A uniform 7.1 m tall aluminum ladder is leaning against a frictionless vertical wall. The ladder has a weight of 272 N. The ladder slips when it makes a 57.0degrees angle with the horizontal floor.
Determine the coefficient of static friction between the ladder and the floor.
See below for my answer to this question posted earlier
To determine the coefficient of static friction between the ladder and the floor, we need to consider the forces acting on the ladder. Let's break down the problem step by step.
1. Start by drawing a free-body diagram of the forces acting on the ladder. On the diagram, draw the weight of the ladder (acting downward at the center of mass of the ladder) and the normal force exerted by the floor (acting perpendicular to the floor).
2. Since the ladder is in equilibrium (not accelerating), we can conclude that the net force in both the horizontal and vertical directions is zero. This means that the sum of the horizontal forces acting on the ladder is zero. In this case, the only horizontal force is the static friction between the ladder and the floor.
3. The static friction force can be calculated using the equation: fs = μs * Fn, where fs is the static friction force, μs is the coefficient of static friction, and Fn is the normal force exerted by the floor.
4. In this problem, the normal force Fn is equal to the weight of the ladder since the ladder is at rest (Fn = mg, where m is the mass of the ladder and g is the acceleration due to gravity). So, Fn = 272 N.
5. We need to find the maximum angle θ when the ladder slips. At this point, the static friction force reaches its maximum value, fs = μs * Fn.
6. The maximum angle θ is related to the coefficient of static friction μs by the equation: μs = tan(θ). This is because the tangent of the angle is equal to the ratio of the opposite side (fs) to the adjacent side (Fn).
7. Rearrange the equation to solve for μs: μs = fs / Fn = tan(θ).
8. In this problem, the maximum angle θ is given as 57.0 degrees. Therefore, μs = tan(57.0 degrees). Use a calculator to find the value of tan(57.0 degrees), which is approximately 1.540.
So, the coefficient of static friction between the ladder and the floor is approximately 1.540.