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Week 3: Work and Energy |
T
M θ
Figure 35: A block is connected to an Acme (massless, unstretchable) string and is pulled so that it exerts a constant tension T on the block at the angle θ.
might be: At what value of the tension does the block begin to move? If it is pulled with exactly that tension, how fast is it moving after it is pulled a horizontal distance L?
We see that in the y-direction, N + T sin(θ) −mg = 0, or N = Fx = T cos(θ) − µsN = 0 (at the point where block barely begins
mg −T sin(θ). In the x-direction, to move).
Therefore: |
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T cos(θ) − µsmg + T µs sin(θ) = 0 |
(261) |
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or |
µsmg |
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T = |
(262) |
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cos(θ) + µs sin(θ) |
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With this value of the tension T, the work energy theorem becomes:
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W = FxL = |
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K |
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(263) |
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where Fx = T cos(θ) − µk(mg − T sin(θ). That is: |
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(T cos(θ) − µk(mg − T sin(θ)) L = |
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mvf2 − 0 (since vi = 0) |
(264) |
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or (after a bit of algebra, substituting in our value for T from the first part): |
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vf = µ cos(θ) + µs sin(θ) − 2µkgL + cos(θ) + µs sin(θ) ¶ |
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(265) |
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2µsgL cos(θ) |
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2µkµsgL sin(θ) |
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Although it is di cult to check exactly, we can see that if µk = µs, vf = 0 (or the mass doesn’t accelerate). This is consistent with our value of T – the value at which the mass will exactly not move against µs alone, but will still move if “tapped” to get it started so that static friction falls back to weaker dynamic friction.
This is an example of how we can combine Newton’s Laws or statics with work and energy for di erent parts of the same problem. So is the next example:
Example 3.2.2: Range of a Spring Gun
Suppose we have a spring gun with a bullet of mass m compressing a spring with force constant k a distance x. When the trigger is pulled, the bullet is released from rest. It passes down a horizontal, frictionless barrel and comes out a distance H above the ground. What is the range of the gun?
If we knew the speed that the bullet had coming out of the barrel, we’d know exactly how to solve this as in fact we have solved it for homework (although you shouldn’t look – see if you can do this on your own or anticipate the answer below for the extra practice and review). To find that speed, we can use the Work-Kinetic Energy Theorem if we can compute the work done by the spring!