what is the magnitude fad of the downward force on section a?
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Question “what is the magnitude fad of the downward force on section a?”
To understand the concept of tension and the relationship
between tension and force. This problem introduces the concept of
tension. The example is a rope, oriented vertically, that is being
pulled from both ends. (Figure 1) Let Fu and Fd (with u for up and
d for down) represent the magnitude of the forces acting on the top
and bottom of the rope, respectively. Assume that the rope is
massless, so that its weight is negligible compared with the
tension. (This is not a ridiculous approximation–modern rope
materials such as Kevlar can carry tensions thousands of times
greater than the weight of tens of meters of such rope.) Consider
the three sections of rope labeled a, b, and c in the figure. At
point 1, a downward force of magnitude Fad acts on section a. At
point 1, an upward force of magnitude Fbu acts on section b. At
point 1, the tension in the rope is T1. At point 2, a downward
force of magnitude Fbd acts on section b. At point 2, an upward
force of magnitude Fcu acts on section c. At point 2, the tension
in the rope is T2. Assume, too, that the rope is at
equilibrium.
Part A
What is the magnitude Fad of the downward force on
section a?
Express your answer in terms of the tension T1.
Part B
What is the magnitude Fbu of the upward force on
section b?
Express your answer in terms of the tension T1.
Part C
The magnitude of the upward force on c, Fcu, and the
magnitude of the downward force on b, Fbd, are equal
because of which of Newton’s laws?
Part D
The magnitude of the force Fbu is ____
Fbd.
Part E
Now consider the forces on the ends of the rope. What is the
relationship between the magnitudes of these two forces?
Now consider the forces on the ends of the rope. What is the
relationship between the magnitudes of these two forces?
Fu>Fd | |
Fu=Fd | |
Fu<Fd |
Part F
The ends of a massless rope are attached to two stationary
objects (e.g., two trees or two cars) so that the rope makes a
straight line. For this situation, which of the following
statements are true?
Check all that apply.
Check all that apply.
The tension in the rope is everywhere the same. | |
The magnitudes of the forces exerted on the two objects by the rope are the same. | |
The forces exerted on the two objects by the rope must be in opposite directions. | |
The forces exerted on the two objects by the rope must be in the direction of the rope. |
Answer
This question requires the following concepts: force equilibrium, free body diagram, and Newton’s third Law of Motion.
Draw the $math_tag_0 free body diagram.
Draw the $math_tag_1 free body diagram
Final, consider part ( F) and confirm that it is consistent with the Newton’s third law and force equilibrium for tension in rope.
Free body diagram (FBD) – This is a way to examine the effects and forces that act on a body. It involves drawing the direction of these forces. After all other bodies have been removed, the free body diagram can be drawn. If a force’s direction is not known, it is assumed. If the value of such force is negative, it can be said the force acts in an opposite direction to what was expected.
You can write the equilibrium condition of the forces on a body as:
\sum F = 0Here, \sum F
Newton’s third law It states, for two bodies in contact: If one body exerts force upon the other, it experiences a response force from the other body equal to its magnitude, but opposite in direction.
General sign convention: Forces acting in the upward direction should be considered positive and forces acting in the downward direction should be considered negative.
(A)
Draw the body diagram for point 1
Here, {F_{ad}}
Force equilibrium condition at point $math_tag_0
\begin{array}{l}\\\sum {{F_{1,a}}} = 0\\\\{T_1} - {F_{ad}} = 0\\\end{array}Here, \sum {{F_{1,a}}}
Rearrangement for $math_tag_5
{F_{ad}} = {T_1}
(B)
Draw the body diagram for point 1
Here, {F_{ad}}
Force equilibrium condition at point $math_tag_0
\begin{array}{l}\\\sum {{F_{1,b}}} = 0\\\\{F_{bu}} - {F_{ad}} = 0\\\end{array}Here, \sum {{F_{1,b}}}
Rearrange for $math_tag_16
{F_{bu}} = {F_{ad}}Substitute $math_tag_18
{F_{bu}} = {T_1}
(C)
Draw the body diagram for point 2
Here, {F_{bd}}
Newton’s third law states, “For every action there is an equal or opposite reaction.” Thus,
{F_{bd}} = {F_{cu}}It is evident that the magnitude of the upward pressure on c is $math_tag_23.
(D)
Draw the 2 body diagram
Here, {F_{cu}}
Force equilibrium condition at point $math_tag_1
\begin{array}{l}\\\sum {{F_{2,c}}} = 0\\\\{F_{cu}} - {T_2} = 0\\\end{array}Here, \sum {{F_{2,c}}}
Because it is a single, massless rope, tension is constant throughout the rope.
Substitute $math_tag_18
{F_{cu}} - {T_1} = 0Substitute $math_tag_21 for the above step
{F_{bd}} - {F_{bu}} = 0Rearrange for $math_tag_16
{F_{bu}} = {F_{bd}}
(E)
Draw the free body diagram for the rope. The $math_tag_0 points are the forces that will act.
Here, {F_u}
Apply force equilibrium condition for the rope.
\begin{array}{l}\\\sum {{F_R}} = 0\\\\{F_u} + {F_{bu}} + {F_{cu}} - {F_{ad}} - {F_{bd}} - {F_d} = 0\\\end{array}Here, \sum {{F_R}}
Substitute $math_tag_21
\begin{array}{l}\\{F_u} + {F_{bu}} + {F_{bd}} - {F_{bu}} - {F_{bd}} - {F_d} = 0\\\\{F_u} - {F_d} = 0\\\end{array}Rearrange $math_tag_36
{F_u} = {F_d}The relationship between the magnitudes and the forces at the ends is $math_tag_42.
(F.1)
Take the following statement: “The tension is the same everywhere in the rope.”
Net force is the result of the forces at the ends of an ideal, weightless rope. This force is equal in magnitude to the rope’s tension. The tension in the rope is constant because it is not massless. It is therefore a truthful statement.
(F.2)
Take the following statement: “The magnitudes and effects of the forces exerted by the rope on the objects are the same.” The tension force applied to the rope is the force that acts on both objects. Because each object pulls at the ends of the rope, the forces acting on them are opposite. Newton’s third law states that the rope pulls them together with the same force. This proves that this statement is true.
(F.3)
Take the following statement: “The forces exerted by the rope on the objects must be in the opposite direction.” Because each object pulls at the rope’s ends, the forces acting on them are opposite. As suggested by Newton’s third law, the rope pulls the objects back. This statement is therefore true.
(F.4)
Take the following statement: “The forces exerted by the rope on the objects must be in the same direction as the rope.” The tension force applied by rope in the direction of the rope is the force that acts upon the objects. This statement is therefore true.
Part A – Ans
The magnitude $math_tag_5
Conclusion
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