A 75.0 kg ladder that is 3.00 m in length is placed against a wall at an angle theta.The center of gravity of the ladder is at a point 1.20 m from the base of the ladder.The coefficient of static friction at the base of the ladder is 0.800.There is no friction between the wall and the ladder.What is the vertical force of the ground on the ladder?
A) 625 N
B) 640 N
C) 735 N
D) 832 N
E) 900 N

Question

A 75.0 kg ladder that is 3.00 m in length is placed against a wall at an angle theta.The center of gravity of the ladder is at a point 1.20 m from the base of the ladder.The coefficient of static friction at the base of the ladder is 0.800.There is no friction between the wall and the ladder.What is the vertical force of the ground on the ladder? 
A) 625 N 
B) 640 N 
C) 735 N 
D) 832 N 
E) 900 N

Quizplus · Accepted Answer

First, we need to find the force exerted by the wall on the ladder, which is perpendicular to the wall, and we can relate this force to the weight and angle of the ladder using trigonometry. The force exerted by the ground on the ladder, which we are trying to find, will be equal and opposite to the force exerted by the wall on the ladder. We can then use the coefficient of static friction and the weight of the ladder to find the maximum force of static friction, which is the force required to keep the ladder from sliding sideways. This force will be in the same direction as the force exerted by the wall on the ladder, and it can be related to the force exerted by the wall and the angle of the ladder using trigonometry again.

Using trigonometry, we can find that the force exerted by the wall on the ladder is F_wall = mg*sin(theta) = (75.0 kg)*(9.81 m/s^2)*sin(theta) and the maximum force of static friction is F_friction = coefficient * F_ground = 0.800 * mg*cos(theta) = 0.800*(75.0 kg)*(9.81 m/s^2)*cos(theta). Since the ladder is not sliding sideways, we know that F_wall <= F_friction. Therefore, we can set these two expressions equal to each other and solve for F_ground:

(75.0 kg)*(9.81 m/s^2)*sin(theta) = 0.800*(75.0 kg)*(9.81 m/s^2)*cos(theta)
tan(theta) = 1/0.800
theta = 38.7 degrees

F_wall = (75.0 kg)*(9.81 m/s^2)*sin(38.7 degrees) = 735 N
F_ground = F_wall = 735 N

Therefore, the vertical force of the ground on the ladder is 735 N, and the best choice is (C).

A 750 Kg Ladder That Is 3