Concept explainers
(a)
To show: The gravitational potential energy of the person–Earth system as a function of the person’s variable height
(a)
Answer to Problem 78P
The gravitational potential energy of the person–Earth system as a function of the person’s variable height
Explanation of Solution
Given info: The mass of the person is
The acceleration due to gravity is
The expression for gravitational potential energy is as follows:
Here,
Substitute
Conclusion:
Therefore, the gravitational potential energy of the person–Earth system as a function of the person’s variable height
(b)
To show: Elastic potential energy of cord as a function of
(b)
Answer to Problem 78P
The elastic potential energy of cord as a function of
Explanation of Solution
Given info: The mass of the person is
The expression for elastic potential energy of spring is as follows:
The cord will stretch by length x only when the person falls more than the length of the cord. Now, the height of the balloon h should be more than the length of the cord l plus the person’s height y, for safe landing.
The expression for extension in spring is as follows:
Here,
Substitute
Substitute
Conclusion:
Therefore, the elastic potential energy of the cord as function of
(c)
To show: The total potential energy of the person-cord–Earth system as a function of
(c)
Answer to Problem 78P
The total potential energy of the person-cord–Earth system as a function of
Explanation of Solution
Given info: The mass of the person is
The expression for total potential energy of the person-cord–Earth system is as follows:
Substitute
Conclusion:
Therefore, the total potential energy of the person-cord–Earth system as a function of
(d)
To draw: The graph of gravitational, elastic, and total potential energies as a function of
(d)
Answer to Problem 78P
The graph of gravitational, elastic, and total potential energies as a function of
Explanation of Solution
Introduction:
The gravitational potential energy above the surface of the earth is directly proportional to the height of the object.
The elastic potential energy is proportional to the square of displacement.
The total potential energy is the sum of all the potential energies in the system.
Given info: The mass of the person is
From part (a), the expression for gravitational potential energy of person as a function of
Table for the above expression is shown below:
The graph of gravitational potential energy with displacement is shown below:
Figure(1)
From part (a), the expression for the elastic potential energy of cord as a function of
The value of elastic potential energy of the cord remains zero till the person does not fall
equal to the length of cord; therefore, the value of the above equation is zero for
Table for the above expression is shown below:
The graph of elastic potential energy with displacement is represented below:
Figure(2)
From part (a), the expression for total potential energy of the person-cord–Earth system as a function of
When the value of
Table for the above expression is shown below:
The graph of total potential energy with displacement is represented below:
Figure(3)
(e)
The minimum height of the person above the ground during his plunge.
(e)
Answer to Problem 78P
The minimum height of the person above the ground during his plunge is
Explanation of Solution
Given info: The mass of the person is
The expression for change in total energy is as follows:
Here,
The value of initial kinetic energy is zero, as the person is at rest.
The length of cord is
Substitute
At minimum height above the ground during plunge, the person comes to rest and the change in kinetic energy is zero, as both values of initial and final kinetic energies are zero.
Substitute
The expression for the roots of the above quadratic equation is as follows:
Substitute
Conclusion:
Therefore, the minimum height of the person above the ground during his plunge is
(f)
Whether potential energy graph shows any equilibrium position and if so the elevation of equilibrium position, whether the equilibrium points are stable or unstable.
(f)
Answer to Problem 78P
The potential energy graph shows that in an equilibrium position at an elevation of
Explanation of Solution
Given info: The mass of the person is
The graph of potential energy shows the equilibrium position at the place where the value of total potential energy is minimum.
The expression for total potential energy is as follows:
Derive the above equation with the height of the person.
For the expression of minima, equate the above expression equal to zero.
The elevation at the point of equilibrium is
The person could not stop at the elevation of equilibrium position as he has kinetic energy that does not allow the person to stay at the elevation of equilibrium position. The equilibrium position is unstable.
Conclusion:
Therefore, the potential energy graph shows that in an equilibrium position at an elevation of
(g)
The jumper’s maximum speed.
(g)
Answer to Problem 78P
The jumper’s maximum speed is
Explanation of Solution
Given info: The mass of the person is
The expression for change in total energy is as follows:
Substitute
The initial kinetic energy of the jumper is zero; hence, the change in the kinetic energy equals the kinetic energy at that position. Velocity is directly proportional to the square root of the kinetic energy; therefore, for maximum value of kinetic energy, the velocity is maximum.
Differentiate the above expression with respect to
Equate
Substitute
Substitute
The height at which the velocity is maximum is
Substitute
Substitute
Substitute
Conclusion:
Therefore, the jumper’s maximum speed is
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Chapter 7 Solutions
Principles of Physics
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