A box sits on a horizontal wooden board. The coefficient of static friction between the box and the board is 0.22. You grab one end of the board and lift it up, keeping the other end of the board on the ground. What is the angle between the board and the horizontal direction when the box begins to slide down the board?

The critical angle is determined by the condition:

From this equation we can find an angle .

To find the angle between the board and the horizontal direction when the box begins to slide down, we can use the concept of equilibrium. At the point of sliding, the forces acting on the box will balance out.

First, let's consider the forces acting on the box:

1. The weight of the box (mg), where m is the mass of the box and g is the acceleration due to gravity.
2. The normal force (N) exerted by the board on the box perpendicular to the board's surface.
3. The static friction force (fs) between the box and the board, opposing the motion.

When the box is just about to slide, the static frictional force reaches its maximum value, which is equal to the product of the coefficient of static friction (μs) and the normal force (N):

fs = μs * N

Since fs is the only horizontal force acting on the box, it must be equal to the component of the weight that is parallel to the board. Now, we can calculate the angle θ:

fs = mg * sin(θ)

μs * N = mg * sin(θ)

N = mg * cos(θ)

From these equations, we can solve for θ:

μs * mg * cos(θ) = mg * sin(θ)

μs * cos(θ) = sin(θ)

Dividing both sides by cos(θ):

μs = tan(θ)

Now we can find the angle θ by taking the inverse tangent of the coefficient of static friction:

θ = arctan(μs)

Given that the coefficient of static friction μs is 0.22, we can calculate the angle θ:

θ = arctan(0.22) ≈ 12.6 degrees

Therefore, the angle between the board and the horizontal direction when the box begins to slide down is approximately 12.6 degrees.

To find the angle at which the box begins to slide down the board, we need to consider the forces acting on the box.

First, let's determine the force of gravity acting on the box. This force is equal to the weight of the box, which can be calculated by multiplying the mass of the box by the acceleration due to gravity (9.8 m/s²).

Next, let's consider the force of static friction between the box and the board. The maximum force of static friction can be found by multiplying the coefficient of static friction (0.22) by the normal force acting on the box. The normal force is equal to the weight of the box when it's on a horizontal surface, which means it's equal to the force of gravity (weight).

In order for the box to begin sliding down the board, the force of gravity acting on the box must be greater than or equal to the maximum force of static friction. Therefore, we set up the following equation:

Force of gravity ≥ Maximum force of static friction

(mass of the box × acceleration due to gravity) ≥ (coefficient of static friction × normal force)

Now, let's express the normal force and the force of gravity in terms of their components in the vertical and horizontal directions. The normal force has no component in the horizontal direction, while the force of gravity has a component in the horizontal direction equal to (force of gravity × sinθ), where θ is the angle between the board and the horizontal direction.

Substituting these values into our equation, we get:

(mass of the box × acceleration due to gravity) ≥ (coefficient of static friction × (force of gravity × cosθ ))

Now, we can cancel out the mass and the acceleration due to gravity from both sides of the equation:

1 ≥ (coefficient of static friction × cosθ )

Rearranging the equation, we have:

coefficient of static friction ≤ 1 / cosθ

Now, the box will start sliding when the coefficient of static friction becomes equal to the expression on the right side of the equation. Solving for θ, we have:

θ = cos⁻¹(1 / (coefficient of static friction))

Plugging in the coefficient of static friction (0.22), we can calculate the angle:

θ ≈ cos⁻¹(1 / 0.22) ≈ 78.3 degrees

Therefore, the angle between the board and the horizontal direction when the box begins to slide down the board is approximately 78.3 degrees.