Concept explainers
A 5-kg fragile glass vase is surrounded by packing material in a cardboard box of negligible weight. The packing material has negligible damping and a force-deflection relationship as shown. Knowing that the box is dropped from a height of 1 m and the impact with the ground is perfectly plastic, determine (a) the amplitude of vibration for the vase, (b) the maximum acceleration the vase experiences in g' s.
(a)
The amplitude of vibration for the vase.
Answer to Problem 19.10P
Amplitude
Explanation of Solution
Given information:
Mass of vase
Height
Velocity at the end of free fall
Assume that the spring is unstretched during the free fall. To better understand we use a simple spring mass model for the motion of the vase and the packing material.
Now, taking slope from the graph.
Calculation:
Now, consider simple harmonic motion:
We can obtain the velocity (v) at any time (t) by differentiating (x) with respect to (t),
When the box hits the ground, let
Then, at
And, velocity,
So, velocity of the vase = velocity at the end of free fall
Now, Natural frequency:
Thus, amplitude of the vase
Conclusion:
The amplitude of the vase is
(b)
The maximum acceleration the vase experiences.
Answer to Problem 19.10P
Acceleration
Explanation of Solution
Given information:
Mass of vase
Height
Velocity at the end of free fall
Assume that the spring is unstretched during the free fall. To better understand we use a simple spring mass model for the motion of the vase and the packing material.
Now, taking slope from the graph.
Calculation:
Now, consider simple harmonic motion:
We can obtain the velocity (v) at any time (t) by differentiating (x) with respect to (t),
When the box hits the ground, let
Then, at
And, velocity,
So, velocity of the vase = velocity at the end of free fall
Now, Natural frequency:
Thus, amplitude of the vase
And, maximum acceleration of the vase,
Conclusion:
Maximum acceleration of the vase is
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Chapter 19 Solutions
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