Calculating ∅ in Simple Harmonic Motion Equation x = xo + Asin(ωt + ∅)

In summary: So:phi = cos-1(0.23674) - 13.5phi = -12.321In summary, to find the value of ∅ in the equation x = xo + Asin(ωt + ∅), take the inverse sine of both sides and solve for ∅. Then, for the equation x(t) = A(cos ωt + ∅), take the inverse cosine of both sides and solve for ∅.
  • #1
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x = xo + Asin(ωt + ∅)
I know all of the values, except for ∅.
But i don't know how to get the value for ∅


The attempt at a solution:
0.349 = 0.367 + 0.413Sin(8.5 * 0.8938 + ∅)
-0.018 = 0.413Sin(7.5973 + ∅)
-0.04358 = Sin(7.5973 + ∅)
From here on, i am confused, I tried using the trigonometric identities.
-0.04358 = Sin7.5973Cos∅+Cos7.5973Sin∅
-0.04358 = 0.1322Cos∅ + 0.9912Sin∅
And once again, i am stuck. I don't know what to do, to get ∅. Please help, thanks.
 
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  • #2
1. solve the equation for phi *before* substituting the values.
2. substitute the values

the opposite of sine is arcsine, on your calculator it is sin^-1 and is often accessed using [inv]+[sin] or similar, so:

arcsin(x-x0)-wt=\phi
 
  • #3
pvpkillerx said:
x = xo + Asin(ωt + ∅)
I know all of the values, except for ∅.
But i don't know how to get the value for ∅The attempt at a solution:
0.349 = 0.367 + 0.413Sin(8.5 * 0.8938 + ∅)
-0.018 = 0.413Sin(7.5973 + ∅)
-0.04358 = Sin(7.5973 + ∅)
From here on, i am confused, [...snip...]
Take the inverse sine of both sides of the equation. The inverse sine is sometimes called ARCSIN, Arcsin, asin, Asin, ASIN, arc sine, or sin-1().

Your calculator should have this function. If not, maybe your computer's calculator has it. For example on Windows 7, go to the accessories start menu folder, and open the calculator. Click on View / Scientific. Select either radians or degrees, depending one which units you are working with for this problem. Enter the "-0.04285" value. Click the inverse button. Then click on sin-1.
 
  • #4
Thanks, i feel stupid now (: Can't believe i didn't see that. Thank you!
 
  • #5
trig functions can be intimidating - a lot gets hidden in those letters.
 
  • #6
I'm having a similar problem. I need to find the phase constant for the function
x(t) = A(cos ωt + phi) so that I can find x when t = 0. When t = 2, x = 0.125. I found the amplitude, but I need the phase constant.

My values: 0.125 = 0.528 cos [(6.75)(2) + phi]

The attempt at a solution
I multiply 6.75 by 2 to get 13.5: 0.125 = 0.528 cos (13.5 + phi)
I then divide by 0.258: 0.23674 = cos (13.5 + phi)
At that point, I don't know what else to do. Any help would be appreciated.
 
  • #7
Your next step is to take the inverse cosine of both sides.
This is called arccos or cos-1.

cos-1(0.23674) = 13.5 + phi

it is the acos function in gnu-octave and is accessed via [inv]+[cos] on most scientific calculators.
 

What is Simple Harmonic Motion?

Simple Harmonic Motion (SHM) is a type of periodic motion in which an object moves back and forth in a regular, repeating pattern. This motion is due to the object being acted upon by a restoring force that is directly proportional to its displacement from its equilibrium position.

What are the characteristics of Simple Harmonic Motion?

The characteristics of Simple Harmonic Motion include a constant period, constant amplitude, and a sinusoidal displacement graph. In addition, the motion is always directed towards the equilibrium position and the acceleration is directly proportional to the displacement and in the opposite direction.

What is the equation for Simple Harmonic Motion?

The equation for Simple Harmonic Motion is x = A sin(ωt + φ), where x is the displacement of the object, A is the amplitude, ω is the angular frequency, t is time, and φ is the phase angle. This equation can also be written as x = A cos(ωt + φ) if the object starts at its maximum displacement instead of its equilibrium position.

What is the relationship between Simple Harmonic Motion and the force acting on the object?

In Simple Harmonic Motion, the restoring force acting on the object is directly proportional to the displacement from the equilibrium position. This can be expressed as F = -kx, where F is the force, k is the spring constant, and x is the displacement. As the object moves further from its equilibrium position, the force increases in the opposite direction, leading to oscillatory motion.

What are some real-life examples of Simple Harmonic Motion?

Some real-life examples of Simple Harmonic Motion include a pendulum swinging back and forth, a mass on a spring bouncing up and down, and a tuning fork vibrating. Other examples include the motion of a swing, the motion of a mass attached to a rubber band, and the motion of a mass floating on a liquid surface. Simple Harmonic Motion can also be observed in the motion of waves, such as ocean waves or sound waves.

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