement.
 
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Related to How to find the velocity of a wave in simple harmonic motion given time

1. How is the velocity of a wave in simple harmonic motion calculated?

The velocity of a wave in simple harmonic motion can be calculated by dividing the wavelength (λ) by the period (T) of the wave. This can be expressed as v = λ/T.

2. What is the equation for calculating the period of a wave in simple harmonic motion?

The period of a wave in simple harmonic motion can be calculated using the equation T = 2π√(m/k), where m is the mass of the object and k is the spring constant.

3. How does the amplitude of a wave affect its velocity in simple harmonic motion?

The amplitude of a wave does not affect its velocity in simple harmonic motion. The velocity of a wave is only dependent on its wavelength and period.

4. Can the velocity of a wave in simple harmonic motion change over time?

No, the velocity of a wave in simple harmonic motion remains constant over time as long as the properties of the medium through which it is traveling do not change.

5. Is the velocity of a wave in simple harmonic motion affected by the mass of the object?

No, the velocity of a wave in simple harmonic motion is not affected by the mass of the object. It is only dependent on the properties of the medium and the restoring force provided by the spring.

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