Applications involving Friction - - Rigid Bodies in Translation and Rotation

Key Concepts: Frictional forces always oppose motion or the tendency toward motion (impending

motion). In applications involving friction you need to determine if friction is sufficient to prevent

sliding, if friction is just enough that sliding impends, or if friction is insufficient to prevent sliding
with the mating surface. So you need to investigate each case and test which case actually occurs.

Strategy: The first and most important step is to construct a complete and accurate free body diagram

of the body (a disk) showing all external forces acting on the body noting that the frictional force

always opposes motion or the tendency toward motion. See the figure below. Align the x-coordinate axis along the direction that sliding might occur. In this figure sliding might occur in the x-direction.

The governing equations of motion are:

ΣF = m a_c ΣM = I_c α

where ΣF are the sum of the external forces acting on the rigid disk

m is the mass of the body located at C, the center of mass of the disk

r is the radius of the disk

a_c = a_cx i + a_cy j is the acceleration of the body in the x and y-directions

ΣM = sum of external forces causing moments about C, the center of mass of the disk

I_c is the mass moment of inertia of the disk about its center of mass = (1/2) mr²

α is the angular acceleration of the disk = α k

g is the acceleration of gravity

F is the frictional force

N is the normal force

let μ be the coefficient of friction

let P be a point of the disk in contact with the mating surface

Examine each case: Case 1: Disk rolls without sliding.

Case 2: Disk rolls and slides in the + x-direction

Case 3: Disk rolls and slides in the ˗ x-direction

Click here to continue.

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