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

Substituting the value of ‘μ’ from equation (2.49) in equation (2.47), we get:

tanθ 2
mr 2 mg cos θ ∗ r

I + 1
g sin θ mr 2 1 g sin θ mr 2 1

a = I = mr 2 ∗ = I ∗ mr 2
I
I + 1 I + 1

mr 2
To simplify the above expression, let us assume that, = x . Therefore, we have:
I

1
a = g sinθ ∗ x ∗
x + 1

Let us simplify the above expression by dividing the numerator and denominator by ‘x’.

1 g sin θ g sinθ g sin θ 2 g sinθ - - - - (2.51)

a = g sin θ 1 = I = mr 2 = 1 = 3
1 + 1 + mr 2 2 1 + 2
x
1 + 2
mr

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:

4gH 2 g sinθ 4gH 3 4gH 9
= 0 + ∗ t = t = ∗ = t = ∗
2
2
3 3 3 2 g sinθ 3 4 g sin θ
3H
Therefore, we have the time of travel to be: t =
2
g sin θ

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