Question

As a logging truck rounds a bend in the road, some logs come loose and begin to roll without slipping down the mountainside. The mountain slopes upward at 24.0 above the horizontal, and we can model the logs as solid uniform cylinders of mass 575 kg and diameter 1.20 m.

Answer 1

The moment of inertia about the point of contact of the logwith slope is
I = 3 M R2 / 2 using the parallel axistheorem.
The torque about the point of contact is T = M g R sin?
T = I a and a = 2 M g R sin ?/ 3 M R2 = 2 g sin ? / (3 R)
a = R a = 2 g sin ? / 3
Since the frictional force provides the torque to produce theangular acceleration about the center of rotation
and the angular acceleration about the center of rotation isthe same as that about the point of contact
F R = T = I a and F = I a /R now using I = 1/2 M R2
F = 1/2 M R2 * 2 g sin ? / (3 R) * 1 / R= M g sin ? / 3
Note that is the difference between the frictional force andthe gravitational force that produces the
linear acceleration or F = M g sin ? - 2 g sin? / 3 = M g sin ? / 3
agreeing with the answer found for F

Answer 2