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Concept explainers
(a)
The gravitational force on a point mass at a distance
(a)
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Answer to Problem 4.24P
The gravitational force on a point mass at a distance
Explanation of Solution
Linear mass density id defined as the mass per unit length.
Here,
Write the expression for force on the point mass due to small segment.
Here,
Substitute the value of
An infinitely long uniform rod is situated on the z axis is shown in the Figure below.
Using the figure shown above and equation (III), write the expression for force along
The z component of the force from –z to z will be cancel each other.
From the figure, write the expression for
Integrate the equation (IV) with the interval minus infinity to plus infinity and Use equation in (V) in (IV), to get the total
Write the expression for distance between point mass and small element from the figure.
Use equation (VII) in (VI).
Use
Solve the equation (IX).
Conclusion:
Therefore, the gravitational force on a point mass at a distance
(b)
Force in terms of rectangular coordinates of the point and verify that
(b)
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Answer to Problem 4.24P
Force in terms of rectangular coordinates of the point is
Explanation of Solution
Write the expression for
Write the expression for unit vector of
Write the expression for gravitational force on a point mass at a distance
Use equation (XI) and (XII) in (XIII).
Write the expression for curl of a function.
Use equation (XIV) in (XV).
From the equation (XVI), it is clear that the force is conservative.
Conclusion:
Therefore, the force in terms of rectangular coordinates of the point is
(c)
Show that
(c)
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Answer to Problem 4.24P
Showed that
Explanation of Solution
Write the expression for curl in cylindrical polar co-ordinates.
Use
Conclusion:
Therefore,
(d)
The potential energy in polar coordinates.
(d)
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Answer to Problem 4.24P
The potential energy is
Explanation of Solution
Potential energy of the system of masses is the negative integral of gravitational force to bring the masses from infinite distance to a distance
Rewrite the equation (XVIII) n polar co-ordinates,.
Use
Conclusion:
Therefore, the potential energy is
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Chapter 4 Solutions
Classical Mechanics
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