Simple Harmonic Motion: Displacement after s seconds

In summary, the conversation is discussing a bungy-jumper hanging from a bungy-cord and undergoing simple harmonic motion. The question is asking for the woman's position after a certain amount of time bouncing. The answer is 14.2m and the correct equation to use is y(t) = X cos(2 π t / T), taking into account the woman's starting position below the rest position. Radians must be used when dealing with simple harmonic motion, and the phase factor φ may not be necessary in this scenario.
  • #1
mexqwerty
13
0
The question is:

A 56.0 kg bungy-jumper hangs suspended from her bungy-cord, at rest. She is displaced from this position by 15.0 m downward, and then released. She bounces up and down, with a period of 5.800 s. Assume the woman undergoes simple harmonic motion, described by
y(t) = X cos(2 π t / T + φ)

Where is the woman after 43.20 s of bouncing? (enter a negative value if she is below her rest position).

The answer is 14.2m.

I've been using x=Xcos(2*pi*t/T) --> 15cos(2pi43.2/5.8) = 10.27m. This equation was in my book. What did I do wrong? Am I supposed to use y(t) = X cos(2 π t / T + φ? In that case, I don't know what φ is...
 
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  • #2
Are you using your calculator properly? (Radians, not degrees.)
 
  • #3
Oooh, thanks. I did not know you had to use radians. So do you have to always have to use radians when dealing with simple harmonic motion?
 
  • #4
mexqwerty said:
Oooh, thanks. I did not know you had to use radians. So do you have to always have to use radians when dealing with simple harmonic motion?
Generally, yes. But you can always convert from one to the other.

Note what's going on in the equation y(t) = X cos(2 π t / T). That ratio t/T tells you what fraction of a period you are dealing with. The 2π tells you that you are dealing with radians, since one complete period is 2π radians.
 
  • #5
Oh, that make sense. Thanks, that was really helpful. =)
 
  • #6
how do you know if the woman is above or below her rest position?
 
  • #7
booooo said:
how do you know if the woman is above or below her rest position?
Set up properly, the equation will tell you.

Or you can just see what fraction of a full period she ends up at.
 
  • #8
Do you mean by using this equation ?y(t) = X cos(2 π t / T + φ)
btw what isφ
 
  • #9
booooo said:
Do you mean by using this equation ?y(t) = X cos(2 π t / T + φ)
Yes.

btw what isφ
φ is a phase factor. But since the motion starts at maximum displacement below the rest position, you may not need it.

What final equation would you use to describe the position as a function of time?
 
  • #10
simply using y(t) = X cos(2 π t / T) ?
is that right
 
  • #11
booooo said:
simply using y(t) = X cos(2 π t / T) ?
is that right
Since she starts out below the rest position, I'd put a minus sign in front of that. Then you'd be fine.
 

1. What is Simple Harmonic Motion?

Simple Harmonic Motion is a type of periodic motion where the restoring force is directly proportional to the displacement from the equilibrium position and acts in the opposite direction of the displacement. This type of motion can be seen in various systems such as a mass on a spring or a pendulum.

2. How is displacement measured in Simple Harmonic Motion?

Displacement in Simple Harmonic Motion is measured as the distance from the equilibrium position to the current position of the object. It is usually represented by the variable 'x' and is measured in meters (m).

3. What is the formula for calculating displacement after s seconds in Simple Harmonic Motion?

The formula for calculating displacement after s seconds in Simple Harmonic Motion is:
x = A * cos(ω*t + φ)
Where:
A = amplitude (maximum displacement from equilibrium)
ω = angular frequency (2π/T)
T = time period (time for one complete oscillation)
φ = phase constant (initial phase angle)
t = time in seconds

4. How does the frequency affect displacement in Simple Harmonic Motion?

The frequency of Simple Harmonic Motion is directly related to the angular frequency (ω) and is inversely related to the time period (T). As the frequency increases, the angular frequency and displacement also increase, resulting in faster oscillations.

5. Can the displacement after s seconds ever be greater than the amplitude in Simple Harmonic Motion?

No, the displacement after s seconds can never be greater than the amplitude in Simple Harmonic Motion. The amplitude represents the maximum displacement from the equilibrium position, so the displacement after s seconds will always be equal to or less than the amplitude.

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