An air-track glider of mass 0.100 kg is attached to the end of a horizontal air

Question

An air-track glider of mass 0.100 kg is attached to the end of a horizontal air track by a spring with force constant 20.0 N/m. initially the spring is unstressed and the glider is moving at 1.50 m/s to the right. With the air track turned off. the coefficient of kinetic friction is muk=0.47. How large would the coefficient of static friction is have to be to keep the glider from springing back to the left when it stops instantaneously? If the coefficient of static friction between the glider and the track is mu s = 0.75. what is the maximum initial speed in that the glider can be given and still remain at rest after it stops instantaneously?

Explanation / Answer

here,
spring constant, k = 20 N/m
velcoity of glider, v = 1.50 m/s
mass of glider, m = 0.100 kg

From Conservation of Energy,
Spring Energy = Energy gained by glider
0.5 * m * v^2 = 0.5 * k *x^2

solving for stretched distance , x
x = sqrt(mv^2/k)
x = sqrt(0.1*1.50^2/20)
x = 0.106 m

Part A:
now From newton Law of motion, Spring force = Frictional Force
k*x = us*mg -------------(1)

solving for Coefficient of static friction, us
us = k*x/mg
us = 20*0.106/(0.1*9.8)
us = 2.163

Part B: if us = 0.75, using eqn 1,
x = us*mg/k
x = 0.75*0.1*9.8/20
x = 0.037 m

From Conservation of Energy,
Spring Energy = Energy gained by glider
0.5 * m * v^2 = 0.5 * k *x^2

solving for velocity of glider, v
v = sqrt(kx^2/m)
v = sqrt(20*0.037^2 / 0.1)
v = 0.523 m/s

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