Question

A box sits on a horizontal wooden ramp. The coefficient of static friction between the box and the ramp is 0.30 You grab one end of the ramp and lift it up, keeping the other end of the ramp on the ground. What is the angle between the ramp and the horizontal direction when the box begins to slide down the ramp? (tutorial: crate on ramp)

Short answer

Answer: The angle between the ramp and the horizontal direction is approximately 73.3°.

Step-by-step solution

Identify the forces acting on the box

Determine the forces that are acting on the box while it is on the ramp. The main forces are the gravitational force acting downward (mg) and the frictional force (f) acting along the ramp. Note that as the box is on the verge of moving, it is still in equilibrium, meaning that the forces along the ramp and perpendicular to the ramp must balance out.

Calculate the maximum static friction

The maximum static friction (F_max) is given by the following formula: F_max = µ_s * N where µ_s is the coefficient of static friction (0.30), and N is the normal force, which is equal to mg * cos(θ).

Write down the balance of forces along the ramp

The box is still in equilibrium, so the forces acting along the ramp must be equal: Frictional force = Gravitational force component along the ramp. F_max = mg * sin(θ)

Substitute F_max in the force balance equation

Now, we will substitute the formula for F_max that we found in step 2 into the force balance equation: µ_s * N = mg * sin(θ)

Simplify the force balance equation with N

As N = mg * cos(θ), we can rewrite the force balance equation: µ_s * (mg * cos(θ)) = mg * sin(θ) Now, we can simplify the equation further by dividing both sides by (mg): µ_s * cos(θ) = sin(θ)

Calculate the angle θ between the ramp and the horizontal direction

Now, we can manipulate the equation in step 5 to solve for θ: tan(θ) = sin(θ) / cos(θ) = 1 / µ_s To find θ, we will use the inverse tangent function: θ = atan(1 / µ_s) θ = atan(1 / 0.30) θ ≈ 73.3° The angle between the ramp and the horizontal direction when the box begins to slide down the ramp is approximately 73.3°.