A 7.68 m, 132 kg uniform ladder rests against a smooth wall. The coefficient of static fric- tion between the ladder and the ground is 0.642, and the ladder makes a 48.4◦ angle with the ground.
How far up the ladder can a 528 kg person climb before the ladder begins to slip? The acceleration of gravity is 9.8 m/s2 .
Answer in units of m
I have tried to find normal force:
I got the weights of the person(5174N) and the ladder(1293.6N). And then that answer would be the force of floor on ladder.
That is correct. Now you have to figure the horizontal forces, they have to sum to zero. You have the wall force, and you have the force of friction on the ground.
Try this: Sum moments about the base, and find wall force in terms of height h of the man. Then ground fricton has to equal that, and you can solve then for h.
Horizontal force: i tried to do 7.68cos 48.4 but maybe I should get the cosine components of the weights.
To find how far up the ladder a 528 kg person can climb before the ladder begins to slip, we need to consider the forces acting on the ladder and use the concept of static friction.
Let's break down the problem step by step:
Step 1: Resolve forces:
The weight of the person can be calculated as:
Weight_person = mass_person * acceleration_due_to_gravity
Weight_person = 528 kg * 9.8 m/s^2
Weight_person = 5174.4 N
The weight of the ladder can be calculated as:
Weight_ladder = mass_ladder * acceleration_due_to_gravity
Weight_ladder = 132 kg * 9.8 m/s^2
Weight_ladder = 1293.6 N
Step 2: Calculate the force exerted on the ladder by the floor:
The normal force (force exerted by the floor on the ladder) can be calculated as:
Normal_force = Weight_person + Weight_ladder
Normal_force = 5174.4 N + 1293.6 N
Normal_force = 6468 N
Step 3: Determine the maximum static friction force:
The maximum static friction force can be calculated using the coefficient of static friction and the normal force:
Maximum_friction_force = coefficient_of_static_friction * Normal_force
Maximum_friction_force = 0.642 * 6468 N
Maximum_friction_force = 4159.976 N
Step 4: Calculate the force exerted by the person on the ladder:
The force exerted by the person on the ladder can be calculated using the angle of the ladder with the ground:
Force_person_on_ladder = Weight_person * sin(angle_of_ladder)
Force_person_on_ladder = 5174.4 N * sin(48.4°)
Force_person_on_ladder = 5174.4 N * 0.7436
Force_person_on_ladder = 3848.8464 N
Step 5: Determine the maximum height the person can climb:
The maximum height the person can climb before the ladder begins to slip is limited by the maximum static friction force. To find the height, we can equate the force exerted by the person on the ladder to the maximum static friction force:
Force_person_on_ladder = Maximum_friction_force
3848.8464 N = 4159.976 N
Step 6: Solve for the distance climbed:
Now, we can solve for the distance climbed by rearranging the equation:
Distance_climbed = (Force_person_on_ladder / Weight_person) * ladder_length
Distance_climbed = (3848.8464 N / 5174.4 N) * 7.68 m
Distance_climbed = 5.715 m (rounded to three decimal places)
Therefore, a 528 kg person can climb approximately 5.715 meters up the ladder before it begins to slip.