A 30 kg block slides down a 20° ramp with an acceleration of 1.2 m/s2. What is the coefficient of kinetic friction between the block and the ramp?
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To find the coefficient of kinetic friction between the block and the ramp, we can start by analyzing the forces acting on the block:
1. Weight (downward force due to gravity): This force can be calculated by multiplying the mass of the block (m = 30 kg) by the acceleration due to gravity (g ≈ 9.8 m/s^2). Therefore, the weight force is W = m * g.
2. Normal force (perpendicular force exerted by the ramp on the block): It acts perpendicular to the ramp surface. On a ramp inclined at an angle θ, the normal force (N) can be calculated by N = m * g * cos(θ). Here, θ is 20°.
3. Friction force: The friction force (F_friction) acts parallel to the surface of contact between the block and the ramp. It can be expressed as F_friction = coefficient of kinetic friction (μ_k) * N. This is the force that opposes the motion of the block down the ramp.
Now, using Newton's second law of motion (F = ma), we can equate the net force with the mass of the block multiplied by its acceleration. The net force is given by F_net = W - F_friction - m * g * sin(θ), where sin(θ) accounts for the component of the weight force acting down the ramp.
Therefore, we have:
m * a = W - F_friction - m * g * sin(θ)
Substituting the values we know:
m = 30 kg
a = 1.2 m/s^2
g ≈ 9.8 m/s^2
θ = 20°
We can plug in these values into the equation and solve for F_friction:
30 kg * 1.2 m/s^2 = (30 kg * 9.8 m/s^2) - F_friction - (30 kg * 9.8 m/s^2 * sin(20°))
Calculating the right side of the equation:
F_friction = (30 kg * 9.8 m/s^2) - (30 kg * 1.2 m/s^2) + (30 kg * 9.8 m/s^2 * sin(20°))
F_friction ≈ 292.4 N
Finally, substituting F_friction = μ_k * N, we can determine the coefficient of kinetic friction:
μ_k = F_friction / N
N = m * g * cos(θ)
μ_k = (30 kg * 9.8 m/s^2 * cos(20°)) / (30 kg * 9.8 m/s^2)
Calculating the right side of the equation:
μ_k ≈ cos(20°)
μ_k ≈ 0.938
Therefore, the coefficient of kinetic friction between the block and the ramp is approximately 0.938.