You have a mass of 69 kg and are on a 57-degree slope hanging on to a cord with a breaking strength of 135 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?

tension=mg*sinTheta+mg*mu*cosTheta

solve for mu
135=69g*sin57 + 69g*cos57*mu
solvefor mu

To determine the required coefficient of static friction between you and the surface, we need to analyze the forces acting on you.

First, let's break down the forces acting parallel to the slope and perpendicular to the slope.

1. Forces parallel to the slope:
- The gravitational force pulling you down the slope is given by the formula: F_parallel = m * g * sin(θ), where m is your mass (69 kg), g is the acceleration due to gravity (9.8 m/s^2), and θ is the angle of the slope (57 degrees). Plugging in the values, we get: F_parallel = 69 kg * 9.8 m/s^2 * sin(57°).

2. Forces perpendicular to the slope:
- The gravitational force acting perpendicular to the slope is given by the formula: F_perpendicular = m * g * cos(θ), where m is your mass (69 kg), g is the acceleration due to gravity (9.8 m/s^2), and θ is the angle of the slope (57 degrees). Plugging in the values, we get: F_perpendicular = 69 kg * 9.8 m/s^2 * cos(57°).

Now, the maximum static friction force (F_friction_max) that can be exerted between you and the surface without you sliding down the slope can be calculated by multiplying the coefficient of static friction (μ) with the normal force (F_perpendicular). Therefore, the equation is:

F_friction_max = μ * F_perpendicular

We can equate F_parallel and F_friction_max, because if the static friction force is equal to or greater than the parallel force, you will be saved from sliding down the slope. So,

F_parallel = F_friction_max

69 kg * 9.8 m/s^2 * sin(57°) = μ * (69 kg * 9.8 m/s^2 * cos(57°))

Now, we can solve the equation to find the value of μ (the coefficient of static friction).

μ = sin(57°) / cos(57°)

Using a calculator, we find:

μ ≈ 1.03

Hence, the coefficient of static friction between you and the surface must be approximately 1.03 for you to be saved from sliding down the slope.