A uniform solid sphere of mass m and radius r is placed on a...

A uniform solid sphere of mass $m$ and radius $r$ is placed on a ramp inclined at an angle $\theta$ to the horizontal. The coefficient of static friction between sphere and ramp is $\mu_{s} .$ Find the maximum value of $\theta$ for which the sphere will roll without slipping, starting from rest, in terms of the other quantities.

Key Concepts

Rolling Without Slipping

This concept refers to the condition where an object’s translational motion and rotational motion are linked by the relation v = r?, ensuring that the point of contact with the surface is momentarily at rest. It is key to analyzing objects that roll without sliding, as it connects linear acceleration with angular acceleration.

Static Friction

Static friction is the force that opposes the initiation of sliding between two surfaces. In the context of rolling objects, it not only prevents slipping but also provides the necessary torque for rotation. The maximum static friction determines the limits within which pure rolling occurs, particularly on inclined surfaces.

Moment of Inertia

Moment of inertia quantifies an object’s resistance to changes in its rotational motion and depends on how its mass is distributed relative to the axis of rotation. It is essential for understanding how an applied torque leads to angular acceleration, especially in bodies like spheres with a defined moment of inertia.

Dynamics on an Inclined Plane

This area of study examines the forces acting on an object on a slope, including gravitational components, normal force, and friction. Understanding the balance of these forces is crucial for determining the conditions required for rolling motion without slipping, such as finding the maximum incline angle for stable rolling.

Transcript

00:01 So the torque provided by friction would be equaling r multiplied by the static force of friction.

00:07 We know that the static force of friction is going to be equal to the coefficient of static friction multiplied by the force normal.

00:15 We can apply newton's second law in the x and y directions and say that then the net force would be equaling to m .g.

00:26 Sign of theta minus the static frictional force.

00:30 And this would be equaling then the mass times the acceleration or net force.

00:37 In the y direction, the force normal minus mg cosine of theta will equal zero because we have translational equilibrium in the y direction.

00:49 And so we can say the force normal would be equaling to mg cosine of theta.

00:57 We know that the torque is equal to the moment of inertia multiplied by the angular acceleration and the angular acceleration is equal to the linear acceleration divided by the radius of the object.

01:10 For the moment of inertia of a sphere, this would be equaling to two -fifths mr squared.

01:19 And so we can say that then the force of static friction would be equaling to the moment of inertia times the angular acceleration divided by r.

01:32 We can then say the static frictional force would be equaling then 2 fifth, m r squared multiplied by a over r divided by r and this is equaling two -fifths times the mass times the acceleration we know that the mass times the acceleration would then be equal to mg sine of theta minus the static frictional force oh or minus two -fifths times the mass times the acceleration...

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