A rock climber hangs freely from nylon rope that is 16 m long and has a diameter of 8.1 mm. If the rope stretches 4.2 cm, what is the mass of the climber?
what is the elastic modulus of the rope?
The problem doesn't say, what's above is all of what I was given
To find the mass of the climber, we can use the concept of the weight of the climber and the elongation of the rope.
Step 1: Calculate the change in length of the rope.
The elongation of the rope is the amount by which it stretches. In this case, the rope stretches 4.2 cm or 0.042 m.
Step 2: Calculate the cross-sectional area of the rope.
The cross-sectional area of a cylindrical object like a rope can be calculated using the formula:
Area = π * (radius)²
Given that the diameter of the rope is 8.1 mm, the radius can be calculated as half of the diameter: 8.1 mm / 2 = 4.05 mm = 4.05 * 10^(-3) m (converting millimeters to meters). Now we can calculate the cross-sectional area:
Area = π * (4.05 * 10^(-3))².
Step 3: Calculate the force applied on the rope.
The force applied to stretch the rope is given by Hooke's Law:
Force = k * elongation,
where k is the stiffness constant of the rope, which depends on its material and other factors. In this case, k is not given, but we can assume it to be constant. So the force applied on the rope is:
Force = k * 0.042.
Step 4: Calculate the weight of the climber.
The weight of an object is given by its mass multiplied by the acceleration due to gravity. In this case, we want to find the mass of the climber, so we can rearrange the equation:
Weight = mass * acceleration due to gravity.
The weight of a climber hanging freely is balanced by the force applied on the rope (by Hooke's Law), so we can equate the two forces:
k * 0.042 = mass * acceleration due to gravity.
Step 5: Calculate the mass of the climber.
Rearranging the equation from Step 4, we get:
mass = (k * 0.042) / acceleration due to gravity.
To calculate the actual value of the mass, we would need to know the stiffness constant of the rope (k). The stiffness constant depends on a variety of factors such as the type and condition of the rope and cannot be determined solely from the given information.