# Does a rope connecting an object to a constant-velocity aircraft fly at an angle?

A massive object is hanging by a massless cord from an aircraft. The aircraft is moving at a constant velocity v.

Even assuming that air resistance exists, shouldn't the cord be perpendicular to the vertical because of inertia? My textbook says it hangs at an angle. • Try it! Tie a rock to a string, dangle the rock from your hand, and move your hand. What happens? – march Oct 10 '16 at 14:31
• A more interesting question is what happens if there is no air resistance. – garyp Oct 10 '16 at 14:36
• Just a question (genuine, non-sarcarstic): is it useful for stackexchange answering questions like this one? – QuantumBrick Oct 10 '16 at 14:38
• You neglected another option - if the object being towed has lift, it could end up above the airplane. But this is a rather poor question. – Jon Custer Oct 10 '16 at 14:40

You have three forces acting on the object. The tension, the weight and the drag. To balance out the tension needs components in both x and y axis so it must hang at an angle.

PS. Look up "Free Body Diagrams" and learn how to construct them. The result is obvious once a good FBD is drawn.

The rope is not rigid. By the time the force due to tension propagates through the rope to the hanging mass the aircraft has already moved forward and thus causing the rope to be slanting at an angle. If the rope was replaced by a rigid rod, it would experience shearing force. However if there is no air resistance and both the mass and aircraft are initially propelled to velocity V the rope/rod will become useless as with no other forces acting on it both the aircraft and mass travel at the same speed. But without air you can't have an aircraft flying so I guess that scenario is ruled out.

• you can have a rocket flying – Bob Bee Oct 10 '16 at 19:40

The answer depends on what assumptions are being made. In the following scenarios, I assume that the wire is light (so its mass can be neglected), flexible (bends with little force), and is so thin that it does not experience significant drag itself.

Airplane moving at constant velocity; no drag

The mass will hang vertically. The only forces on the mass are vertical : gravitational attraction to the Earth and tension in the wire. These forces are balanced. There are no horizontal forces on the hanging mass. If the wire did not hang vertically then there would be a horizontal component of tension in the wire which is not balanced by any other force, so the mass would accelerate forward.

Airplane moving at constant velocity; there is drag

Now there is a horizontal force on the mass, which must be balanced by a horizontal component of tension in the wire. The wire cannot be vertical; it must be inclined.

Airplane moving at constant acceleration; no drag

The hanging mass is accelerating forward, so there must be a net horizontal force on it. This can only be provided by the tension in the wire, which must therefore be inclined to the vertical.

Airplane moving at constant acceleration; there is drag

The horizontal component of tension in the wire must now do 2 things : (1) balance the drag on the hanging mass, and (2) provide enough unbalanced force to accelerate the mass. The wire will hang at an even greater angle to the vertical.