Find the amplitude and phase shift of an oscillating spring?

[mybluesock]

Homework Statement

A 205 g mass attached to a horizontal spring oscillates at a frequency of 4.60 Hz. At t =0s, the mass is at x= 6.00 cm and has v_x =- 34.0 cm/s. Determine Amplitude and Phase Shift.

Homework Equations

x(t)=Acos($$\omega$$*t)
$$\omega$$=(2*$$\pi$$)/T
T=1/f

The Attempt at a Solution

x(t)=Acos($$\omega$$*t), so A=x/cos($$\omega$$*t).
$$\omega$$=(2*$$\pi$$)/T
T=1/f, or 1/4.60Hz. So $$\omega$$=28.903 rad/s.
A=(.06m)/cos(28.903rad/s*0s)=.06m.

I think this isn't working because I don't know the phase shift, but I'm not sure how to find the phase shift if I don't know A. Is there another formula I need?

 

[anathema]

Thank you for posting your question. I am a scientist and I would be happy to help you with your problem.

To determine the amplitude and phase shift of the oscillating mass attached to a horizontal spring, we can use the following equations:

x(t) = A*cos(ωt + φ)

v(t) = -A*ω*sin(ωt + φ)

where A is the amplitude, ω is the angular frequency, t is time, and φ is the phase shift.

From the given information, we know that the mass has a frequency of 4.60 Hz, which corresponds to an angular frequency of ω = 2πf = 2π*4.60 Hz = 28.903 rad/s. We also know that at t = 0s, the mass is at x = 6.00 cm and has a velocity of v_x = -34.0 cm/s.

Plugging these values into the equations, we get:

x(t) = A*cos(28.903t + φ)

v(t) = -A*28.903*sin(28.903t + φ)

We can use the given initial conditions to solve for A and φ. At t = 0s, x = 6.00 cm, so:

6.00 cm = A*cos(28.903*0 + φ) = A*cos(φ)

Similarly, at t = 0s, v = -34.0 cm/s, so:

-34.0 cm/s = -A*28.903*sin(28.903*0 + φ) = -A*28.903*sin(φ)

Dividing the second equation by the first, we get:

34.0/6.00 = 28.903*sin(φ)/cos(φ) = 28.903*tan(φ)

Solving for φ, we get:

φ = arctan(34.0/6.00)/28.903 = 0.415 rad

Now, we can use this value of φ to solve for A:

6.00 cm = A*cos(0.415) = A*0.920 = 5.52 cm

Therefore, the amplitude of the oscillation is A = 5.52 cm and the phase shift is φ = 0.415 rad.

I hope this helps. Let me know if you have any further questions or