How do you find the initial phase angle?

[nukeman]

Homework Statement

A mass is hanging on a vertical spring, and oscillates in SHM with an amplitude of 10.0 cm, and period 0.500 s. The graph shows its motion as a function of time. At t = 0, the mass is found at x = -7.50 cm below the equilibrium position.

Assuming that x(t) = A cos (ωt + φ), find the value of the initial phase angle, φ.

Homework Equations

The Attempt at a Solution

How do I get started in finding the initial phase angle, φ?

 

[Delphi51]

Set x(0) = -7.5 cm and solve for the angle. You know the A and the w, so only one unknown.

 

[nukeman]

Set x(0) = -7.5 cm and solve for the angle. You know the A and the w, so only one unknown.

Im sorry I don't quite understand. It has to be x(t) = A cos (ωt + φ)

So don't I have to figure out A, ω, and t ? then just solve for φ ?

How come you just put x(0) = -7.5cm ? (I understand why you put x(0))

Thanks very much for the help!

 

[Delphi51]

Sorry, I should have written x(0) = A cos(ω*0 + φ) = -7.5 cm.
You WILL have to calculate numbers for ω and A before you can solve for φ.

 

[asteriatic]

As a scientist, the initial phase angle, φ, can be found by using the given information and equations for simple harmonic motion. First, we can use the equation x(t) = A cos (ωt + φ) to represent the motion of the mass on the spring. We know that the amplitude, A, is equal to 10.0 cm and the period, T, is equal to 0.500 s. We can also determine the angular frequency, ω, by using the equation ω = 2π/T, which gives us ω = 12.57 rad/s.

Next, we can use the given information that at t = 0, the mass is found at x = -7.50 cm below the equilibrium position. This means that at t = 0, the mass has a displacement of -7.50 cm from the equilibrium position. Plugging this into our equation for x(t), we get -7.50 cm = 10.0 cm cos (ω(0) + φ), which simplifies to -0.75 = cos φ.

To solve for φ, we can use the inverse cosine function to find the angle whose cosine is -0.75. This gives us an initial phase angle, φ, of approximately 135 degrees or 2.36 radians.

In conclusion, the initial phase angle, φ, can be found by using the given information and equations for simple harmonic motion. It is important to note that the initial phase angle can vary depending on the starting position of the mass, but it can always be determined using the given information.