Concept explainers
A 5-kg collar C is released from rest in the position shown and slides without friction on a vertical rod until it hits a spring with a constant of k = 720 N/m that it compresses. The velocity of the collar is reduced to zero, and the collar reverses the direction of its motion and returns to its initial position. The cycle is then repeated. Determine (a) the period of the motion of the collar, (b) the velocity of the collar 0.4 s after it was released. (Note: This is a periodic motion, but it is not simple harmonic motion.)
Fig. P19.15
(a)
The period
Answer to Problem 19.15P
The period
Explanation of Solution
Given Information:
The mass (m) of the collar C is 5 kg.
The spring constant (k) is
The value of acceleration due to gravity (g) is
The vertical distance (h) between the collar and the spring is 0.5 m.
Calculation:
Calculate the natural circular frequency
Substitute
Calculate the initial velocity
Substitute
Calculate the free fall time
Substitute
Measure the displacement (x) from the position of static displacement of the spring.
Calculate the weight (W) of the collar C using the relation:
Substitute 5kg for m and
Calculate the static displacement
Substitute
Show the displacement of the collar and spring while impact as in Figure (1).
Write the equation of motion for simple harmonic motion as below:
Substitute 5 kg for m and
Write the expression for displacement (x):
Differentiate the above equation.
When time (t) is 0 the displacement (x) is equal to
Substitute 0 for t and
Rewrite the above equation,
Substitute 0 for t,
Calculate the phase angle
Substitute
Calculate the amplitude
Substitute
Hence, from time of impact, the ‘time of flight’ is the time necessary for the collar to come to rest on its downward motion. The time required for collar to rest return is
At time
Calculate the time
Substitute 0 for
Substitute
Calculate the period of motion
Substitute
Therefore, the period
(b)
The velocity
Answer to Problem 19.15P
The velocity
Explanation of Solution
Given Information:
The mass (m) of the collar C is 5 kg.
The spring constant (k) is
The value of acceleration due to gravity (g) is
The vertical distance (h) between the collar and the spring is 0.5 m.
Calculation:
Calculate the velocity
Rewrite equation (2).
Substitute
Therefore, the velocity
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Chapter 19 Solutions
Vector Mechanics for Engineers: Statics and Dynamics
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