A table is supported against a wall forming an angle with the vertical. The coefficient of static friction between the table and the floor is (μe). The frictional force of the wall on the board may be neglected.
Find an expression for the maximum value that the angle can take without the table slipping on the floor.
Can someone guide me? Thank you.
Certainly! To find the maximum value that the angle can take without the table slipping on the floor, we need to consider the forces acting on the table.
The table is supported against a wall, so the only force acting perpendicular to the table is the normal force (N) exerted by the floor on the table.
The other force we need to consider is the gravitational force (mg) acting vertically downward, where m is the mass of the table and g is the acceleration due to gravity.
Since the table is at an angle with the vertical, we can resolve the gravitational force into two components: one that acts perpendicular to the table (mg cosθ) and one that acts parallel to the table (mg sinθ), where θ is the angle the table makes with the vertical.
The maximum angle at which the table will not slip can be determined by comparing the maximum static frictional force (F_max) with the component of the gravitational force parallel to the table.
The maximum static frictional force (F_max) can be calculated using the equation F_max = μeN, where μe is the coefficient of static friction between the table and the floor, and N is the normal force.
Since the normal force is the same as the component of the gravitational force perpendicular to the table, we have N = mg cosθ.
Therefore, the equation for the maximum angle (θ_max) can be expressed as:
F_max = μe * N = μe * mg * cosθ_max
Comparing this with the component of the gravitational force parallel to the table (mg sinθ), we have:
F_max = μe * mg * cosθ_max >= mg * sinθ
Simplifying the equation, we get:
μe * cosθ_max >= sinθ
To find the maximum value of θ, we need to find the value of θ_max that satisfies this inequality.
One way to solve this inequality is to take the inverse sine of both sides:
sin^(-1)(μe * cosθ_max) >= θ
So the maximum value that the angle (θ) can take without the table slipping on the floor is given by:
θ_max = sin^(-1)(μe * cosθ_max)
I hope this explanation helps! Let me know if you have any further questions.