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## Homework Statement

A block of mass

*m*starts from rest at a height

*h*and slides down a frictionless plane inclined at angle

*θ*with the horizontal, as shown below. The block strikes a spring of force constant

*k*.

Find the distance the spring is compressed when the block momentarily stops. (Let the distance the block slides before striking the spring be

*l*. Use the following as necessary:

*m*,

*θ*,

*k*,

*l*, and

*g*.)

## Homework Equations

Conservation of Energy:

GPE : U =

*mgh*

K = (1/2)

*mv*

^{2}

Spring = (1/2)

*kx*

^{2}

## The Attempt at a Solution

My Diagram:

Let H =

*l**sinθ

Let h =

*x**sinθ

Moment 1: E = mg(H+h) = mgsinθ(

*l*+

*x*)

Moment 2: E = (1/2)mv

^{2}+ mgh = (1/2)mv

^{2}+ mg

*x**sinθ

Moment 3: E = (1/2)kx

^{2}

-------- Attempts to get rid of x

^{2}led to

x = h/sin

(1/2)k(h/sinθ)

^{2}= mgsinθ(

*l*+

*x*) ;;;

(kh

^{2}/2sin

^{2}θ) = mgsinθ(

*l*+

*x*) ;;;

(kh

^{2}/2) = mg

*l*sin

^{3}θ + mg

*x*sin

^{3}θ ;;;

(kh

^{2}/2) - mg

*l*sin

^{3}θ = mg

*x*sin

^{3}θ ;;;

x = k*h

^{2}/2mgsin

^{3}θ -

*l*