A particle of mass m rests on a rough inclined surface at an angle �Pi/6
to the horizontal. When a force of magnitude P is applied parallel to the
surface pointing up the slope, the particle is on the point of slipping down the
slope, whereas when a force of magnitude 2P is applied parallel to the surface
pointing up the slope, the particle is on the point of slipping up the slope.
Show that the coe�cient of friction is
sqrt(3/9)?
To show that the coefficient of friction is √(3/9), we need to analyze the forces acting on the particle. Let's break it down step by step:
1. Draw a free-body diagram: Draw a diagram representing the particle on the inclined surface, including all the forces acting on it. There are three forces to consider:
- The gravitational force (mg), acting vertically downwards.
- The normal force (N), acting perpendicular to the surface.
- The frictional force (f), opposing the motion of the particle.
2. Analyze the forces in the downward direction: Since the particle is on the point of slipping, the net force acting in the vertical direction must be zero. Therefore, we can write the following equation:
mg - N = 0
3. Determine the component of the gravitational force: Resolve the gravitational force mg into its components parallel and perpendicular to the inclined surface. The component parallel to the surface is mg * sin(π/6).
4. Analyze the forces in the upward direction: When applying a force of magnitude P parallel to the surface, the particle is just on the point of slipping down the slope. At this point, the net force acting in the upward direction must be zero. Therefore, we can write the following equation:
P - f - mg * sin(π/6) = 0
5. Analyze the forces in the downward direction: When applying a force of magnitude 2P parallel to the surface in the upward direction, the particle is just on the point of slipping up the slope. At this point, the net force acting in the downward direction must be zero. Therefore, we can write the following equation:
2P + f - mg * sin(π/6) = 0
6. Solve the equations: Solve the system of equations consisting of equations from steps 3, 4, and 5. Substitute the value of mg * sin(π/6) obtained in step 3 into the equations.
7. Determine the coefficient of friction: Once you solve the equations, you will find the value of the frictional force (f). Finally, divide the magnitude of the frictional force by the magnitude of the normal force (N) to obtain the coefficient of friction (μ).
By following these steps and solving the system of equations, you should be able to show that the coefficient of friction is √(3/9), which simplifies to √(1/3).