A ladder has a weight of 30-lbs and a length of 26-ft. Determine the maximum distance D it can be
placed from the smooth wall and not slip. The coefficient of static friction between the floor and the
pole is 0.3.
To determine the maximum distance D the ladder can be placed from the smooth wall without slipping, we need to consider the forces acting on the ladder.
The ladder experiences two forces: the weight acting vertically downward and the friction force acting horizontally at the base of the ladder. For the ladder to remain in equilibrium, the sum of the vertical forces and horizontal forces must be zero.
First, let's determine the vertical force acting on the ladder due to its weight. The weight of the ladder is given as 30 lbs. We need to convert this weight to a force by multiplying it by the acceleration due to gravity (approximately 9.8 m/s^2).
Weight = mass x acceleration due to gravity
The mass of the ladder can be calculated using the formula:
Weight = mass x gravity
30 lbs = mass x 9.8 m/s^2
Solving for mass:
mass = 30 lbs / 9.8 m/s^2
Next, let's determine the friction force acting horizontally at the base of the ladder. The friction force can be calculated using the formula:
Friction force = coefficient of static friction x normal force
The normal force is the force exerted by the floor on the ladder perpendicular to its surface and can be calculated by multiplying the mass of the ladder by the acceleration due to gravity.
Normal force = mass x gravity
Now we can calculate the maximum friction force:
Maximum friction force = coefficient of static friction x normal force
Finally, the maximum distance D can be calculated using the following equation:
Maximum distance D = (maximum friction force) / (weight of the ladder)
Let's plug in the values:
Weight = 30 lbs
Coefficient of static friction = 0.3
mass = 30 lbs / 9.8 m/s^2
normal force = mass x gravity
maximum friction force = coefficient of static friction x normal force
maximum distance D = (maximum friction force) / (weight of the ladder)
It's important to note that using consistent units is crucial for accurate calculations. In this example, we assumed pounds (lbs) for weight and feet (ft) for length. However, converting the values to a consistent unit, such as Newtons (N) for force and meters (m) for length, would provide more accurate results.