A uniform ladder of mass m = 40 kg and length l = 10 m is leaned against a smooth vertical wall. A person of mass M = 80 kg stands on the ladder a distance x = 7 m from the bottom, as measured along the ladder. The foot of the ladder is d = 1.2 m from the bottom of the wall. What are the force exerted by the wall and the normal reaction exerted by the floor on the ladder?
Select one:
321.7 N, 1133.5 N
210.3 N, 1200 N
90.1 N, 1177.2 N
70.3 N, 2133.6 N
321.7 N, 1133.5 N
To find the force exerted by the wall and the normal reaction exerted by the floor on the ladder, we can use the principles of equilibrium.
Step 1: Calculate the torque exerted on the ladder by the person's weight.
The torque exerted by an object is given by the formula:
Torque = Force x Lever Arm
In this case, the force is the weight of the person, which is equal to M x g, where M is the mass of the person and g is the acceleration due to gravity (9.8 m/s^2).
Lever Arm = Distance from the point of rotation to the force
In this case, the point of rotation is the point where the ladder touches the ground, and the distance is given as d = 1.2 m.
So, the torque exerted on the ladder by the person's weight is:
Torque = (M x g) x d = (80 kg x 9.8 m/s^2) x 1.2 m
Step 2: Calculate the torque exerted on the ladder by the ladder's weight.
Similarly, the torque exerted by the ladder's weight can be calculated by using the ladder's center of mass, which is located at the midpoint of the ladder.
Lever Arm = Distance from the point of rotation to the center of mass
In this case, the point of rotation is the point where the ladder touches the ground, and the distance is half of the ladder's length, l/2.
Torque = (m x g) x (l/2) = (40 kg x 9.8 m/s^2) x (10 m/2)
Step 3: Calculate the torque exerted on the ladder by the wall.
Since the ladder is in equilibrium, the sum of all torques must be zero. Therefore, the torque exerted on the ladder by the wall is equal to the sum of the torques exerted by the person and the ladder's weight.
Now we can set up an equation:
Torque exerted by the person - Torque exerted by the ladder's weight = 0
(M x g) x d - (m x g) x (l/2) = 0
Step 4: Solve for the force exerted by the wall.
Dividing both sides of the equation by d, we get:
(M x g) - (m x g x (l/2d)) = 0
(M x g) - (m x g x (l / 2d)) = 0
Now we can substitute the given values:
(80 kg x 9.8 m/s^2) - (40 kg x 9.8 m/s^2 x (10 m / (2 x 1.2 m)))
Simplifying this expression will give us the force exerted by the wall.
Step 5: Calculate the normal reaction exerted by the floor on the ladder.
To find the normal reaction exerted by the floor on the ladder, we can use the principle of vertical equilibrium. The sum of all vertical forces acting on the ladder must be zero.
The vertical forces acting on the ladder are the weight of the ladder and the weight of the person, which are balanced by the normal reaction exerted by the floor.
So, the normal reaction exerted by the floor is equal to the sum of the weights of the ladder and the person.
Normal reaction = (m x g) + (M x g)
Now, substitute the given values and calculate the normal reaction exerted by the floor on the ladder.
By following these steps and calculations, you will be able to determine the correct answer among the given options.