Force problem with non-massless rope

[azure kitsune]

Homework Statement

A block of mass M is pulled along a horizontal frictionless surface by a rope of mass m. A horizontal force F is applied to one end of the rope. (a) Show that the rope must sag, even if only by an imperceptible amount. Then, assuming that the sag is negligible, find (b) the acceleration of rope and block, (c) the force on the block from the rope, and (d) the tension in the rope at its midpoint.

Homework Equations

F=ma

The Attempt at a Solution

For (a), it is because the rope has mass so gravity acts on it. Since no other vertical force acts on the rope, it's net force is downwards and therefore must sag.

For (b): From force = mass * acceleration:

a = \dfrac{F}{m+M}

For (c), the pull of the rope is the only force acting on the block. It is also accelerating with the block:

F = M\cdot a = \dfrac{M\cdot F}{m+M}

So far, everything makes sense to me. However, I am not sure how to approach part (d) because this is the first problem where the rope has mass, so I don't know how to deal with it. Can anyone help?

 

[LowlyPion]

Homework Statement

A block of mass M is pulled along a horizontal frictionless surface by a rope of mass m. A horizontal force F is applied to one end of the rope. (a) Show that the rope must sag, even if only by an imperceptible amount. Then, assuming that the sag is negligible, find (b) the acceleration of rope and block, (c) the force on the block from the rope, and (d) the tension in the rope at its midpoint.

Homework Equations

F=ma

The Attempt at a Solution

For (a), it is because the rope has mass so gravity acts on it. Since no other vertical force acts on the rope, it's net force is downwards and therefore must sag.

For (b): From force = mass * acceleration:

a = \dfrac{F}{m+M}

For (c), the pull of the rope is the only force acting on the block. It is also accelerating with the block:

F = M\cdot a = \dfrac{M\cdot F}{m+M}

So far, everything makes sense to me. However, I am not sure how to approach part (d) because this is the first problem where the rope has mass, so I don't know how to deal with it. Can anyone help?

Your answer for a) isn't quite complete. Because there is a vertical component down from weight, there must be corresponding components of the tension in the positive y direction to offset it, or it would be in motion downward.

For d) you have answered what the Force is pulling from the end of the rope and you have answered what the Force from the rope pulling the block, so ... what do you figure it is half way in between?

 

[azure kitsune]

Would this be a more complete answer for (a)?:

Assume that the rope is perfectly horizontal. However, gravity is acting on the rope, yet it is not moving downwards. Therefore there must be an corresponding upward force acting against gravity. But the rope is perfectly horizontal, so there is no upward force from the tension of the rope, and there are no other vertical forces acting on the rope. Contradiction?

For (d), I'm guessing it's the average? But I can't think of any reason using physics principles.

 

[LowlyPion]

Would this be a more complete answer for (a)?:

Assume that the rope is perfectly horizontal. However, gravity is acting on the rope, yet it is not moving downwards. Therefore there must be an corresponding upward force acting against gravity. But the rope is perfectly horizontal, so there is no upward force from the tension of the rope, and there are no other vertical forces acting on the rope. Contradiction?

For (d), I'm guessing it's the average? But I can't think of any reason using physics principles.

That answer shows a better understanding for a)

For d) think about if you had cut the cable in half and connected it back together with a small linkage. What force would the half alone exert on the other half and the block?