A 50.0 kg mountain climber is rappelling down a steep mountain face. Her foot makes contact with the rock at point A. Her rope attaches to her harness at point B, which is located at x1 = 0.77 m and y1 = 0.42 m relative to point A. Her centre of mass is found at point C, which has an x coordinate x2 = 0.92 m relative to point A.
the tension in the rope if the rope makes an angle of θ = 37° with respect to the horizontal.
What is the minimum coefficient of static friction between her foot and the rock to prevent her from slipping?
To find the minimum coefficient of static friction between the climber's foot and the rock, we need to analyze the forces acting on the climber.
1. Gravity force: The weight of the climber can be calculated using the mass and acceleration due to gravity (g = 9.8 m/s^2). The gravity force is given by Fg = m * g.
2. Tension force: The tension in the rope pulls the climber downward. We can decompose this force into horizontal and vertical components. The vertical component is T * sin(θ), and the horizontal component is T * cos(θ).
3. Friction force: The friction force prevents the climber from slipping down the rock. It acts in the opposite direction of the motion and is equal to (μ * N), where μ is the coefficient of static friction and N is the normal force.
The normal force is the perpendicular force exerted by the rock on the climber's foot. In this case, the normal force is equal to the vertical component of the tension force plus the gravity force. N = T * sin(θ) + Fg.
Now, let's set up the equation:
T * cos(θ) - μ * (T * sin(θ) + Fg) = 0
Simplifying the equation:
T * cos(θ) - μ * T * sin(θ) - μ * Fg = 0
We can solve this equation for μ:
μ = (T * cos(θ) - μ * T * sin(θ)) / Fg
Now, substitute the values:
- T = tension in the rope (which we need to calculate)
- θ = 37°
- Fg = m * g (mass of the climber is 50 kg)
To find the tension in the rope, we can use the fact that the climber is in equilibrium. This means that the sum of the forces in the horizontal and vertical directions must be zero:
Horizontal equilibrium: T * cos(θ) = 0
Vertical equilibrium: T * sin(θ) + Fg = 0
Substituting the values:
T * cos(37°) = 0
T * sin(37°) + (50 kg * 9.8 m/s^2) = 0
Solve these two equations simultaneously to find the tension, T.
Finally, substitute the value of T in the equation for μ to find the minimum coefficient of static friction, μ.