A solid sphere of uniform density starts from rest and rolls without slipping down an inclined plane with angle? = 30o. the sphere has mass m = 8 kg and radius r = 0.19 m. the coefficient of static friction between the sphere and the plane is? = 0.64. what is the magnitude of the frictional force on the sphere?

Question

Physics

Santos Powell

20 July, 12:43

A solid sphere of uniform density starts from rest and rolls without slipping down an inclined plane with angle? = 30o. the sphere has mass m = 8 kg and radius r = 0.19 m. the coefficient of static friction between the sphere and the plane is? = 0.64. what is the magnitude of the frictional force on the sphere?

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Answers (1)

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Roy Gibbs · Answer 1

Let a be the linear acceleration of the center of mass, fs the magnitued of the frictional force, I be the rotational inertia of the ball, M be the mass, R the moment arm, and g be the gravity.

For any body rolling along an incline of angle O with the horizontal, a is:

a = - g sin O / (1 + (I / MR^2))

and

fs = - I (a / R^2)

Thus, substituting the given value to the formulas above, we get (assuming g = 9.8 m/s^2 and I = 2/5 MR^2 for a solid sphere)

a = - (9.8 m/s^2) sin30 / (1 + ((2/5MR^2) / (MR^2)))

= - (9.8 m/s^2) sin 30 / (1 + (7/5))

= - 3.5 m/s^2

and

fs = - (2/5MR^2) a / R^2

= - (2/5M) a

= - (2 * 8 / 5) (-3.5)

= 11.2 N