A block is sitting on a ramp. The block feels a force of friction sufficient to keep it from sliding down the ramp, which makes an angle of 30 degrees w.r.t. horizontal. What is the minimum coefficient of static friction between the block & ramp?

0.5

To determine the minimum coefficient of static friction between the block and the ramp, we need to consider the forces acting on the block.

1. First, let's consider the forces acting parallel to the ramp. We can break down the force of gravity acting on the block into two components: one parallel to the ramp and one perpendicular to the ramp. The component parallel to the ramp is given by F_parallel = m * g * sin(θ), where m is the mass of the block, g is the acceleration due to gravity (approximately 9.8 m/s^2), and θ is the angle of the ramp (30 degrees).

2. The frictional force opposing the motion of the block is given by F_friction = μ * F_normal, where μ is the coefficient of static friction and F_normal is the normal force acting perpendicular to the ramp. In this case, since the block is not sliding down the ramp, the frictional force is equal in magnitude to the component of gravity parallel to the ramp, so F_friction = F_parallel.

3. The normal force is equal in magnitude and opposite in direction to the perpendicular component of gravity, which is given by F_perpendicular = m * g * cos(θ).

4. Since we have established that F_friction = F_parallel, substituting the values we have, we get:
μ * F_normal = F_parallel
μ * (m * g * cos(θ)) = m * g * sin(θ)
μ = (m * g * sin(θ)) / (m * g * cos(θ))
μ = tan(θ)

5. Plugging in the value of θ (30 degrees), we get:
μ = tan(30 degrees)
μ ≈ 0.577

Therefore, the minimum coefficient of static friction between the block and the ramp is approximately 0.577.