A person using a ladder supported on vertical wall is 3/4 of the way up the ladder. If the person climbing the ladder has a weight of 980 newtons and the ladder is 4.89 meters long, how far from the wall can the base of the ladder be placed, and not slip? The coefficient of friction between the base of the ladder and the ground is 0.34. Assume that there is no friction between the ladder and the wall and that the ladder is effectively weightless.
To find the maximum distance from the wall where the base of the ladder can be placed without slipping, we need to consider the forces acting on the ladder.
Step 1: Identify the forces acting on the ladder:
- Weight of the climber: 980 newtons (acting downward)
- Normal force from the ground: acts perpendicular to the ground
- Frictional force between the base of the ladder and the ground: acts parallel to the ground
Step 2: Determine the forces causing the ladder to slip:
The ladder will slip if the maximum frictional force (frictional force = coefficient of friction * normal force) is less than or equal to the component of the weight of the climber acting parallel to the ground.
Step 3: Calculate the component of the weight parallel to the ground:
Since the person is 3/4 of the way up the ladder, the distance from the base of the ladder to the person is 3/4 * 4.89 = 3.67 meters.
The component of the weight acting parallel to the ground is given by 980 * (3.67 / 4.89).
Step 4: Calculate the maximum frictional force:
The maximum frictional force is given by the coefficient of friction * normal force.
Step 5: Equate the maximum frictional force with the component of the weight:
Set the maximum frictional force equal to the component of the weight and solve for the normal force.
Step 6: Calculate the maximum distance from the wall:
Using the calculated normal force, divide the component of the weight by the normal force to determine the maximum distance from the wall.