a mass of 3kg rest on horizontal plane. the plane gradually inclined until at a angel 0=20 degrees with the horizontal, the mass just begins to slide. what is the coefficent of static friction between the block and the surface?
To find the coefficient of static friction between the block and the surface, you need to use the concept of equilibrium. When the mass is just about to slide, the force of static friction is equal to the force exerted parallel to the inclined plane.
Here's how you can solve it step by step:
1. Draw a free-body diagram for the mass on the inclined plane. The forces acting on the mass are the gravitational force (mg) acting vertically downward and the static friction force (fs) acting parallel to the inclined plane.
2. Resolve the gravitational force into components. Since the plane is inclined at an angle of θ = 20 degrees with the horizontal, the component of the weight acting perpendicular to the plane is mg * cos(θ), and the component acting parallel to the plane is mg * sin(θ).
3. Write the equation for equilibrium in the direction perpendicular to the plane. The sum of the forces in this direction should be zero:
mg * cos(θ) = Normal force
4. Write the equation for equilibrium in the direction parallel to the plane. The sum of the forces in this direction should be zero:
fs = mg * sin(θ)
5. Now, since the mass is just about to slide, the static friction force (fs) is at its maximum value, which is given by:
fs = μs * Normal force,
where μs is the coefficient of static friction.
6. Substitute the equation for normal force from step 3 into the equation for fs from step 5:
μs * (mg * cos(θ)) = mg * sin(θ)
7. Solve the equation for μs:
μs = (mg * sin(θ)) / (mg * cos(θ))
μs = tan(θ)
So, the coefficient of static friction between the block and the surface is equal to the tangent of the angle of inclination:
μs = tan(20 degrees)
Using a scientific calculator, you can find that tan(20 degrees) is approximately 0.364.
Therefore, the coefficient of static friction between the block and the surface is approximately 0.364.