Explanation Given information: Mass of collar m = 5 Κ g Spring constant k = 720 Ν / m To simplify the analysis we measure the displacement from the position of static displacement of the spring, under the weight of the collar: Thus, static deflection is given by the equation: δ s t = W k δ s t = 5 × 9.81 720 δ s t = 0.068125 m Then, m x ¨ + k x = 0 5 x ¨ + 720 x = 0 Where, x is measured positively up from the position of static deflection. The equation of a simple harmonic motion is given as: x = x m sin ( ω n t + ϕ ) We can obtain the velocity (v) at any time (t) by differentiating (x) with respect to (t), Since, v = d x d t v = d d t ( x m sin ( ω n t + ϕ ) ) v = ω n x m cos ( ω n t + ϕ ) Natural frequency ω n = k m ω n = 720 5 ω n = 12 r a d / s And, velocity of free fall of collar v o = 2 g h v o = 2 × 9.81 × 0.5 v o = 3.132 m / s Now, to calculate x m a n d ϕ , impose the conditions at impact and count the time from there. Thus, Free-fall time of collar: t 1 = 2 × 0.5 g t 1 = 1 9 To determine (b) The velocity of collar.

12th Edition

ISBN: 9781259977305

Author: BEER, Ferdinand P. (ferdinand Pierre), Johnston, E. Russell (elwood Russell), Cornwell, Phillip J., SELF, Brian P.

Publisher: Mcgraw-hill Education,

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Chapter 19.1, Problem 19.15P

To determine

(a)

The time period and motionof collar.

To determine

(b)

The velocity of collar.

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Chapter 19.1, Problem 19.14P

Chapter 19.1, Problem 19.16P

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(a) The time period and motionof collar.

Solution Summary: The author explains that to simplify the analysis, we measure the displacement from the position of static displacement of the spring.