What is the tension in the rope at this point?

[bcarter]

How do you solve this problem?
Arlene is to walk across a high wre strung horizontally between two buildings 10.0meters apart. The sag in in the rope when she is at the mid point is 10 degrees. If her mass is 50 kgs, what is the tension in the rope at this point?

 

[Doc Al]

By identifying all the forces acting on the woman (draw a picture) and then applying the conditions for equilibrium (that the net force on her must be zero).

 

[quicknote]

The tension in the rope at this point can be determined by using the principles of statics and trigonometry. First, we need to draw a free body diagram of Arlene standing at the midpoint of the rope. We can see that there are two forces acting on her: the weight of Arlene pulling downwards and the tension force in the rope pulling upwards. The angle of 10 degrees indicates that the rope is not perfectly horizontal, so we need to split the weight force into its vertical and horizontal components.

Next, we can use the trigonometric relationship of sine to calculate the vertical component of the weight force. This will be equal to the tension force in the rope, as the rope is in equilibrium. We can then use the vertical component of the weight force and the angle of 10 degrees to calculate the horizontal component of the weight force. Adding this to the horizontal component of the tension force, we can use the Pythagorean theorem to find the magnitude of the tension force in the rope.

Using the given values of Arlene's mass and the distance between the two buildings, we can plug these into our calculations and find the tension in the rope at the midpoint. It is important to note that this calculation assumes that the rope is massless and that there are no other external forces acting on the system.

In summary, to solve this problem we need to use the principles of statics and trigonometry to analyze the forces acting on Arlene at the midpoint of the rope. By breaking down the weight force into its vertical and horizontal components and using the trigonometric relationship of sine, we can calculate the tension force in the rope.