A large fake cookie sliding on a horizontal surface is attached to one end of a

Question

A large fake cookie sliding on a horizontal surface is attached to one end of a horizontal spring with spring constant k = 410 N/m; the other end of the spring is fixed in place. The cookie has a kinetic energy of 19.0 J as it passes through the spring's equilibrium position. As the cookie slides, a frictional force of magnitude 9.00 N acts on it. (a) How far will the cookie slide from the equilibrium position before coming momentarily to rest? (b) What will be the kinetic energy of the cookie as it slides back through the equilibrium position?

Explanation / Answer

The initial energy state of the system is 19 J. At equilibrium state, the spring contains only the KE.

As the cookie slides, the energy is converted from KE into spring PE and frictional heat. If there were no friction, the cookie would move until PE = 19J

where spring constant k = 410N/m

The work done by friction is F × d. So the stopping point is when the work done by friction plus the spring PE sums to the initial energy.

PE (spring) = KE – W (friction)
KE = PE (spring) + F× d

25 J = .5 * 410N/m * x^2 + 9 N x
205 N/m x^2 + 9 N x - 25 N m = 0
This has a positive root at x = 0.281 m. Since that's less than the F=0 case of 0.304 m.

To cross verify

E(tot) = 1/2 × 410N/m × (0.281 m)2 + 9 N (0.281 m)
E(tot) = 16.18 J + 2.52J
E(tot) = 19 J

b) The same thing happens, so again we subtract the energy done by friction.
KE = PE (spring) – W (friction)
KE = 16.18 J - 2.52J
KE = 14 J

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