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Friction - - Rigid Bodies in Translation and Rotation

Example: A rigid disk of radius r = 4 m and mass m = 4 kg starts at rest on a horizontal surface

with a coefficient of friction μ = 1/4. Two forces P and D then act on the disk. P = ˗ 5 i + j N

and D = 5 i + 20 j N. Force P acts through the mass center, C. Force D acts at point Q directly

to the right of C. Find the acceleration of its center of mass, C, and its angular acceleration.

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

of the disk.

First examine case 1 where the disk rolls without slipping.

ΣF = m a_cx ˗ P_x ˗ F + D_x = m a_Cx (1)

ΣF = m a_cy = 0 N ˗ mg + P_y + D_y = 0 (2)

ΣM_C = I_c α ˗ F r + D_y r = (1/2) m r² α (3)

If the disk rolls without slipping, a_C = a_P + a_C_/P|n + a_C_/P|t = α k x r j = ˗ r α i (4)

Since ω = 0 (starts from rest) and both a_P + a_C_/P|n are zero.

By (4) and (1) ˗ P_x ˗ F + D_x = ˗ m r α Known quantities are P_x , D_x , D_y

By (3) ˗ F + D_y = (1/2) m r α Eliminate α and solve for F

The result for the given data is that F = 13.3333 N

Next calculate N and μ N = F_max . For the data N = 19 N and F_max = 4.74 N

Since F is needed to prevent sliding and F_max is less than F, this case fails and the disk must roll