A uniform ladder of mass m and length L leans at an angle θ against a frictionless wall. If the coefficient of static friction between the ladder and the ground is 0.43, what is the minimum angle at which the ladder will not slip?
Do i use the equation
Fn- Mg= 0
yes, you start with that. I would sum moments about any point and those moments to zero. And finally, the sum of forces in the horizontal are zero also.
To determine the minimum angle at which the ladder will not slip, we need to consider the forces acting on the ladder.
1. Normal force (Fn): The perpendicular force exerted by the ground on the ladder.
2. Weight (Mg): The downward force due to the ladder's mass.
3. Frictional force (Fs): The force opposing the ladder from slipping.
Since the ladder is not slipping, the maximum static friction force (Fsmax) should be equal to the force component parallel to the ground (Fs):
Fsmax = μs * Fn,
where μs is the coefficient of static friction between the ladder and the ground.
The force component parallel to the ground can be determined as follows:
Fs = Mg * sin(θ),
where θ is the angle the ladder makes with the ground.
Therefore, we have:
Fsmax = μs * Fn = Mg * sin(θ).
Since the ladder is in equilibrium, the vertical forces must cancel each other out:
Fn - Mg * cos(θ) = 0.
Solving this equation for Fn, we get:
Fn = Mg * cos(θ).
Now, substituting Fn in terms of Mg * cos(θ) in the expression for Fsmax, we get:
Mg * cos(θ) * μs = Mg * sin(θ).
Dividing both sides by Mg, we obtain:
cos(θ) * μs = sin(θ).
Finally, solving for θ, we have:
θ = arctan(μs).
Therefore, the minimum angle at which the ladder will not slip is given by the arctan(μs) or inverse tangent of the coefficient of static friction. In this case, the minimum angle is:
θ = arctan(0.43).
To determine the minimum angle at which the ladder will not slip, you need to consider the forces acting on the ladder. The equation you mentioned, Fn - Mg = 0, is not sufficient in this case.
Here's the step-by-step process to solve the problem:
1. Identify the forces: The forces acting on the ladder are the weight (mg), the normal force (Fn) exerted by the ground, and the frictional force (f) between the ladder and the ground.
2. Break down the weight: Divide the weight into two components: one parallel to the ladder (mg sinθ) and the other perpendicular to the ladder (mg cosθ).
3. Sum the forces: Considering the equilibrium condition (no net force), the vertical forces equation is: Fn - mg cosθ = 0. This is due to the vertical equilibrium of the ladder.
4. Determine the frictional force: The maximum static frictional force is given by the coefficient of static friction (μs) multiplied by the normal force. Therefore, f ≤ μsFn.
5. Calculate the forces along the ladder: The horizontal forces equation is: f - mg sinθ = 0. This equation represents the static frictional force balancing the component of the weight along the ladder.
6. Substitute and simplify: Substitute the equation for f from step 4 into the equation from step 5. This gives you μsFn - mg sinθ = 0.
7. Solve for θ: Rearrange the equation to solve for the angle θ: sinθ = μsFn / mg.
8. Substitute the known values: Replace the given coefficient of static friction (μs), mass of the ladder (m), and other known quantities into the equation from step 7.
9. Calculate the minimum angle: Use inverse sine (sin^-1) to find the minimum angle.
By following these steps, you can find the minimum angle at which the ladder will not slip.