Concept explainers
Review. A 65.0-kg bungee jumper steps off a bridge with a light bungee cord tied to her body and to the bridge. The outstretched length of the cord is 11.0 m. The jumper reaches the bottom of her motion 36.0 m below the bridge before bouncing back. We wish to find the time interval between her leaving the bridge and her arriving at the bottom of her motion. Her overall motion can be separated into an 11.0-m tree fall and a 25.0-m section of simple harmonic oscillation. (a) For the free-fall part, what is the appropriate analysis model to describe her motion? (b) For what time interval is she in free fall? (c) For the simple harmonic oscillation part of the plunge, is the system of the bungee jumper, the spring, and the Earth isolated or nonisolated? (d) From your response in part (c) find the spring constant of the bungee cord. (c) What is the location of the equilibrium point where the spring force balances the gravitational force exerted on the jumper? (f) What is the angular frequency of the oscillation? (g) What time interval is required for the cord to stretch by 25.0 m? (h) What is the total time interval for the entire 36.0-m drop?
(a)
The appropriate analysis model of jumper’s motion for the free fall part.
Answer to Problem 15.30P
The jumper’s motion has the constant acceleration.
Explanation of Solution
A free falling object is an object that is falling under the sole influence of gravity.
Any object that is being acted upon only by the force of gravity is said to be in a state of free fall. The free fall phase follows the parabolic behavior. Since the only gravity acted on the free fall, the acceleration is constant.
Conclusion:
Therefore, the jumper’s motion has the constant acceleration.
(b)
The time required for free fall.
Answer to Problem 15.30P
The time required for free fall is
Explanation of Solution
The mass of bungee jumper is
The equation for the kinematic is,
Here,
Substitute
Conclusion:
Therefore, the time required for free fall is
(c)
Weather the system of the bungee jumper, the spring and the earth is isolated or non-isolated for simple harmonic oscillation.
Explanation of Solution
When a system is isolated, it means that it is separated from its environment in such a way that no energy flows on or out of the system. The non-isolated system interacts with its environment and exchanges the energy.
The energy of the system of the bungee jumper, the spring and the earth is exchanged only with each other not outside from the system. Since the earth and spring act on the jumper, the system is isolated.
Conclusion:
Therefore, the system the bungee jumper, the spring and the earth is isolated.
(d)
The spring constant of the bungee cord.
Answer to Problem 15.30P
The spring constant of the bungee cord is
Explanation of Solution
Write an expression of the law of conservation of the energy for the bungee jumper
Here,
Substitute
Conclusion:
Therefore, the spring constant of the bungee cord is
(e)
The location of the equilibrium point where the spring force balances the gravitational force exerted on the jumper.
Answer to Problem 15.30P
The location of the equilibrium point where the spring force balances the gravitational force exerted on the jumper is
Explanation of Solution
The equation for the equilibrium point from the lorded is,
Here,
The equilibrium point is calculated as,
Here,
Substitute
Substitute
The amplitude of the motion is,
Substitute
Conclusion:
Therefore, the location of the equilibrium point where the spring force balances the gravitational force exerted on the jumper is
(f)
The angular frequency of the oscillation.
Answer to Problem 15.30P
The angular frequency of the oscillation is
Explanation of Solution
The formula to calculate angular frequency of the oscillation is,
Substitute
Conclusion:
Therefore, the angular frequency of the oscillation is
(g)
The time interval required for the cord to stretched by
Answer to Problem 15.30P
The time interval required for the cord to stretched by
Explanation of Solution
The expression for the position of a particle in simple harmonic motion is,
Substitute
Substitute
Conclusion:
Therefore, the time interval required for the cord to stretched by
(h)
The total time interval for the entire
Answer to Problem 15.30P
The total time interval for the entire
Explanation of Solution
The total time interval for the entire
Substitute
Conclusion:
Therefore, the total time interval for the entire
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