Understanding the Phase Constant in Simple Harmonic Motion

In summary, the displacement of a mass oscillating on a spring is described by the equation x(t) = xmcos(ωt + ). The phase constant, which determines the initial displacement and velocity when t = 0, can be found by sketching a cosine curve and determining its initial value, matching it to the initial condition of the spring and mass, and finding the angular offset from zero.
  • #1
1MileCrash
1,342
41

Homework Statement



The displacement of a mass oscillating on a spring is given by x(t) = xmcos(ωt + ). If the initial displacement is zero and the initial velocity is in the negative x direction, then the phase constant is:

Homework Equations





The Attempt at a Solution



How do I start? The book just tells me that the phase constant depends on displacement and velocity when t = 0, but doesn't say how.
 
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  • #2
Sketch a cosine curve. What's its initial value? Where on the curve would match the initial condition of the spring and mass? What's (angular) the offset from zero?
 
  • #3
gneill said:
Sketch a cosine curve.

OK

What's its initial value?

1

Where on the curve would match the initial condition of the spring and mass?

Huh??
 
  • #4
Does the mass start at a maximum extension like the cosine function does?
 
  • #5
No, initial displacement is 0. So, I need to find where cosx equals 0?
 
  • #6
1MileCrash said:
No, initial displacement is 0. So, I need to find where cosx equals 0?

Not only that, but where it's going through zero and going negative, just like the mass' displacement.
 
  • #7
Still have no clue on this.
 
  • #8
Have a gander:

attachment.php?attachmentid=40822&stc=1&d=1320887764.jpg
 

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FAQ: Understanding the Phase Constant in Simple Harmonic Motion

What is the phase constant of SHM?

The phase constant of SHM (Simple Harmonic Motion) is the initial phase angle at which an oscillating object starts its motion. It is represented by the symbol φ and is measured in radians.

How is the phase constant related to the initial conditions of a SHM system?

The phase constant is determined by the initial conditions of a SHM system, such as the initial displacement, velocity, and acceleration of the object. It helps to describe the starting position and direction of the motion.

What is the difference between phase constant and phase angle?

The phase constant and phase angle are often used interchangeably, but they have different meanings. The phase constant is a fixed value that describes the initial phase of an oscillating object, while the phase angle can change over time and represents the current phase of the motion.

How does the phase constant affect the motion of an object in SHM?

The phase constant determines the starting position and direction of the motion, and it also affects the amplitude and frequency of the oscillations. A change in the phase constant can result in a phase shift or change in the starting position of the motion.

Can the phase constant be negative?

Yes, the phase constant can be negative. It depends on the starting position and direction of the motion. A negative phase constant indicates that the object starts its motion in the opposite direction of the equilibrium position.

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