Concept explainers
(a)
The total energy of the child stick Earth system .
(a)
Answer to Problem 61AP
The total energy of the child-stick Earth system is
Explanation of Solution
Refer to Figure P8.61; Total energy of the system remains conserved at every point in the system.
Consider point A to calculate total energy. As the boy is momentarily at rest at point A Kinetic energy for the system is zero.
Write the expression for elastic potential energy stored in the spring.
Here,
Write the expression for potential energy.
Here,
Write the expression for kinetic energy of the system.
Here,
Write the expression for total energy at point A.
Here,
Substitute
Here,
Substitute
Conclusion:
Substitute
Thus, the total energy of the child-stick Earth system is
(b)
The value of
(b)
Answer to Problem 61AP
The value of
Explanation of Solution
Total energy at point A is equal to total energy at point C as the energy of system is conserved.
Write the expression for total energy at point C.
Here,
Kinetic energy at C is zero as child again momentarily comes to rest at the top of the jump and elastic potential energy at C is also zero because string is no more compressed.
Substitute
Here,
Substitute
Write the expression for conservation of energy.
Substitute
Rearrange above equation for
Conclusion:
Substitute
Thus, the value of
(c)
The speed of the child at point B.
(c)
Answer to Problem 61AP
Thus, the speed of the child at point B is
Explanation of Solution
When
Write the expression for total energy at point C.
Here,
Substitute
Here,
Substitute
Write the expression for conservation of energy.
Substitute
Rearrange above equation for
Conclusion:
Substitute
Thus, the speed of the child at point B is
(d)
The value of
(d)
Answer to Problem 61AP
The value of
Explanation of Solution
To calculate the value of
Write the expression for total energy at any point.
Here,
Rearrange above equation for
Substitute
Substitute
Simplify above expression.
Differentiate the equation with respect to
Differentiate equation (XII) with respect to
Equate it to zero.
Rearrange above equation for
Conclusion:
Substitute
As
Thus, the value of
(e)
The child’s maximum upward speed.
(e)
Answer to Problem 61AP
The child’s maximum upward speed is
Explanation of Solution
Consider point D where velocity is maximum and it is the point where we have maximum kinetic energy.
Write the expression for total energy at point D.
Here,
Write the expression for conservation of energy between point A and D.
Substitute
Rearrange above equation for
Substitute
Here,
Multiply above equation by
Rearrange above equation for
Conclusion:
Substitute
Thus, the child’s maximum upward speed is
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Chapter 8 Solutions
Physics For Scientists And Engineers With Modern Physics, 9th Edition, The Ohio State University
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