Conservation Of Energy, Block Sliding Down Incline Onto Spring.

[Creepypunguy]

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)mv²
Spring = (1/2)kx²

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)mv² + mgh = (1/2)mv² + mgx*sinθ

Moment 3: E = (1/2)kx²

-------- Attempts to get rid of x² led to
x = h/sin
(1/2)k(h/sinθ)² = mgsinθ(l+ x) ;;;
(kh²/2sin²θ) = mgsinθ(l+ x) ;;;
(kh²/2) = mglsin³θ + mgxsin³θ ;;;
(kh²/2) - mglsin³θ = mgxsin³θ ;;;

x = k*h²/2mgsin³θ - l

 

[Doc Al]

Moment 1: E = mg(H+h) = mgsinθ(l+ x)

Moment 2: E = (1/2)mv² + mgh = (1/2)mv² + mgx*sinθ

Moment 3: E = (1/2)kx²

This is good.

-------- Attempts to get rid of x² led to
x = h/sin
(1/2)k(h/sinθ)² = mgsinθ(l+ x) ;;;
(kh²/2sin²θ) = mgsinθ(l+ x) ;;;
(kh²/2) = mglsin³θ + mgxsin³θ ;;;
(kh²/2) - mglsin³θ = mgxsin³θ ;;;

x = k*h²/2mgsin³θ - l

Don't try to express things in terms of "h"--that's an unknown directly related to "x".

Hint: Just compare moments 1 and 3. Solve that equation!

 

[ehild]

As h=xsinθ, your last equation does not give x explicitly in terms of the known quantities, k, m, θ, L, and g.

ehild

 

[Creepypunguy]

So by solving equations 1 and 3 I got
mglsinθ+mgxsinθ = (1/2)kx2
0 = (k/2)x2 - mgsinθx - mglsinθ
quadratic formula

x = (mgsinθ ± √[mgsinθ(mgsinθ+2kl)] ) / k

 

[ehild]

It is all right now.

ehild