Question

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 1

Concepts and reason

The concepts required to solve this question are, free body diagram, force equilibrium, and Newton’s third law of motion.

Initially, draw the free body diagram of point $1$ for section (a) of the rope, apply force equilibrium condition in vertical direction, and determine the magnitude of the downward force at point $1$ . Next, draw the free body diagram of point $1$ for section (b) of the rope, apply force equilibrium, and determine the magnitude of the upward force at point $1$ . And then, draw the free body diagram of point $2$ for the section b and section c, and use the definition of the Newton’s third law to show the given condition.

Further, draw the free body diagram of the point $2$ on section c of the rope, apply force equilibrium, and determine the magnitude of the upward force ${F_{bu}}$ . And then, draw the free body diagram of the rope, apply the force equilibrium condition for the entire rope, and determine the relation between the two forces that act at ends of the rope.

Finally, for part (F), consider the given statements and verify if it agrees with the concept of the force equilibrium for the tension in the rope, and definition of the Newton’s third law.

Fundamentals

Free Body Diagram (FBD): It is a method to analyze the reactions and external forces acting on a given body by sketching the direction of the forces on the body. The free body diagram is sketched after all the other adjoining bodies have been removed. Generally, the direction of a force is assumed if it is not predefined. Upon continuing the problem, if the value of such a force is obtained as negative, it can be said that the force acts in the direction opposite to what was assumed to.

The equilibrium condition for the forces on a body can be written as,

$\sum F = 0$

Here, $\sum F$ is the net force acts on the body.

Newton’s third law: It states that, for two bodies in contact, if one of the body exerts force on the other body, it experiences a reaction force from the other body equal in magnitude but opposite in the direction.

General sign convention for the force: The forces that act in the upward direction are taken as positive, and the forces that act in the downward direction are taken as negative.

(A)

Draw the free body diagram of point $1$ for the section a.

Here, ${F_{ad}}$ is the downward force acts at point $1$ on the section a, and ${T_1}$ is the tension in rope at point $1$ .

Apply force equilibrium condition at point $1$ .

$\begin{array}{l}\\\sum {{F_{1,a}}} = 0\\\\{T_1} - {F_{ad}} = 0\\\end{array}$

Here, $\sum {{F_{1,a}}}$ is the net force at point $1$ for the section a.

Rearrange for ${F_{ad}}$ .

${F_{ad}} = {T_1}$

(B)

Draw the free body diagram of point $1$ for the section b.

Here, ${F_{ad}}$ is the downward force that acts at point $1$ , and ${F_{bu}}$ is the upward force that acts at point $1$ on section b.

Apply force equilibrium condition at point $1$ .

$\begin{array}{l}\\\sum {{F_{1,b}}} = 0\\\\{F_{bu}} - {F_{ad}} = 0\\\end{array}$

Here, $\sum {{F_{1,b}}}$ is the net force at point $1$ for the section b.

Rearrange for ${F_{bu}}$ .

${F_{bu}} = {F_{ad}}$

Substitute ${T_1}$ for ${F_{ad}}$ .

${F_{bu}} = {T_1}$

(C)

Draw the free body diagram of point $2$ for the section b and section c.

Here, ${F_{bd}}$ is the downward force that acts at point $2$ on section c, and ${F_{cu}}$ is the upward force that acts at point $2$ on section c.

Newton’s third law states that, for every action, there is an equal and opposite reaction. Thus,

${F_{bd}} = {F_{cu}}$

From the above, it is inferred that the magnitude of the upward force on c, that is, ${F_{cu}}$ , and the magnitude of the downward force on b, that is, ${F_{bd}}$ are equal, as could be inferred directly from Newton’s third law of motion.

(D)

Draw the free body diagram of the point $2$ on section c of the rope.

Here, ${F_{cu}}$ is the upward force acts at point $2$ on the section c, and ${T_2}$ is the tension in rope at point $2$ .

Apply force equilibrium condition at point $2$ .

$\begin{array}{l}\\\sum {{F_{2,c}}} = 0\\\\{F_{cu}} - {T_2} = 0\\\end{array}$

Here, $\sum {{F_{2,c}}}$ is the net force at point $2$ for the section c.

Since, it is single massless rope, the tension in the rope is everywhere the same.

Substitute ${T_1}$ for ${T_2}$ .

${F_{cu}} - {T_1} = 0$

From above step, substitute ${F_{bd}}$ for ${F_{cu}}$ , and ${F_{bu}}$ for ${T_1}$ .

${F_{bd}} - {F_{bu}} = 0$

Rearrange for ${F_{bu}}$ .

${F_{bu}} = {F_{bd}}$

(E)

Draw the free body diagram of the rope. Consider the all forces that act on the points $1$ and $2$ within sections a, b, and c.

Here, ${F_u}$ is the upward force that acts on the upper end of the rope, and ${F_d}$ is the downward force that acts on the lower end of the rope.

Apply force equilibrium condition to 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, $\sum {{F_R}}$ is the net force acts on the rope.

Substitute ${F_{bd}}$ for ${F_{cu}}$ , and ${F_{bu}}$ for ${F_{ad}}$ .

$\begin{array}{l}\\{F_u} + {F_{bu}} + {F_{bd}} - {F_{bu}} - {F_{bd}} - {F_d} = 0\\\\{F_u} - {F_d} = 0\\\end{array}$

Rearrange for ${F_u}$ .

${F_u} = {F_d}$

Hence, the relationship between the magnitudes of the two forces that act at the ends of the rope is ${F_u} = {F_d}$ .

(F.1)

Consider the statement, “The tension in the rope is everywhere the same”.

The forces on the two ends of an ideal, massless rope results in net force on the rope. The magnitude of this force is equal to the tension in the rope. As the rope is massless, the tension is everywhere the same. Hence, it is a true statement.

(F.2)

Consider the statement, “The magnitudes of the forces exerted on the two objects by the rope are the same”. The force that acts on the two objects is the tension force applied by the rope. The forces that act on the two objects are in opposite direction as each object tends to pull the ends of the rope. So, the rope pulls them back with same magnitude of force by Newton’s third law. Hence, this statement is true.

(F.3)

Consider the statement, “The forces exerted on the two objects by the rope must be in opposite direction”. The forces that act on the two objects are in opposite direction as each object tends to pull the ends of the rope. So, the rope pulls them back as suggested by the Newton’s third law of motion. Hence, this statement is true.

(F.4)

Consider the statement, “The forces exerted on the two objects by the rope must be in the direction of the rope”. The force that acts on the two objects is the tension force applied by the rope, in the direction of the rope. Hence, this statement is true.

Ans: Part A

The magnitude ${F_{ad}}$ of the downward force on section a is ${T_1}$ .

Answer 2

Part a)

As rope is in equilibrium, therefore net force at point 1 in part a will be zero

Part b)

At point 1, part 'a' will apply force on part 'b', hence as action reaction force

Part c)

as they are action - reaction force

Part d)

Magnitude of as part 'b' is in equilibrium, therefore net force acting on the part 'b' is equal to zero.

Part e)

Part f)

The tension in the rope is everywhere the same.