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Unformatted text preview: n at point C. In this region there is a coefficient of kinetic friction, so the change in energy must be equal to the work done by the nonconservative force. Hence we can write ΔE = ΔKE + ΔPE = W NC Since the block starts at the y=0 position and ends on the y=0 position, the change in potential energy must be zero. To find the work done by the nonconservative force, again we draw a freebody diagram for the block. d
N € ∑F y =0 FN − mg = 0 fk €
mg
€ From the freebody diagram, we can see that the displacement d is in the opposite direction the frictional force, se we can write
W NC = f k d cos(180 0 ) = − f k d = −µk mgd In the second line we used the definition of the frictional force.
€ Putting everything together we can write
1 21 2 mv − mv 0 = −µk mgd 2 2 € Solving for the velocity gives
2 v = v 0 − 2µk mgd Section C to delta x. Here there is no frictional force, so the system is conservative. The initial velocity here is just the velocity above, € so we can write ΔE = ΔKE + ΔPE = 0
Since the spring is maximally compressed, this tells us that the final velocity of the block must be zero. So we have € 1 1 2 − mv 2 + k (Δx ) = 0 2 2 Solving for the spring constant gives
€ mv 2 k= 2 (Δx ) = 2 mg( R − µk d ) (Δx ) 2 612.5 N / m In the second line we used the two expression for the velocities.
€ For the second way we just use the entire range, using the same definition for the y=0 position. So we can write ΔE = ΔKE + ΔPE = W NC
Since the block starts at rest and ends at rest, the change in kinetic energy is zero. So we can write € 12 kx − mgR = −µk mgd 2 Solving for the spring constant yields € 2 mg(R − µk d ) k= = 612.5 N / m. 2 x Same as first way! €...
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This note was uploaded on 01/02/2010 for the course PHY 101 taught by Professor Pralle during the Fall '08 term at SUNY Buffalo.
 Fall '08
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