Conservartion of Energy - spring problem

[Nynex]

Homework Statement

A large cookie sliding on a horizontal surface is attached to one end of a horizontal spring with spring constant k = 380 N/m; the other end of the spring is fixed in place. The cookie has a kinetic energy of 19.5 J as it passes through the position where the spring is unstretched. As the cookie slides, a frictional force of magnitude 10.0 N acts on it.

(a) How far will the cookie slide from the position where the spring is unstretched before coming momentarily to rest?

Homework Equations

Unsure

The Attempt at a Solution

All the spring potential energy is converted into kinetic energy when the spring reaches its equilibrium position.

Then, the spring undergoes kinetic friction, releasing thermal energy, which means all the energy at the equilibrium position will be converted into thermal energy.

Therefore:

20 = 10d
d = 2

Is this correct? I'm unsure, it seems like I'm missing something...

 

[Doc Al]

Is this correct? I'm unsure, it seems like I'm missing something...

No. You forgot about spring PE as the spring is stretched. Initially, at the unstretched position, there is only KE. That energy ends up as both thermal energy and spring PE.

 

[Nynex]

Kinetic Energy = Thermal Energy + Potential Spring Energy
19.5 = 10d + 1/2kd²
0 = 180d² + 10d -19.5
d = 0.30 m, -0.36 m

Is this correct? I think it makes sense...the spring isn't displaced as far to the right due to friction.

 

[Doc Al]

Kinetic Energy = Thermal Energy + Potential Spring Energy
19.5 = 10d + 1/2kd²
0 = 180d² + 10d -19.5
d = 0.30 m, -0.36 m

Is this correct? I think it makes sense...the spring isn't displaced as far to the right due to friction.

Yes, perfectly correct. Note: Only one of the solutions to that quadratic is physically meaningful for this problem. Which one?

 

[Nynex]

It would be 0.30 m, it's initially compressed by 0.36 m to the left, the friction is what stops the spring from fully stretching out by 0.36 m to the right. Thanks for all the help!

 

[Doc Al]

It would be 0.30 m,

Right.

it's initially compressed by 0.36 m to the left, the friction is what stops the spring from fully stretching out by 0.36 m to the right.

Not really. A negative value for d makes no sense, since that would mean that the friction actually increases the energy. So we just ignore that solution to the quadratic.