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Document Title
Fundamentals of Stress and Vibration Chapter Title
[A Practical guide for aspiring Designers / Analysts] 2. Engineering Mechanics

tanθ
condition for perfect rolling − μ ≥
mr 2
I + 1

tanθ
condition for slipping − μ <
mr 2
I + 1

Example 1: find the time taken by a solid cylinder to roll down an inclined plane if it is released
with zero initial velocity.

[Fig 2.63: cylinder rolling down an inclined plane]

Solution: considering the conservation of energy, as there is no loss of energy.
Potential Energy = Transational Kinetic Energy + Rotational Kinetic Energy

1 1 1 1 mr 2 v 2 3
2
2
2
2
mgH = mv + Iω = mv + ∗ ∗ = mv
2 2 2 2 2 r 2 4
Further simplifying the above expression, we get:
4 4gH
2
v = gH = v = - - - - (2.50)
3 3
The acceleration of the cylinder could be computed using equations (2.47) and (2.49), we have:

μ mg cos θ r 2
a = - - - - (2.47)
I

tanθ
μ = 2 - - - - (2.49)
mr + 1
I

Page 72 QP No. SSC/Q4401, Version 1.0, NSQF Level 7, Compliant with Aero and Auto Industries,