(30 points) As shown in Figure 1, a mass m is on a frictionless horizontal secti
ID: 1770464 • Letter: #
Question
(30 points) As shown in Figure 1, a mass m is on a frictionless horizontal section of a track at an elevation h he mass is then compressed a distance xo against a massless spring with spring constant k and then released rom rest. The mass then slides down a frictionless ramp and then through a horizontal section of the track that has a short section with friction. Because of the passage of the mass through the section with friction, there is an increase in thermal energy AEth of the mass-track system. The mass then slides up an identically sloped frictionless ramp until it encounters another frictionless horizontal section of track with identical elevation h. The mass then encounters another massless spring with identical spring constant k which compresses a distance x1 before bringing the mass momentarily to rest. The compression of the spring then causes the mass to b pushed back along the track in the opposite direction. The cycle repeats as the mass slides back and forth between the springs. a) Using the law of conservation of energy, derive an equation for the increase in thermal energy AEth of the mass-track system due to friction after a single passage of the mass through the section with friction. Your equation for Eth m ust be in terms of k, xo, and X1 only. f conservation of energy and your answer to a), derive an equation for the maximum number t the mass can cross the section with friction such that the mass will still be able to reach the top of times n that f either slope (i.e. find n such that the velocity of the mass when it reaches the top of a slope is zero). Your in terms of xo and x1 only. Have "physics faith" that many variables will cancel p to 10 points) Derive an equation for the maximum number of times n that the mass can tion with friction before being brought permanently to rest at the bottom horizontal portion of the track. Your equation for n must be in terms of k, xo, X1, m, h, and g only rn ground level section with friction Figure 1. Diagram for ProblemExplanation / Answer
a] Increase in thermal energy = decrease in potential energy
delta Eth = 0.5 kx0^2 - 0.5 kx1^2
b] Energy lost in one crossing = 0.5 kx0^2 - 0.5 kx1^2
Total initial spring energy = 0.5 kx0^2
number n = total spring energy/Energy lost in one crossing
= [0.5 kx0^2]/[0.5 kx0^2 - 0.5 kx1^2]
= x0^2/[x0^2 - x1^2]
c] number n = total energy/Energy lost in one crossing
= [0.5 kx0^2+ mgh ]/[0.5 kx0^2 - 0.5 kx1^2]
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