a bkock placed on a rough inclined plane of length 4m just begins to slide down when the upper end of the plane is 1m high from the ground.calculate the coefficient of static friction.
the angle of the board is arcsin(.25)
mu=tangent(arcsin(.25))
force down the plane=mgSinTheta
force normal: mgCosTheta
friction up plane: mg*mu*cosTheta
movement occurs when force down >friction or
mgSinTheta=>mg*mu*CosTheta or
mu=TanTheta
To calculate the coefficient of static friction, we need to use the concept of equilibrium. The block just begins to slide when the force of gravity pulling it downhill overcomes the static friction holding it in place.
Let's break down the problem step by step:
1. Draw a diagram: Sketch a rough inclined plane with a block on it. Mark the length of the plane as 4m and the vertical height from the upper end to the ground as 1m.
2. Identify the relevant forces: The main forces acting on the block are the gravitational force (mg) pulling it downhill and the normal force (N) perpendicular to the inclined plane.
3. Resolve the forces: Since the block is on the verge of sliding, the force of static friction (fs) is at its maximum. The component of the gravitational force pulling the block downhill is given by mgsinθ, where θ is the angle of inclination.
4. Apply Newton's second law: In the vertical direction, the sum of the forces is equal to zero because the block is not moving vertically. Therefore, N - mgcosθ = 0. This equation tells us that the normal force is equal to mgcosθ.
5. Calculate the force of static friction: In the horizontal direction, the sum of the forces is equal to the force of static friction. Therefore, fs = mgsinθ.
6. Substitute values and solve: We have the equation fs = mgsinθ. The mass (m) cancels out, so we can calculate the coefficient of static friction (μs) using the formula μs = fs / N. By substituting mgcosθ for N, we get μs = mgsinθ / mgcosθ.
7. Simplify the equation: Cancel out the mass (m) from both the numerator and the denominator. The equation becomes μs = sinθ / cosθ.
8. Calculate the angle of inclination: We are given the vertical height (1m) and the length of the plane (4m). Using trigonometry, the angle of inclination θ can be determined as θ = arctan(1/4).
9. Substitute the angle and calculate the coefficient of static friction: Using the value of θ, substitute it back into the equation μs = sinθ / cosθ. Calculate the coefficient of static friction by evaluating this equation.
I hope this step-by-step explanation helps you understand how to calculate the coefficient of static friction on a rough inclined plane!