Final answer:
To determine if a rod held at a certain angle to the horizontal will not slip, we compare the friction force to the weight component parallel to the incline. The maximum angle for which the rod will not slip is given by θ = tan^(-1)(μs), where μs is the coefficient of friction.
Step-by-step explanation:
To determine if the lower end of a rod held at an angle θ to the horizontal will not slip when released, we need to compare the friction force to the component of the weight of the rod parallel to the incline. If the friction force is greater than or equal to the weight component, the rod will not slip.
The friction force is given by fs ≤ μsN, where μs is the coefficient of friction and N is the normal force. The normal force can be calculated as N = Mgcosθ, where M is the mass of the rod and g is the acceleration due to gravity.
The weight component parallel to the incline is given by Mgsinθ. So, the condition for the rod not to slip is fs ≥ Mgsinθ.
Now, we can substitute fs and N into the inequality and solve for μs:
μsN ≥ Mgsinθ
μs(Mgcosθ) ≥ Mgsinθ
μs ≥ tgθ
Therefore, the maximum angle of the incline above the horizontal for which the rod will not slip is θ = tan^(-1)(μs).