Distance spring decompresses when friction is involved

[becky_marie11]

Homework Statement

A block of mass m = 4.30 kg slides along a horizontal table with speed v0 = 8.50 m/s. At x = 0 it hits a spring with spring constant k = 68.00 N/m and it also begins to experience a friction force. The coefficient of friction is given by μ = 0.200. How far did the spring compress when the block first momentarily comes to rest?
m=4.30 kg
v_o=8.50m/s
k=68.00N/m
μ=.200
x=?

Homework Equations

K_f-K_i=W
K=1/2mv²
F_s=-kx
F_fric=μN_f
W=Fd

The Attempt at a Solution

First I found the work of the whole system (friction and spring forces) by using
K_f-K_i=W.
For the initial kinetic energy, I used
K=1/2mv²=155J.
For the final kinetic energy, I used 0, because I'm measuring the distance after the mass came to a stop. So, plugging the numbers into the Work equation, I simply get
W=-155J
I'm pretty sure I'm on the right track so far?
Then, using
W=Fd (just multiplying the forcexdistance because both vectors are in the same direction)
I plugged in:
W=(-kx+μN_f)x
I work it out to a quadratic by plugging the numbers in...
-68x²+8.44366x+155=0
I solve then for x (using the quadratic root finder on my calculator, so I'm not screwing up the quadratic..), but both of the answers it gives me are incorrect..
What am I doing wrong? Help please!

 

[Doc Al]

Then, using
W=Fd (just multiplying the forcexdistance because both vectors are in the same direction)

Careful: That only works when the force is constant. The spring force is not constant.

Hint: What's the energy stored in a compressed spring?

Also: Careful with signs. The work done on the mass by both the spring and the friction force is negative.

 

[tiny-tim]

welcome to pf!

hi becky! welcome to pf!

isn't it 1/2 kx² ? ​

 

[xlava]

hi becky! welcome to pf!

isn't it 1/2 kx² ? ​

W = ∫Fdx (F dot dx, technically)
Fspring = kx

W = ∫kxdx
Wspring = kx²/2

Just to show where it came from, since W does NOT equal Fx, that is not a definition, it is a derivation of work for a constant force.

 

[becky_marie11]

Thank you guys! I finally figured out the right answer! But when I solve for the quadratic equation, why is the distance vector only distributed to the friction force? This is what I tried to do...
W=(-1/2kx^2+friction force)*x
and this is what is right...
W=-1/2kx^2+friction force*x
Is it because the work of the spring is already in terms of "work" and the friction force isn't?

 

[tiny-tim]

hi becky!

This is what I tried to do...
W=(-1/2kx^2+friction force)*x
and this is what is right...
W=-1/2kx^2+friction force*x
Is it because the work of the spring is already in terms of "work" and the friction force isn't?

yes!

work done is force "dot" distance, and 1/2 kx² is 1/2 kx (average force) times x (distance)!

 

[becky_marie11]

Thank you!