For safety in climbing, a mountaineer uses a nylon rope that is 55 m long and 0.9 cm in diameter. When supporting a 88-kg climber, the rope elongates 1.4 m. Find its Young's modulus.
To find the Young's modulus of the nylon rope, we can use Hooke's Law, which states that the elongation or deformation of an object is directly proportional to the force applied to it.
First, let's determine the force applied to the rope when supporting the climber. We know the weight of the climber is 88 kg, so we can calculate the force (F) using the acceleration due to gravity (g), which is approximately 9.8 m/s²:
F = m * g
F = 88 kg * 9.8 m/s²
F ≈ 862.4 N
Now, we have the force applied to the rope (862.4 N) and the elongation of the rope (1.4 m). According to Hooke's Law, the elongation (ΔL) is proportional to the force (F) applied and the Young's modulus (Y):
ΔL = (F * L) / (A * Y)
Where:
ΔL is the elongation of the rope (1.4 m)
F is the force applied to the rope (862.4 N)
L is the original length of the rope (55 m)
A is the cross-sectional area of the rope (πr²)
Y is the Young's modulus (to be determined)
To solve for Y, we need to find the cross-sectional area (A) of the rope. The diameter (d) of the rope is given as 0.9 cm, so we can calculate the radius (r) by dividing the diameter by 2:
r = d / 2
r = 0.9 cm / 2
r = 0.45 cm
r = 0.0045 m (converting cm to m)
Now, we can calculate the cross-sectional area (A) using the formula:
A = πr²
A = π * (0.0045 m)²
A ≈ 0.000063617 m²
Now, let's rearrange the Hooke's Law formula to solve for Y:
Y = (F * L) / (A * ΔL)
Substituting the values we have:
Y = (862.4 N * 55 m) / (0.000063617 m² * 1.4 m)
Calculating the value:
Y ≈ 2.4 x 10^9 N/m²
So, the Young's modulus of the nylon rope is approximately 2.4 x 10^9 N/m².