Concept explainers
(a)
The angle of the rope with the vertical.
(a)
Answer to Problem 44PQ
The angle of the rope with the vertical is
Explanation of Solution
Assume the man as a particle of mass
The following figure gives the direction of all forces on the man.
Apply equilibrium condition of forces along
Here,
Write the expression for net forces along
Here,
From figure1, expand equation (I) using all forces along
Here,
Rearrange above equation to get
From figure1, expand equation (II) using all forces along
Here,
Rearrange above equation to get
Divide equation (IV) by (III) to get
In problem it is allowed to ignore effect of air drag force.
Substitute
Conclusion:
Substitute
Therefore, the angle of the rope with the vertical is
(b)
The expression for the angle made by the rope with the vertical as a function of time by considering drag force due to air.
(b)
Answer to Problem 44PQ
The expression for the angle made by the rope with the vertical as a function of time by considering drag force due to air is
Explanation of Solution
Write the expression for the drag force.
Here,
Since man is modeled as a particle of dimeter
Write the expression for the velocity of the particle at each instant of time.
Here,
Conclusion:
Substitute
Substitute
Substitute
Substitute
Therefore, the expression for the angle made by the rope with the vertical as a function of time by considering drag force due to air is
(c)
Whether terminal speed is a meaningful concept for this situation and explain the reason.
(c)
Answer to Problem 44PQ
Terminal speed is not a meaningful concept for this situation since helicopter provides net acceleration at every time and never reaches a zero net force situation.
Explanation of Solution
Terminal speed is constant speed attained by a body moving in a fluid so that drag force is proportional to velocity of the body. Consider situation of a body moving through a fluid. The forces acting are gravitational force and drag force. Gravitational force is a constant force whereas drag force depends on velocity of the body at each instant of time. At particular point when drag force equal to gravitational force the body takes constant velocity.
In this case drag force depends on square of velocity of the body. As the drag force increases the helicopter exerts more force to provide constant acceleration
(d)
The effect on tension if the helicopter continues to accelerate and the result for a real rope in this situation.
(d)
Answer to Problem 44PQ
The tension in the helicopter is inversely proportional to cosine of angle made by the rope with the vertical. As helicopter accelerates, angle made by the rope with the vertical increases. Thus, tension in the rope increases as the helicopter accelerates.
Explanation of Solution
Rearrange equation (III) to get
Therefore, tension in the rope continues to increase as the helicopter accelerates. A real rope has certain limit to withstand tension. After a particular value it will break.
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Chapter 6 Solutions
Physics for Scientists and Engineers: Foundations and Connections
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