(a)
The appropriate analysis model for the system when the balloon remains stationary.
(a)
Answer to Problem 58P
A particle in equilibrium model is the appropriate analysis model for the system when the balloon remains stationary.
Explanation of Solution
An analysis model is a simplified version of any physical system that strips away the unnecessary aspects of the situation. There are different kinds of analysis models such the particle under constant velocity, particle under constant acceleration and particle in equilibrium etc.
In the given situation, the balloon remains stationary which implies the balloon has zero acceleration. The net force on the balloon in any direction is zero. The balloon is in equilibrium. The particle in equilibrium is the appropriate analysis model for the given physical situation.
Conclusion:
Thus, a particle in equilibrium model is the appropriate analysis model for the system when the balloon remains stationary.
(b)
The force equation for the balloon.
(b)
Answer to Problem 58P
The force equation for the balloon is
Explanation of Solution
Write the equation for equilibrium.
Here,
Take the upward direction to be
Write the equation for
Here,
Put the above equation in equation (I).
Conclusion:
Therefore, the force equation for the balloon is
(c)
The expression for the mass of the segment of string.
(c)
Answer to Problem 58P
The expression for the mass of the segment of string is
Explanation of Solution
Write the equation for the buoyant force.
Here,
Write the equation for the weight of the helium.
Here,
Write the equation for density of the helium.
Here,
Rewrite the above equation for
Put the above equation in equation (IV).
Write the equation for the weight of the balloon.
Here,
Write the equation for the weight of the segment of the string above the ground.
Here,
Put equations (III), (V), (VI) and (VII) in equation (II) and rewrite it for
Write the equation for the volume of the balloon.
Here,
Conclusion:
Put equation (IX) in equation (VIII).
Therefore, the expression for the mass of the segment of string is
(d)
The numerical value of the mass
(d)
Answer to Problem 58P
The numerical value of the mass
Explanation of Solution
Equation (X) can be used to determine the numerical value of the mass
Conclusion:
The density of air is
Substitute
Therefore, the numerical value of the mass
(e)
The numerical value of the length
(e)
Answer to Problem 58P
The numerical value of the length
Explanation of Solution
Write the equation for the mass
Here,
Rewrite the above equation for
Conclusion:
Substitute
Therefore, the numerical value of the length
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Chapter 15 Solutions
Principles of Physics
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