Concept explainers
A 300-g mass at the end of a Hookean spring vibrates up and down in such a way that it is 2.0 cm above the tabletop at its lowest point and 16 cm above at its highest point. Its period is 4.0 s. Determine (a) the amplitude of vibration, (b) the spring constant, (c) the speed and acceleration of the mass when it is 9 cm above the tabletop, (d) the speed and acceleration of the mass when it is 12 cm above the tabletop.
(a)
The amplitude of the spring vibration when the mass of
Answer to Problem 31SP
Solution:
Explanation of Solution
Given data:
The mass of
Time period is
Formula used:
The amplitude of an hormonic motion is the half of the distance between the end points of the motion.
Mathematically,
Here,
Explanation:
The amplitude is the maximum displacement of the spring from the equilibrium position.
Recall the expression for the amplitude:
Substitute
Conclusion:
Therefore, the amplitude of the vibration is
(b)
The spring constant of the spring when the mass of
Answer to Problem 31SP
Solution:
Explanation of Solution
Given data:
The mass of
Time period is
Formula used:
The formula for time period of a simple spring mass system in SHM is expressed as
Here,
Explanation:
Consider the expression for time period of the spring–mass system for the
Rearrange for
Substitute
Conclusion:
Therefore, the spring constant is
(c)
The speed and magnitude of the acceleration of the mass when the mass of
Answer to Problem 31SP
Solution:
Explanation of Solution
Given data:
The mass of
Time period is
Formula used:
The expression for velocity of mass in SHM at a location
Here,
The expression for magnitude of acceleration of a mass in terms of displacement and time period can be written as
Here,
Explanation:
Consider the expression for velocity of the mass
Substitute
Understand that the given position of the body hung to spring will be at equilibrium position. Therefore, the value of
Recall the expression for
Substitute
Conclusion:
The speed is
(d)
The speed and magnitude of the acceleration of the mass when the mass of
Answer to Problem 31SP
Solution:
Explanation of Solution
Given data:
The mass of
Time period is
Formula used:
The expression for magnitude of acceleration of a mass in terms of displacement and time period can be written as
Here,
The expression for velocity of mass in SHM at a location
Here,
Explanation:
Consider the expression for velocity of the mass
Substitute
Recall the expression for
Understand that the position of the body hung to spring will be at
Substitute
Conclusion:
The velocity
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