Solve Tension and Angles Homework: Find Smallest \theta

[Payne0511]

Homework Statement

An adventurous archaeologist crosses between two rock cliffs by slowly going hand-over-hand along a rope stretched between the cliffs. He stops to rest at the middle of the rope . The rope will break if the tension in it exceeds 2.80×10^4 N, and our hero's mass is 91.0 kg.

What is the smallest value the angle \theta can have if the rope is not to break?

Homework Equations

Tension=(M*g)/cos(theta)

The Attempt at a Solution

I was able to find the tension on the rope to be 2260 N. from there I thought I could set the above equation >= to the maximum tension of the rope 28000 N and solve for theta, which gave me 1.54, but that was clearly wrong. Not too sure what else to try here..

thanks

 

[kuruman]

How is theta defined, with respect to the vertical or horizontal? However it is defined, draw a free body diagram. There are three forces acting on the mass, the weight and two pieces of the rope. Your expression does not have a factor of 2 in it.

 

[tomcue]

for any help



To solve for the smallest value of theta, we need to consider the forces acting on the rope. The weight of the archaeologist, 91.0 kg, exerts a downward force of (91.0 kg)(9.8 m/s^2) = 891.8 N on the rope. This force must be balanced by the tension in the rope, which is directed upwards. Therefore, we can set up the following equation: Tension = weight / cos(theta). Plugging in the values, we get:
Tension = (891.8 N) / cos(theta)
Now, we know that the tension must be less than or equal to the maximum tension of 2.80×10^4 N. Therefore, we can set up the following inequality:
Tension <= 2.80×10^4 N
Substituting in the expression for tension, we get:
(891.8 N) / cos(theta) <= 2.80×10^4 N
Rearranging, we get:
cos(theta) >= (891.8 N) / (2.80×10^4 N)
Solving for theta, we get:
theta <= cos^-1(891.8 N / 2.80×10^4 N) = 1.67 radians
Therefore, the smallest value of theta that will not break the rope is 1.67 radians, or approximately 95.7 degrees.