A uniform 5.83 kg solid sphere of radius 0.295 m that is free to roll on a horiz

Question

A uniform 5.83 kg solid sphere of radius 0.295 m that is free to roll on a horizontal surface has a light axle passing through its diameter as shown. The axle is connected to a like yoke and in turn a light rope as shown. A constant tension of 4.75 N is applied ot the rope and the sphere starts to roll without slipping on the rough surface.

a) What is the angular acceleration of the sphere in rad/s2 ? (Hint: Find the acceleration of the sphere's center of mass first.)

b) What is the magnitude of the static frictional force that acts on the sphere in Newtons? (?s is NOT needed for this part of the question.)

c) If the coefficient of static friction is 0.177, what is the maximum tension in Newtons that could be applied to the rope for which the sphere would still roll without slipping? (i.e. If the tension is greater than this value, the sphere would start to slip.)

Assume g = 10.0 m/s2.

Hint: Here is where you finally use fs ? fs,max. The left hand side is determined by your 'general expression' for fs from part b), and the relationship between T and acm from part a). Together, these equations and the inequality imply an upper bound for the Tension Tmax . If you were to apply a larger force than this maximum, the sphere wouldn't roll it would slide.)

Explanation / Answer

a) acceleration of centre of mass, a = F/m = 0.82 m/s^2

angular acceleration , = a/r = 2.76 rad/s^2

b) static frictional force = 4.75 N

c) F max = fs / mu = 26.83 N

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