You have a mass of 93 kg and are on a 56-degree slope hanging on to a cord with a breaking strength of 156 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 between you and the surface, we can analyze the forces involved.
First, let's consider the forces acting on you on the slope. There are three forces at play:
1. The force of gravity pulling you downwards, which can be calculated as the product of your mass (93 kg) and the acceleration due to gravity (9.8 m/s^2). Thus, the force of gravity is 93 kg * 9.8 m/s^2 = 911.4 N.
2. The normal force exerted by the surface perpendicular to the slope. This can be found by multiplying the force of gravity by the cosine of the angle of the slope. Therefore, the normal force is 911.4 N * cos(56°).
3. The force of static friction, which opposes the motion and keeps you from sliding down the slope.
For you to be saved from the fire, the force of static friction must be equal to or greater than the force component pulling you downwards, which is given by the equation:
Force of static friction ≥ Force component pulling you downwards
The force component pulling you downwards can be obtained by multiplying the force of gravity by the sine of the angle of the slope:
Force component pulling you downwards = 911.4 N * sin(56°)
Now, we can set up the inequality:
Force of static friction ≥ 911.4 N * sin(56°)
Substituting the breaking strength of the cord (156 N) for the force of static friction:
156 N ≥ 911.4 N * sin(56°)
Now, we can solve the equation for the coefficient of static friction:
Coefficient of static friction = 156 N / (911.4 N * sin(56°))
Calculating this value will give you the required coefficient of static friction to two decimal places.