worker stands still on a roof sloped at an angle of 30° above the horizontal. He is prevented from slipping by a static frictional force of 560 N. Find the mass of the worker.

485

To find the mass of the worker, we need to consider the forces acting on the worker and use the relationship between force, mass, and acceleration.

Here are the steps to solve the problem:

Step 1: Draw a free-body diagram. This diagram shows all the forces acting on the worker.
- There are two forces acting on the worker: the weight (mg) pulling vertically downward and the static friction force (F_friction) acting horizontally along the slope.
- The angle between the slope and the horizontal is given as 30°.

Step 2: Resolve the weight force into components.
- The weight force mg can be split into two components: one perpendicular to the slope (mg⋅cosθ) and one parallel to the slope (mg⋅sinθ).

Step 3: Set up equations based on Newton's second law.
- In the vertical direction, the forces are balanced, so the worker is not moving up or down, and the equation is:
∑Fy = 0 → mg⋅cosθ = 0

- In the horizontal direction, the static friction force cancels out the component of the weight force parallel to the slope, so the equation is:
∑Fx = 0 → F_friction = mg⋅sinθ

Step 4: Solve for the mass.
- Substitute the given values into the equation for the static friction force:
F_friction = 560 N
θ = 30°

- Rearrange the equation to solve for the mass (m):
mg⋅sinθ = 560 N
m = 560 N / (g⋅sinθ)

- Finally, substitute the value of g (acceleration due to gravity, approximately 9.8 m/s²) and solve for m:
m = 560 N / (9.8 m/s²⋅sin30°)

Calculating this, we find that the mass of the worker is approximately 108 kg.