Determining angular frequency and amplitude for SHM

[Elvis 123456789]

Homework Statement

A mass "m" is attached to a spring of constant "k" and is observed to have an amplitude "A" speed of "v₀" as it passes through the origin.

a) What is the angular frequency of the motion in terms of "A" and "v₀"?

b) Suppose the system is adjusted so that the mass has speed "v₁" at position "x₁" and speed "v₂" at position "x₂". Find the angular frequency, "ω₀", and amplitude "A" in terms of these given quantities. Simplify your results.

Homework Equations

x = A*sin(ω₀*t -φ)

v = A*ω₀*cos(ω₀*t -φ)

The Attempt at a Solution

so for the first part

x(t₀) = A*sin(ω₀*t₀ -φ) = 0

therefore φ = ω₀*t₀

v(t₀) = A*ω₀*cos(ω₀*t₀ -φ) = v₀

therefore ω₀ = v₀/A

for part b.)

x₁ = A*sin(ω₀*Δt₁)
v₁ = A*ω₀*cos(ω₀*Δt₁)

x₂ = A*sin(ω₀*Δt₂)
v₂ = A*ω₀*cos(ω₀*Δt₂)

where Δt₁ = t₁ - t₀
and Δt₂ = t₂ - t₀theres four unknowns and four equations so I know I should be able to solve for ω₀ and A, but I can't seem to untangle all these trig functions. any hints?

 

[gneill]

I suggest taking a look at an energy conservation approach. Both the KE and PE expressions square their variable so position and velocity directions don't matter there.

 

[Elvis 123456789]

I suggest taking a look at an energy conservation approach. Both the KE and PE expressions square their variable so position and velocity directions don't matter there.

Ok so using conservation of energy

KE₁ + PE₁ = KE₂ + PE₂

1/2*m*v₁² + 1/2*k*x₁² = 1/2*m*v₂² + 1/2*k*x₂²

canceling the halves and dividing by "m" on both sides

v₁² + ω₀²*x₁² = v₂² + ω₀²*x₂²

then rearranging for ω₀
ω₀ = sqrt [ (v₁² - v₂²)/(x₂² - x₁²)

for the amplitude I use the fact that the total energy at any time is equal to 1/2*k*A²

so KE₁ + PE₁ = 1/2*k*A²

plugging in and rearranging I get

A = sqrt [ x₁² + v₁² * (x₂² - x₁²)/(v₁² - v₂²) ]

does this look okay?

 

[gneill]

That looks good.

For the amplitude you could also have chosen just one of the given (x,v) pairs to find the total energy. The resulting expression might be a bit cleaner...

 

[Elvis 123456789]

That looks good.

For the amplitude you could also have chosen just one of the given (x,v) pairs to find the total energy. The resulting expression might be a bit cleaner...

I don't quite understand what you mean. I chose just x₁ and v₁ but x₂ and v₂ factor in because I had to divide by the mass again to get the angular frequency since they didn't want the answer in terms of "k"

 

[gneill]

You're right, I missed that stipulation. Mind you, if you find $\omega_o$ first, you should be able to use that in the amplitude expression. You should be able to reach:

$A = \sqrt{\left(\frac{v}{\omega_o}\right)^2 + x^2}$

 

[Elvis 123456789]

You're right, I missed that stipulation. Mind you, if you find $\omega_o$ first, you should be able to use that in the amplitude expression. You should be able to reach:

$A = \sqrt{\left(\frac{v}{\omega_o}\right)^2 + x^2}$

Yes, that's the expression that I arrived at before I plugged in the corresponding values. Thanks for the help :)