A uniform ladder of mass M and length L leans at an angle theta against a frictionless wall. If the coefficient of static friction between the ladder and the ground is U, determine a formula for the minimum angle at which the ladder will not slip.
On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed v_0 and a "reverse" spin of angular speed w_0. A kinetic friction force acts on the ball as it initially skids across the table.
a) Using conservation of angular momentum, find the critical angular speed w_c such that, if w_0 = w_c, kinetic friction will bring the ball to a complete (as opposed to momentary) stop.
b)f w_0 is 20% smaller than w_c i.e., w_0 = 0.80w_c, determine the ball's center of mass velocity (v_CM) when it starts to roll without slipping.
To determine the minimum angle at which the ladder will not slip, let's analyze the forces acting on the ladder.
There are three forces to consider:
1. The weight of the ladder, which acts vertically downward and can be expressed as W = Mg, where M is the mass of the ladder and g is the acceleration due to gravity.
2. The normal force exerted by the ground on the ladder, which acts perpendicular to the ground and cancels out the weight of the ladder.
3. The friction force between the ladder and the ground, which acts parallel to the ground and opposes the motion or tendency of motion.
Since the ladder is not slipping, there will be static friction between the ladder and the ground. The maximum possible static friction force is given by fs = μsN, where μs is the coefficient of static friction and N is the normal force.
Now, let's consider the forces acting in the horizontal direction. There are two forces to consider:
1. The component of the weight of the ladder acting parallel to the ground, which is given by Wsinθ.
2. The friction force acting in the opposite direction, which is denoted as fs.
To determine the minimum angle at which the ladder will not slip, we need to find the maximum angle for which the force of friction fs is still large enough to prevent slipping. In other words, we want to find the maximum angle for which fs is equal to or greater than Wsinθ.
Setting fs = Wsinθ, we have:
μsN = Wsinθ
Since N = Mg, we can substitute it into the equation:
μsMg = Wsinθ
Now, W can be expressed as W = Mg, so the equation becomes:
μsMg = (Mg)sinθ
Canceling out Mg:
μs = sinθ
To find the minimum angle θ, we need to take the inverse sine of both sides:
θ = sin^(-1)(μs)
Therefore, the formula for the minimum angle at which the ladder will not slip is:
θ = sin^(-1)(μs)