# Spring + Friction system equation

1. Nov 25, 2003

### Millenium

The question is An object of mass m is traveling on a horizontal surface. There is a coefficient of kinetic friction u between the object and the surface. The object has speed when it reaches x = 0 and encounters a spring. The object compresses the spring, stops, and then recoils and travels in the opposite direction. When the object reaches x = 0 on its return trip, it stops.

The question then asks me to find the spring constant k.

The answer to the problem has to be in terms of the following symbols:
u = coefficient of kinetic friction
m = mass of block
g = acceleration due to gravity
v = initial velocity of the block

I used conservation of energy
E(final) = E(initial) + W(nonconservative forces)

E(final) = 0 because the block is at rest (i think)

E(initial) if trying to find k could probably only be found if E(initial) = 1/2*k*x^2 right?

and then the W(nonconservative forces) = -m*g*u*x

then in order to find x I'd have to find how far the block would go with initial velocity v along the surface with u being the friction. So I'd get x = (v^2)/(2*u*g).

And then plugging everything back in and solving for k i get:

(4*g^2*m*u^2)/(v^2)

I figured that'd be the answer but when I put it into the online answer it says that its off by a multiplicative factor. Where did I go wrong?

2. Nov 25, 2003

### Staff: Mentor

This confuses me a bit. Let me state it my way and see if we agree:
There are three points of interest: initial (speed = v; x = 0), compressed (v = 0; x = x), final (v = 0, x = 0).
Einitial = 1/2 m v2. (I assume v is the speed of the object when it first reaches x=0.)
Ecompressed = 1/2 k x2
Efinal = 0 (it's not moving and the spring is unstretched).

First compare Ecompressed with Efinal:
Ecompressed = Efinal + Wagainst friction;
I get: 1/2 k x2 = 0 + &mu;mgx
(solve that for x)

Next compare Einitial with Ecompressed;
Einitial = Ecompressed + Wagainst friction;
I get: 1/2 m v2 = 1/2 k x2 + &mu;mgx

Plug in the value for x from the first equation and solve for k.

That's fine, just be careful with signs when you use it.

Not sure what you're doing here. Seems like you're finding how far the object would slide before coming to rest if it started with initial speed v. This is not relevant. (See my analysis above.)
Go over it again; you're almost there.

3. Nov 25, 2003

### Millenium

thank you very much I just equated that it should be (8*g^2*m*u^2)/(v^2) and it was right.