A box of mass m is pushed against a spring of negligible mass and force constant
ID: 1458912 • Letter: A
Question
A box of mass m is pushed against a spring of negligible mass and force constant k, compressing it a distance x. The box is then released and travels up a ramp that is at an angle above the horizontal. The coefficient of kinetic friction between the box and the ramp is k, where k<1. The box is still moving up the ramp after traveling a distance s>|x| along the ramp. Calculate the angle for which the speed of the box after traveling distance s is a minimum.
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Explanation / Answer
Let's call the energy at "s" the "final" KE + PE. Initially, we have some spring energy U, show more Let's call the energy at "s" the "final" KE + PE.
Initially, we have some spring energy U,
and in between we have some work done by friction.
U - work = KE + PE
½kx² - µmgs*cos = ½mv² + mgs*sin
v² = (k/m)x² - 2µgs*cos - 2gs*sin
v = [(k/m)x² - 2gs(sin + µcos)]
"speed ... is a minimum" suggests taking the derivative wrt . But dv/d is minimized where
cos - µ*sin = 0
= arctan(1/µ)
the lower the friction, the higher the angle.
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