Consider a heavy rope with mass per unit length 2.10 kg/m; a length L=18.00 m of the rope is hanging vertically from a support, as shown. If the support and rope are placed in an elevator accerlerating downward with a=0.800 m/s^2, what is the tension in the rope at point P? (at point P, mass is 12.6)

Question

Consider a heavy rope with mass per unit length 2.10 kg/m; a length L=18.00 m of the rope is hanging vertically from a support, as shown. If the support and rope are placed in an elevator accerlerating downward with a=0.800 m/s^2, what is the tension in the rope at point P? (at point P, mass is 12.6)

This is what I did:

F=ma
F=(12.6)(0.8)
F=10.08

Sum of F = T-mg = 10.08 N
Sum of F = T-(12.6)(9.81) = 10.08 N
Therefore, T is equal to 134 N.

It was the wrong answer, could someone help me?

Answer 1

To find the tension in the rope at point P, you need to consider the forces acting on the rope.

First, let's calculate the weight of the rope. The mass per unit length of the rope is given as 2.10 kg/m, and the length of the rope hanging vertically is 18.00 m. The total mass of the rope is therefore (2.10 kg/m) * (18.00 m) = 37.80 kg.

The weight of the rope is given by W = mg, where m is the mass of the rope and g is the acceleration due to gravity, which is approximately 9.81 m/s^2. Therefore, W = (37.80 kg) * (9.81 m/s^2) = 370.84 N.

Now, let's analyze the forces at point P. There are two forces acting on the rope at this point: the tension force T and the weight force W.

Since the elevator is accelerating downward with an acceleration of -0.800 m/s^2 (negative because it's in the opposite direction of the positive direction), there is an additional force due to this acceleration acting on the rope. This force is given by F = ma, where m is the mass of the object and a is the acceleration. At point P, the mass is 12.6 kg, and the acceleration is -0.800 m/s^2. Therefore, the force due to acceleration is F = (12.6 kg) * (-0.800 m/s^2) = -10.08 N.

Now, let's apply Newton's second law to find the tension force T at point P. The sum of the forces acting on the rope is equal to the mass of the rope times its acceleration. In this case, the sum of forces is T - W - F, where T is the tension force, W is the weight force, and F is the force due to acceleration.

So we have: T - W - F = m * a

Plugging in the values we have calculated: T - (370.84 N) - (-10.08 N) = (37.80 kg) * (0.800 m/s^2)

T - 370.84 N + 10.08 N = 30.24 N

T = 390.00 N

Therefore, the tension in the rope at point P is 390.00 N.