A box sits on a horizontal wooden board. The coefficient of static friction between the box and the board is 0.41. 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?
njnj
22.29
22deg
Mg = Force of box = Normal force,Fn.
Fp = Mg*sin A. = Force parallel with incline.
Fs = u*Fn = 0.41Fn = 0.41Mg = Force of static friction.
Fp-Fs = M*a
Mg*sin A-0.41Mg = M*0 = 0.
Divide both sides by Mg:
sin A-0.41 = 0
A = 24.2 Degrees.
To find the angle between the board and the horizontal direction when the box begins to slide down, we need to consider the forces acting on the box.
When the box is about to slide down, the static friction force between the box and the board is at its maximum value. The maximum static friction force can be calculated using the coefficient of static friction and the normal force.
The normal force is the force exerted by the board on the box, which is equal to the weight of the box (mg), where m is the mass of the box and g is the acceleration due to gravity.
The maximum static friction force (F(static max)) can be calculated as F(static max) = μ(static) * N, where μ(static) is the coefficient of static friction and N is the normal force.
In this case, we need to find the angle when the box is just about to slide. This occurs when the maximum static friction force is equal to the component of the weight of the box parallel to the board.
Let's assume the angle between the board and the horizontal direction is θ. The component of the weight of the box parallel to the board is mg * sin(θ).
Therefore, we have F(static max) = mg * sin(θ).
Since the maximum static friction force is given by F(static max) = μ(static) * N = μ(static) * mg, we can equate these two equations:
mg * sin(θ) = μ(static) * mg.
Dividing both sides by mg, we get:
sin(θ) = μ(static).
Now we can find the angle θ by taking the inverse sine (sin⁻¹) of μ(static):
θ = sin⁻¹(μ(static)).
Substituting the given coefficient of static friction (μ(static) = 0.41) into this equation, we can find the angle:
θ = sin⁻¹(0.41).