A man stands on a slope of 20 degrees.His weight is 65kg. Given that the coefficient of static friction is 1.5, Calculate the maximum angle that the slope can have to the horizontal before he begins to slide.

Also how does the pressure on his feet change with the changing slope angle?

When the slope angle A is so large that he begins to slide,

downslope force = friction force
M g sin A = Us* M g cos A
(Us is the static friction coefficient).
sinA/cosA = tanA = 1.5
A = 56.3 degrees

Note that the man's weight and the initial angle are not needed for the answer.

Force on feet(normal to surface)
= M g cos A,
which decreases with increasing angle

To calculate the maximum angle that the slope can have before the man begins to slide, we need to consider the forces acting on the man.

First, let's break down the weight of the man into its components. The weight can be split into two forces: the force acting perpendicular to the slope (normal force) and the force acting parallel to the slope (gravitational force).

The gravitational force acting on the man can be calculated using the formula:

F_gravity = mass * gravity

where mass is the weight of the man (65 kg) and gravity is the acceleration due to gravity (approximately 9.8 m/s^2).

F_gravity = 65 kg * 9.8 m/s^2 = 637 N

The force of friction (F_friction) depends on the coefficient of static friction (µ_s) and the normal force (N). The formula to calculate the maximum static friction is:

F_friction = µ_s * N

Since the maximum static friction force needs to balance the gravitational force acting parallel to the slope to prevent sliding, we can equate the two forces:

F_friction = F_gravity

µ_s * N = 637 N

To find the normal force (N), we need to calculate the component of the weight acting perpendicular to the slope. This can be done using trigonometry.

N = weight * cos(angle)

where the angle is the angle of the slope (20 degrees).

N = 65 kg * 9.8 m/s^2 * cos(20 degrees) = 611 N

Substituting this value of N into the equation for friction:

µ_s * 611 N = 637 N

Finally, we can solve for the maximum angle (θ_max) by rearranging the equation:

θ_max = arccos(F_friction / (µ_s * N))

θ_max = arccos(637 N / (1.5 * 611 N))

θ_max ≈ 48.3 degrees

Therefore, the maximum angle that the slope can have to the horizontal before the man begins to slide is approximately 48.3 degrees.

Regarding the pressure on the man's feet, the pressure is defined as the force exerted perpendicular to a surface divided by the area over which the force is applied. As the slope angle changes, the normal force acting on the man's feet changes.

Since pressure is calculated by dividing the normal force by the area, the pressure on the man's feet will change as the slope angle changes. As the slope angle increases, the normal force decreases because more of the man's weight is acting parallel to the slope rather than perpendicular to it. Consequently, the pressure on the man's feet will decrease as the slope angle increases.