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Phy 231
Your 'cq_1_15.1' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** CQ_1_15.1_labelMessages **
A rubber band begins exerting a tension force when its length is 8 cm. As it is stretched to a length of 10 cm its tension increases with length, more or less steadily, until at the 10 cm length the tension is 3 Newtons.
Between the 8 cm and 10 cm length, what are the minimum and maximum tensions?
answer/question/discussion: ->->->->->->->->->->->-> :
Min tension at 'ds = 8 cm
MaX Tension at 'ds = 10 cm
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Assuming that the tension in the rubber band is 100% conservative (which is not actually the case) what is its elastic potential energy at the 10 cm length?
answer/question/discussion: ->->->->->->->->->->->-> :
PE @ 10 cm = ForceAve * 'ds = 1.5 N * 2 cm = 3.0 N cm
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If all this potential energy is transferred to the kinetic energy of an initially stationary 20 gram domino, what will be the velocity of the domino?
answer/question/discussion: ->->->->->->->->->->->-> :
KE = 0.03 J
KE = 1/2 * m * v^2
0.03 J = 0.5 * 0.02 kg * v^2
v = 'sqrt ( (2 * 0.03 J) / 0.02 kg) = 1.73 m/s
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If instead the rubber band is used to 'shoot' the domino straight upward, then how high will it rise?
answer/question/discussion: ->->->->->->->->->->->-> :
Using the above info, with no influence by gravity the domino would have a velocity of 1.73 m/s, but gravity is pulling down on the domino as the band is
returning to equilibrium.
a = -9.80 m/s^2 KE = 0.03 J Vo = 0 m/s vf = ?
vf = 'sqrt ( 2*a*'ds) = 'sqrt ( 2*(-9.80 m/s^2) * 0.02 m) = -0.6261 m/s.
Adding this to the band velocity without gravity I will get the final velocity of the domino at the end of the band return.
vf = Vband + Vgrav = 1.73 m/s + (-0.6261 m/s) = 1.1039 m/s
This result is my new vo for projectile motion.
vo = 1.1039 m/s a = -9.80 m/s vf = 0 cm/s
vAve = 1.1039 m/s / 2 = 0.55195 m/s
vf = vo + a*'dt
vo / a = 'dt = (1.1039 m/s) / -9.80 m/s^2 = .112643 sec
'ds = vAve * 'dt = 0.55195 m/s * .112643 sec = 0.062173 m = 6.22 cm
The domino will rise 6.22 cm.
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This can be calculated more efficiently by just considering the relevant energy relationships.
The domino-and-rubber-band system has an initial potential energy of .03 J with KE zero. At the maximum height its KE is again zero, so (assuming the rubber band to be idea) its potential energy would again be .03 J. The initial elastic PE would be converted to gravitational PE.
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The gravitational PE of a .02 kg mass at height 6 cm would be about .2 N * .06 m = .012 J, not .03 J. I believe the mass would rise to a height closer to 15 cm.
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For University Physics students:
Why does it make sense to say that the PE change is equal to the integral of the force vs. position function over an appropriate interval, and what is the appropriate interval?
answer/question/discussion: ->->->->->->->->->->->-> :
With Force on the vertical axis, and position on the horizontal axis, I came up with the integral this way.
Force Ave = (ForceFinal + ForceInitial) / 2 = forceAve
My integral is bounded by the position, which is position final, and position initial = sf to s0
Integral form(sf(upper) to s0(Lower)) of (Vave * 'ds)
'dPE = 'dForce * 'ds.
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Good, but I don't think your last calculation works. Check my notes.
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