# Simple harmonic motion period with velocity defined

• PeterRV
In summary, the period of a simple harmonic motion (SHM) is given by T=2π√((A^2-L^2)/vL)), where A is the amplitude, L is the distance from equilibrium, and vL is the speed at that point. By substituting the values of x and v from the equations for SHM, it is possible to relate them and solve for the period. This can also be done using the conservation of energy equation, resulting in a formula for k that includes the mass, speed, and amplitude of the motion.

## Homework Statement

A mass m is sliding back and forth in a simple harmonic motion (SHM) with an amplitude A on a horizontal frictionless surface. At a point a distance L away from equilibrium, the speed of the plate is vL (vL is larger than zero).

## Homework Equations

What is the period of the SHM?

## The Attempt at a Solution

a_x=-kx/m -> vX= (-kx^2)/(2m)
k = (-vX*2m)/(x^2)
T=2π*√(m/k)=2π*√(m/((-vX*2m)/(x^2))
T=2π√((-x^2)/vX))
Filling in point at distance L from equilibrium, I get:

T=2π√((-L^2)/vL))

The correct answer is T=2π√((A^2-L^2)/vL)), but I cannot imagine where the A^2 comes from.

Any help is appreciated!

For simple harmonic motion,
x=Asin(wt+Φ)
and v=dx/dt = Awcos(wt+Φ)

Can you try relating x and v somehow?

erisedk said:
For simple harmonic motion,
x=Asin(wt+Φ)
and v=dx/dt = Awcos(wt+Φ)

Can you try relating x and v somehow?

Yes, x=∫vdt, or is that not what you meant with relating x and v?

Nope, I meant try substituting the value of (wt+Φ) from the first equation into the second one.

erisedk said:
Nope, I meant try substituting the value of (wt+Φ) from the first equation into the second one.
I am terribly sorry, but my native language is not English and I do not know what you mean with the value of (wt+Φ). I do not have the frequency in the data?

(wt+Φ) = arcsin(x/A)
What is cos(wt+Φ)?
After figuring out cos(wt+Φ), substitute it into v=Awcos(wt+Φ). You'll get an expression that relates x and v.

Ik think you have to apply conservation of energy, it will result in:
1/2*k*L^2 + 1/2*m*v^2=1/2*k*A^2
k =(m*v^2)/(A^2 -L^2)