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

šƒ’Ž‡ ʹǣ ϐ‹† –Š‡ –‹‡ ”ƒ–‹‘ ˆ‘” ƒ Š‘‘’ǡ •Ž‹†‹‰ †‘™ ƒ ‹ Ž‹‡† ’Žƒ‡ ƒ† ”‘ŽŽ‹‰ †‘™ ƒ
‹ Ž‹‡† ’Žƒ‡ǡ ƒ• •Š‘™ ‹ ȏ ‹‰ ʹǤ͸ͶȐǤ

[Fig 2.64 : Hoop rolling down an inclined plane]

Solution: for the hoop rolling down the inclined plane, the acceleration is given by:

g sin θ g sinθ g sin θ

a = I = mr 2 2
1 + 2 1 + 2
mr mr
considering the conservation of energy, as there is no loss of energy.

Potential Energy = Transational Kinetic Energy + Rotational Kinetic Energy

1 1 1 1 v 2
2
2
2
2
2
mgH = mv + Iω = mv + ∗ mr ∗ = mv
2 2 2 2 r 2
Further simplifying the above expression, we get:
v = gH = v = gH
2
To compute the time of travel, let us use the expression [v = U+at], where, the initial velocity ‘U’ is
zero.

Substituting the value of ‘v’ and ‘a’ in the expression [v = U+at], we get:

g sin θ 2 4
gH = 0 + ∗ t = t = gH ∗ = t = gH ∗
2
2
2 g sin θ g sin θ
H
Therefore, we have the time of travel to be: t = 2 - - - - (2.52)
2
g sin θ

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