Science & Mathematics » Physics » Physics: You have one object of each of these shapes, all with mass 0.840 kg : a uniform solid cylinder, a thin-walled hollow cylinder...?

# Physics: You have one object of each of these shapes, all with mass 0.840 kg : a uniform solid cylinder, a thin-walled hollow cylinder...?

You have one object of each of these shapes, all with mass 0.840 kg : a uniform solid cylinder, a thin-walled hollow cylinder, a uniform solid sphere, and a thin-walled hollow sphere. You release each object from rest at the same vertical height h above the bottom of a long wooden ramp that is inclined at 31.0 ∘ from the horizontal. Each object rolls without slipping down the ramp. You measure the time t that it takes each one to reach the bottom of the ramp; (Figure 1) shows the results.

A- solid sphere
B- solid cylinder
C- hollow sphere
D- hollow cylinder

All objects have the same kinetic energy at the bottom of the ramp. Object D has the greatest rotational kinetic energy at the bottom of the ramp.

a) What minimum coefficient of static friction is required for all four objects to roll without slipping?

• mg = weight of ball objects = (0.840)(9.81) = 8.24 N
weight component acting parallel and down incline = mg(sin 31°) = 4.244 N
weight component acting Normal to incline = mg(cos 31°) = 7.063 N
minimum value of µs:
acceleration along ramp = g(sin 31°) = 9.81(sin 31°) = 5.052 m/s² *
f = static friction force on object D = µs(7.063)
fa = static friction acceleration on object D = µs(7.063)/0.840 = 8.408(µs) m/s²
µs = 5.052/8.408 = 0.601 ANS

*could also be obtained by taking 4.244/m = 4.244/0.840 = 5.052 m/s²
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• For each round object, I = k*m*R² where for:
Solid sphere, k = 0.4
Solid cylinder, k = 0.5
Hollow sphere, k = 0.667
Hollow cylinder, k = 1.0

a) For each shape, µs(min) = [k/(1+k)]*tanΘ
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• µs=0.300
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