A ladder can fall for two reasons. If it is set too steep and the climber gets their mass to the left of the ladder's base, the ladder likely will fall over backwards. If the ladder is set at too shallow of an angle the required force of friction between the ladder and the ground might be too great and the base of the ladder will slip. Assume that there is no friction between the ladder and the wall and that the ladder is effectively weightless. The coefficient of friction between the base of the ladder and the ground is 0.49. The person using the ladder will be 76% of the way up the ladder. If the person climbing the ladder has a weight of 970 newtons and the ladder is 5.7 meters long, how far from the wall can the base of the ladder be placed and not slip?
Distance from wall =
sum moments about the top of the ladder.
Is this a weightless ladder? the ladder weight is not given.
clockwise positive, and my sketch has clockwise toward the wall.
d is the distance from base to wall
Theta is the base angle, so d/5.7=sinTheta
sum moments about the top wall junction point with the ladder.
W*(.24*5.7*970g sinTheta)-970g*.76*5.7*cosTeta*mu=0
(the sum of the moments about any point is equal to zero)
divde through by cosTheta, then solve for tanTheta, then d=5.7cosTheta
To find the distance from the wall at which the base of the ladder will not slip, we need to consider the forces acting on the ladder.
1. Weight of the climber: The climber's weight is given as 970 newtons. This force acts downwards at a distance of 76% of the ladder's length, which is equal to 0.76 * 5.7 meters.
2. Frictional force: The coefficient of friction between the base of the ladder and the ground is given as 0.49. The frictional force acts in the opposite direction of motion, which in this case would be upwards.
To prevent the ladder from slipping, the frictional force must be greater than or equal to the component of the climber's weight parallel to the ground.
Step 1: Calculate the component of the climber's weight parallel to the ground:
Weight_parallel = Climber's weight * sin(theta)
theta = angle of the ladder with respect to the ground = arctan(0.76)
Step 2: Calculate the frictional force:
Frictional_force = Coefficient of friction * Normal force = Coefficient of friction * Climber's weight * cos(theta)
Step 3: Set up the inequality to find the maximum distance from the wall:
Frictional_force >= Weight_parallel
Step 4: Substitute the values and solve for the maximum distance:
Coefficient of friction * Climber's weight * cos(theta) >= Climber's weight * sin(theta)
Coefficient of friction * cos(theta) >= sin(theta)
0.49 * cos(arctan(0.76)) >= sin(arctan(0.76))
Now, you can use a scientific calculator or an online trigonometric calculator to evaluate this inequality.
After evaluating the inequality, you can find the maximum distance from the wall at which the base of the ladder can be placed without slipping.