A uniform ladder 5 m long weighing 200 N
friction between the bottom of the ladder and the
its base 3m from the wall. The coefficient of static
ground is 0.45. How far, measured along the
ladder, can a 600 N man climb before the ladder
is leaning against a frictionless vertical wall with
starts to slip?
To find out how far a 600 N man can climb before the ladder starts slipping, we need to determine the maximum friction force that the ground can provide before the ladder starts slipping.
First, let's calculate the weight of the ladder using the given information. The weight of the ladder is 200 N.
Next, let's calculate the normal force exerted by the ground on the ladder. The normal force is equal to the weight of the ladder because the ladder is at rest in the vertical direction. Therefore, the normal force is also 200 N.
Now, let's calculate the maximum friction force the ground can provide using the coefficient of static friction. The equation for static friction is Fs ≤ μs * N, where Fs is the static friction force, μs is the coefficient of static friction, and N is the normal force. Plugging in the values, we have Fs ≤ 0.45 * 200 N = 90 N.
Since the ladder is leaning against a frictionless vertical wall, the friction force will not help prevent the ladder from slipping. Hence, the maximum force that can support the ladder in the horizontal direction is 0 N.
Now, let's consider the forces acting on the ladder. We have the weight of the ladder (200 N) acting downward, the normal force (200 N) acting upward, and the force exerted by the man (600 N) acting upward.
The net force on the ladder in the horizontal direction will be the difference between the force exerted by the man and the maximum friction force that the ground can provide. So, the net force in the horizontal direction is 600 N - 90 N = 510 N.
To calculate how far the man can climb before the ladder starts slipping, we need to determine the point of contact of the ladder with the ground when the ladder is about to slip. Let's assume that the man climbs x meters up the ladder.
Using the concept of moments, we can equate the clockwise and counterclockwise moments about the point of contact between the ladder and the ground.
The clockwise moment is equal to the weight of the ladder multiplied by its distance from the point of contact, which is 200 N * 3 m = 600 Nm.
The counterclockwise moment is equal to the force exerted by the man multiplied by his distance from the point of contact, which is 600 N * x m = 600x Nm.
Setting the clockwise moment equal to the counterclockwise moment, we get 600 Nm = 600x Nm.
Simplifying, we find x = 1 meter.
Therefore, the man can climb 1 meter along the ladder before it starts slipping against the frictionless vertical wall.