Tension in a Massless Rope

Part A = T_{1}

Part B = T_{2}

Part C = 3rd

Part D = equal to

Part E = F_{u} = F_{d}

Part F =

- 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.

**Solutions Below:**

- At point 1, a downward force of magnitude F
_{ad}acts on section a. - At point 1, an upward force of magnitude F
_{bu}acts on section b. - At point 1, the tension in the rope is T
_{1}. - At point 2, a downward force of magnitude F
_{bd}acts on section b. - At point 2, an upward force of magnitude F
_{cu}acts on section c. - At point 2, the tension in the rope is T
_{2}.

Assume, too, that the rope is at equilibrium.

Part A

_{ad}of the downward force on section a? Express your answer in terms of the tension T

_{1}.

We know from the given information that at point 1 downward force of magnitude F_{ad} acts on section a and that the tension at point 1 is T_{1. Since the two are the same thing, we know that:}

F_{ad = }T_{1}

Part B

_{bu}of the upward force on section b? Express your answer in terms of the tension T

_{1}. We know that the downward force (Part A) is the same as the upwards force – so our answer remains the same:

F_{bu} =T_{1}

Part C

_{cu}, and the magnitude of the downward force on b, F

_{bd}, are equal because of which of Newton’s laws?

Recall Newton’s 3rd law: To every action there is always an equal and opposite reaction. So the answer is:

Newton’s 3rd law

Part D

_{bu}is ____ F

_{bd}.

It is important to realize that F_{bu} and F_{bd} are not a Newton’s third law pair of forces. Instead, these forces are equal and opposite due to the fact that the rope is stationary (a_{b}=0) and massless(m_{b}=0). By applying Newtons first or second law to this segment of rope you obtain:

F_{b net} = F_{bu} – F_{bd} = m_{b}a_{b} = 0, since m_{b} = 0 and a_{b} = 0. Note that if the rope were accelerating, these forces would still be equal and opposite because m_{b} = 0.

So the answer is:

equal to

Part E

The forces on the two ends of an ideal, massless rope are always equal in magnitude. Furthermore, the magnitude of these forces is equal to the tension in the rope.

Therefore:

F_{u} = F_{d}

Part F

All of the below are true:

- 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.