A worker stands still on a roof sloped at an angle of 23° above the horizontal. He is prevented from slipping by a static frictional force of 540 N. Find the mass of the worker.
m g sin 23 is the component of the worker's weight down the roof slope.
That equals the friction force 540 N
Solve M g sin23 = 540
to get the mass, M.
He's a pretty heavy guy, it seems.
497
M*9.8*sin23=540
M*9.8=540/sin23 =1382.02
M=1382.02/9.8 =141.02kg
M=141.02kg
210.99
Well, I hope the worker isn't clowning around up there! Let's get to the math. We can start by breaking down the available information.
The static frictional force acts parallel to the slope, preventing the worker from slipping. In this case, the static frictional force is equal to the force of gravity acting down the slope.
Now, we can use trigonometry to relate the angle and the normal force acting on the worker. Since the normal force acts perpendicular to the slope, it can be calculated using the equation:
N = mg * cos(θ)
Where N is the normal force, m is the mass of the worker, g is the acceleration due to gravity (9.8 m/s^2), and θ is the angle of the slope in radians.
Since the static frictional force is equal to the normal force, we have:
F_friction = N = mg * cos(θ)
Given that the static frictional force is 540 N and the angle of the slope is 23°, we can rearrange the equation to solve for the mass (m):
m = F_friction / (g * cos(θ))
m = 540 N / (9.8 m/s^2 * cos(23°))
Now, let me do the math.
To find the mass of the worker, we need to use the force of static friction equation:
F_friction = μ * N
Where:
F_friction is the force of static friction (given as 540 N)
μ is the coefficient of static friction
N is the normal force
The coefficient of static friction depends on the materials in contact. Let's assume a value of μ = 0.7, which is a common value for rubber-soled shoes on dry surfaces.
Now, the normal force (N) is the component of the worker's weight perpendicular to the surface. We can calculate it using the equation:
N = mg * cos(θ)
Where:
m is the mass of the worker (what we want to find)
g is the acceleration due to gravity (approximated as 9.8 m/s²)
θ is the angle of the roof (given as 23°)
Now, substitute the values into the equation and solve for m:
540 N = 0.7 * m * 9.8 m/s² * cos(23°)
Rearrange the equation:
m = 540 N / (0.7 * 9.8 m/s² * cos(23°))
Now, calculate the value:
m ≈ 40.43 kg
Therefore, the mass of the worker is approximately 40.43 kg.