You have a mass of 68 kg and are on a 51-degree slope hanging on to a cord with a breaking strength of 152 newtons. What must be the coefficient of static friction to 2 decimal places between you and the surface for you to be saved from the fire?
To determine the coefficient of static friction required to save you from the fire, we need to consider the forces acting on you.
First, let's analyze the forces parallel to the inclined plane (the slope):
The force of gravity acting vertically downwards can be divided into two components: one perpendicular to the slope and one parallel to the slope.
1. Perpendicular Force (N):
The perpendicular component of the gravitational force pushing you into the slope is equal to the normal force acting on you. It can be calculated using the equation:
N = m * g * cos(θ)
where m is your mass (68 kg), g is the acceleration due to gravity (9.8 m/s²), and θ is the angle of the slope (51 degrees).
Plugging in the values:
N = 68 kg * 9.8 m/s² * cos(51°)
2. Parallel Force (F):
The parallel force acting on you is the force of gravity pulling you down the slope, which can be calculated using the equation:
F = m * g * sin(θ)
Plugging in the values:
F = 68 kg * 9.8 m/s² * sin(51°)
Now, let's calculate the maximum static friction force (fs max) that can be exerted by the surface before you start sliding down the slope. The maximum static friction force is equal to the product of the normal force and the coefficient of static friction (μs):
fs max = μs * N
In this case, fs max needs to be less than or equal to the breaking strength of the cord (152 N) to ensure your safety. Therefore, we have:
μs * N ≤ 152 N
Now, substitute N with its value from the earlier calculations:
μs * (68 kg * 9.8 m/s² * cos(51°)) ≤ 152 N
Simplifying the equation:
μs ≤ (152 N) / (68 kg * 9.8 m/s² * cos(51°))
Finally, calculate the coefficient of static friction (μs) by taking the inverse cosine (cos⁻¹) of both sides of the equation:
μs ≤ cos⁻¹((152 N) / (68 kg * 9.8 m/s² * cos(51°)))
Using a calculator or appropriate software, evaluate the expression with the given values to obtain the coefficient of static friction required to save you from the fire on the slope. Round the result to two decimal places.