A ladder leans against a wall 5 meters above the ground so a homeowner can clear his gutters of leaves from last fall. The bottom of the ladder just begins to slip when it is 2.4 meters from the wall. What is the coefficient of static friction between the bottom of the ladder and the ground?
weight of ladder = w
assume no friction between wall and ladder so
Force down on ground = w = force upon ladder from ground
now there is a horizontal force from the wall on the top of the ladder, call it F
ladder weight w down at center of ladder
take moments about ladder at ground
F * 5/2 = W (2.4/2)
so
F = (2.4 /5) W = 0.48 W
lo and behold, that F is balanced by the friction force at the base of the ladder
so mu = 0.48
To find the coefficient of static friction, we need to analyze the forces acting on the ladder at the moment it begins to slip.
Let's consider the forces involved:
1. Weight (W) - It acts vertically downward due to gravity and can be calculated using the formula W = m * g, where m is the mass of the ladder and g is the acceleration due to gravity (approximately 9.8 m/s^2).
2. Normal Force (N) - It acts perpendicularly to the surface of contact between the ladder and the ground. In this case, it is equal in magnitude and opposite in direction to the weight of the ladder, which can be represented as N = W.
3. Frictional Force (f) - It acts parallel to the ground, opposing the tendency of the ladder to slip. The magnitude of the frictional force can be obtained using the formula f = μ * N, where μ is the coefficient of static friction.
When the ladder is on the verge of slipping, the frictional force is at its maximum, given by f = μ * N, where N is equal to the weight of the ladder.
The weight of the ladder can be calculated as follows:
W = m * g = mass of the ladder * acceleration due to gravity
The vertical distance between the ground and the point where the ladder touches the wall is 5 meters. Using similar triangles, we can find that the horizontal distance is 2.4 meters.
Using the Pythagorean theorem, we can calculate the length of the ladder:
ladder length^2 = 2.4^2 + 5^2
Once we have the length of the ladder, we can calculate the weight of the ladder.
Finally, we can solve for the coefficient of static friction using the equation f = μ * N, where f is the weight of the ladder.
By substituting the calculated weight of the ladder into the equation, we can find the coefficient of static friction (μ).
Note: Please provide the mass of the ladder for a more accurate calculation.