Two mountain climbers are working their way up a glacier when one falls into a crevasse. The icy slope can be considered frictionless. Sue's weight is pulling Paul up the 45* slope. If Sue's mass is 66kg and if she falls 2m in 10s starting from rest, find the tension in the rope joining them and Paul's mass.
To find the tension in the rope and Paul's mass, we can use Newton's laws of motion and the concept of gravitational force.
First, let's calculate the acceleration of Sue using the kinematic equation:
Δd = v₀t + (1/2)at²
Where:
Δd is the change in distance, which is 2m (the distance she falls),
v₀ is the initial velocity, which is 0 m/s (starting from rest),
t is the time, which is 10s,
a is the acceleration.
Rearranging the equation to solve for acceleration (a):
a = (2Δd) / t²
= (2 * 2m) / (10s)²
= 0.4 m/s²
Now, let's determine the tension in the rope. The tension (T) in the rope can be found by using the equation:
T = m * g * sin(θ)
Where:
m is the mass of Sue,
g is the acceleration due to gravity (approximately 9.8 m/s²),
θ is the angle of the slope, which is 45°.
Substituting the known values:
T = (66kg) * (9.8 m/s²) * sin(45°)
≈ 453.05 N
So, the tension in the rope between Sue and Paul is approximately 453.05 Newtons.
Now, to find Paul's mass, we can use the equation for Newton's second law of motion:
T = m * g * sin(θ)
Rearranging the equation to solve for Paul's mass (m):
m = T / (g * sin(θ))
= 453.05 N / (9.8 m/s² * sin(45°))
m ≈ 66kg
Therefore, Paul's mass is approximately 66kg.