# Phase Difference with Initial Conditions for SHM

## Homework Statement

A mass-spring system with a natural frequency of 3.6 Hz is started in motion with an initial displacement from equilibrium of 6.1 cm and an initial velocity of 0.7 m/s. What is the value of ϕ?
(Question aside: Finding the amplitude of the resulting function?)

x=Acos(ωt+ϕ)

## The Attempt at a Solution

Well, I want to say that if we didn't have a initial velocity, then the phase would be 0 rad, because it would simply start at a maximum, and with the cos function, a maximum at time=0 would be 0 phase, so the introduction of the initial condition is having me at a stand still.

I would also like to find the amplitude of the resulting motion, and was wondering where to start on this, since the initial displacement cannot be taken to be the amplitude because of the initial velocity in the system.

andrewkirk
Homework Helper
Gold Member
The system is underdetermined (unsolvable) because we don't know the mass.

If we had the mass, we could proceed as follows:

1. Use the frequency and the mass to calculate the spring constant.
2. use the spring constant to write a formula for the potential energy as a function of displacement from the equilibrium point.
3. write an equation expressing the potential energy at maximum displacement equal to potential energy at initial displacement plus initial KE
4. solve to get the amplitude
5. You now have ω and A, so you can use those in your equation above, together with t=0 and x(0)=6.1cm to calculate the phase ϕ.

To see why it's underdetermined, consider that, if the mass is very light, and the initial velocity is outwards, the mass will be very close to its maximum amplitude, as the kinetic energy will be exhausted by even a very small further extension of the spring. If the mass is heavy, it will travel much farther out before the KE runs out. These scenarios correspond to different phases.

Last edited:
1. Use the frequency to calculate ##\sqrt{\frac{k}{m}}## where ##k## is the spring constant.
2. use the spring constant to write a formula for the potential energy as a function of displacement from the equilibrium point.
3. write an equation expressing the potential energy at maximum displacement equal to potential energy at initial displacement plus initial KE
4. solve to get the amplitude
5. You now have ω and A, so you can use those in your equation above, together with t=0 and x(0)=6.1cm to calculate the phase ϕ.

Wouldn't you have to have the mass of the system that is oscillating in order to find the spring constant of the system; this equation;
ω_0 = root(k/m) with ω_0 being the natural frequency.

andrewkirk
Homework Helper
Gold Member
Wouldn't you have to have the mass of the system that is oscillating in order to find the spring constant of the system; this equation;
ω_0 = root(k/m) with ω_0 being the natural frequency.
Quite right. I must have been typing a correction on this at the same time as you were pointing it out.

Quite right. I must have been typing a correction on this at the same time as you were pointing it out.

--So how would I go about finding the spring constant using the natural frequency with the equation we both suggested?--

EDIT: Did not see your edit. Dismiss comment above.

Don't see how this is particularly unsolvable, since it is on a assignment that I am working on, will check back later.

rude man
Homework Helper
Gold Member
--So how would I go about finding the spring constant using the natural frequency with the equation we both suggested?--

EDIT: Did not see your edit. Dismiss comment above.

Don't see how this is particularly unsolvable, since it is on a assignment that I am working on, will check back later.
The original problem, finding φ, is solvable. The "question aside" is not.
Solve the 2nd order linear, constant-coefficient ODE with the two initial conditions included. Wind up with x = A cos(ωt + φ). ω is known, A is not determinable.

The original problem, finding φ, is solvable. The "question aside" is not.
Solve the 2nd order linear, constant-coefficient ODE with the two initial conditions included. Wind up with x = A cos(ωt + φ). ω is known, A is not determinable.

I do understand how to find the phase value now, thanks for the input!