Simple harmonic motion interpretation problem

In summary, the expression ##t= \frac{1}{\omega} cos^{-1}(x/A)## comes from the SHM equation ##x = Acos(\omega t)##, where ##A## is the amplitude, ##t## is time, and ##x## is displacement. When ##x = 0##, ##t = \frac{\pi}{2\omega}##, but this does not necessarily mean there is no movement. This is because ##\cos^{-1}## has multiple roots and can result in a displacement equal to the amplitude, not zero. To have ##x=0## at ##t=0##, the equation should use ##\sin## instead of ##\cos##
  • #1
Celso
33
1
I'm in trouble trying to understand the expression ##t= \frac{1}{\omega} cos^{-1}(x/A)## that comes from ##x = Acos(\omega t)##, in which ##A## is the amplitude, ##t## is time and ##x## is displacement.
When ##x = 0##, ##t = \frac{\pi}{2\omega} ##, shouldn't it be 0 since there was no movement?
 
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  • #2
##\cos^{-1}## has multiple roots.
 
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  • #3
Dale said:
##\cos^{-1}## has multiple roots.
It has a root in ##x/A = 1##, but in that case the distance would be equal to the amplitude, not zero
 
  • #4
You seem to be using a form of the SHM equation that treats ##t=0## as a time when ##x## is a maximum. If you want ##x=0## at ##t=0## you need to use ##\sin##, not ##\cos##.
 
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  • #5
Yes, @Ibix is right. In this equation x is the displacement from equilibrium, not the displacement from t=0. It starts at the peak.
 
  • #6
ah that's my mistake, thank you guys
 

Related to Simple harmonic motion interpretation problem

1. What is simple harmonic motion?

Simple harmonic motion is a type of periodic motion in which an object moves back and forth in a straight line with a constant frequency and amplitude. This type of motion is often seen in systems that have a restoring force and no friction, such as a mass on a spring or a pendulum.

2. How is simple harmonic motion interpreted in real-world situations?

In real-world situations, simple harmonic motion can be seen in various systems such as the motion of a pendulum, the vibrations of a guitar string, or the oscillations of a car suspension. It is also used in engineering and design to create stable and efficient systems.

3. 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, and φ is the phase constant. This equation can also be written as x = A*cos(ωt + φ) depending on the starting position of the object.

4. How is the period of simple harmonic motion calculated?

The period of simple harmonic motion is the time it takes for one complete cycle of motion. It can be calculated using the equation T = 2π/ω, where T is the period and ω is the angular frequency. The period is typically measured in seconds.

5. How does the mass and spring constant affect simple harmonic motion?

The mass and spring constant both play a role in determining the frequency and period of simple harmonic motion. A larger mass will result in a lower frequency and longer period, while a larger spring constant will result in a higher frequency and shorter period. These factors also affect the amplitude of the motion.

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