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## Homework Statement

A 5.0 kg block hangs from a spring with spring constant 2000N/m. The block is pulled down 5 cm from equilibrium position and given an initial velocity of 1.0 m/s back toward equilibrium. What are the

a.) frequency

b.) amplitude

c.) total mechanical energy of the motion

## Homework Equations

oscillatory motion x(t) = Acos(ωt + ø

_{0})

f = ω/(2∏)

ω = √(k/m)

## The Attempt at a Solution

frequency found by ω = √(k/m)

ω = √(2000/5) = 20

f = 20/(2∏) = 10/∏

which I believe was correct.

The only way to find the amplitude is to relate it to the velocity, correct? Here I can't simply use .5kA

^{2}= .5m(V

_{max})

^{2}because at V

_{max}there should also be some gravitational potential. Used the motion equation and it's derivative:

x(t) = Acos(ωt + ø

_{0})

v(t) = -ωAsin(ωt + ø

_{0})

x(0) = .05 = A cos(ø

_{0})

=>ø

_{0}= cos

^{-1}(.05/A)

edit: here I made an error v(0) should be -20Asin(ø

_{0})

v(0) = -1 = -20A/sin(ø

_{0})

substituting...

-1 = -20Asin(cos

^{-1}(.05/A))

=>sin

^{-1}(1/(20A)) = cos

^{-1}(.05/A)

which is the same thing as saying (.05)

^{2}+ (.05)

^{2}= A

^{2}right? If so, A = .0707, which I can't verify at the moment but seems reasonable.

Where I get stuck is here:

The energy should be entirely kinetic at the bottom of the stretch given by .5k(A + Δx)

^{2}where Δx was the initial stretch of the spring caused by gravity before it was pulled. So I figured I could set it equal to the initial condition when the spring is displaced .05 m from equilibrium and the initial velocity is 1.0 m/s:

.5k(A + Δx)

^{2}= .5k(.05 + Δx)

^{2}+ mg(A-.05) + .5mV

_{0}

^{2}

Then I solved for Δx, in order to plug it back into .5k(A + Δx)

^{2}and I won't bore you with all that here but I've done it twice now and Δx comes out -.1549, which doesn't seem right. Plugging it into the equation negative or positive comes out as 26.6 and 223.6 respectively, and I believe the answer was supposed to be 5 as total mechanical energy.

After doing it again with the correct amplitude I got 8.94, a lot closer to 5.0 but still off...

If anyone sees my errors here I'd be super grateful!

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