A 15 kg uniform ladder leans againt a wall in the atrium of a large hotel. The ladder is 8m long, and it makes an angle as 60 degree with the floor. The coefficient of static friction between the floor and the ladder is u+ 0.45. Assume the wall is frictionless.
a) How far along the ladder can a 60kg painter climb before the ladder starts to slip?
b) Therefore, at where height above the ground can the same painter climb before the ladder starts to slip ?
To determine how far along the ladder a painter can climb before the ladder starts to slip, we need to consider the forces acting on the ladder.
Let's start by breaking down the forces acting on the ladder:
1. The weight of the ladder: The weight acts downward from the center of mass of the ladder. The formula for weight is W = m * g, where m is the mass of the ladder and g is the acceleration due to gravity (approximately 9.8 m/s²). Given that the ladder weighs 15 kg, the weight is W = 15 kg * 9.8 m/s² = 147 N.
2. The normal force: The normal force is the force exerted by the floor on the ladder in the vertical direction. It counteracts the weight of the ladder. Since the ladder is in equilibrium (not moving), the normal force equals the weight of the ladder. Therefore, the normal force is also 147 N.
3. The frictional force: The frictional force on the ladder opposes its impending motion along the floor. The formula for the maximum static frictional force is Fmax = u * N, where u is the coefficient of static friction and N is the normal force. In this case, the coefficient of static friction is given as 0.45, and the normal force is 147 N. Thus, the maximum static frictional force is Fmax = 0.45 * 147 N = 66.15 N.
Since the ladder is leaning against a wall, we have a right-angled triangle formed by the ladder, the floor, and the wall. The angle between the ladder and the floor is given as 60 degrees.
Now, let's calculate how far along the ladder the painter can climb before the ladder starts to slip (let's call this distance "x").
We can determine the forces acting in the horizontal and vertical directions. In the vertical direction, we have:
Sum of forces in the vertical direction = 0
Normal force - weight of the painter - weight of the ladder = 0
147 N - (60 kg * 9.8 m/s²) - 147 N = 0
-588 N = 0 (Since the forces are balanced)
This indicates that the ladder is not in equilibrium in the vertical direction, which means the painter has exceeded the ladder's weight limit (the ladder would start to tip over before it slips vertically).
So, the answer to part (a) is that the painter cannot climb along the ladder.
Now, let's move on to part (b) and find the height above the ground where the painter can climb before the ladder starts to slip.
The ladder will start to slip when the horizontal component of the force due to the painter's weight exceeds the maximum static frictional force.
In the horizontal direction, we have:
Sum of forces in the horizontal direction = 0
Force due to painter's weight = Maximum static frictional force
Weight of the painter * sin(60 degrees) = Fmax
60 kg * 9.8 m/s² * sin(60 degrees) = 66.15 N
Now, we can solve for the height above the ground (let's call it "h") at which the painter can climb:
h = x * sin(60 degrees)
From the earlier equation:
66.15 N = 60 kg * 9.8 m/s² * sin(60 degrees)
66.15 N = 588 N * sin(60 degrees)
sin(60 degrees) = 66.15 N / 588 N
sin(60 degrees) ≈ 0.1125
Using the trigonometric identity sin(60 degrees) = √3/2, we have:
0.1125 ≈ √3/2
√3 ≈ 0.1125 * 2
√3 ≈ 0.225
Now, we can find the height above the ground:
h = x * √3
h = x * 0.225
Therefore, the height above the ground at which the painter can climb before the ladder starts to slip is approximately 0.225 times the distance along the ladder.
To determine where the ladder starts to slip, we need to analyze the forces acting on it. Let's break it down step-by-step.
Step 1: Calculate the weight of the ladder and the painter.
The weight of the ladder can be calculated using the formula: weight = mass × acceleration due to gravity.
The weight of the ladder is:
Weight of ladder = 15 kg × 9.8 m/s^2 = 147 N.
The weight of the painter is:
Weight of painter = 60 kg × 9.8 m/s^2 = 588 N.
Step 2: Calculate the normal force acting on the ladder.
The normal force, N, is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force is exerted by the floor on the ladder.
Since there is no vertical acceleration, the normal force must be equal to the weight of the ladder and the painter combined.
Therefore, N = Weight of ladder + Weight of painter = 147 N + 588 N = 735 N.
Step 3: Calculate the frictional force.
The maximum static friction force, F_friction, can be calculated using the formula: F_friction = u × N, where u is the coefficient of static friction.
Given that the coefficient of static friction, u, is 0.45, we can calculate the frictional force as:
F_friction = 0.45 × 735 N = 330.75 N.
Step 4: Determine the climbing range of the painter.
To prevent the ladder from slipping, the force applied by the painter (F_applied) must not exceed the available frictional force.
The vertical component of the force applied by the painter is F_vertical = Weight of painter × sin(60°).
F_vertical = 588 N × sin(60°) = 509.4 N.
The maximum horizontal component of the force applied by the painter is F_horizontal = Weight of painter × cos(60°).
F_horizontal = 588 N × cos(60°) = 294 N.
Since the maximum available frictional force is 330.75 N, the horizontal component of the force applied by the painter (294 N) is less than the maximum available frictional force.
Therefore, the painter can climb the entire length of the ladder (8 m) before it starts to slip.
Step 5: Calculate the height above the ground where the painter can climb before the ladder starts to slip.
To calculate the height, we need to consider the vertical component of the force applied by the painter and the ladder's angle with the floor.
The ladder's angle with the floor is 60°, so the vertical component of the force applied by the painter (F_vertical) will help support the weight of the ladder.
The height above the ground where the ladder starts to slip can be calculated using the formula: height = ladder length × sin(angle).
Height = 8 m × sin(60°) = 6.93 m.
Therefore, the painter can climb to a height of approximately 6.93 meters above the ground before the ladder starts to slip.
I am going to say g = 10 m/s^2
base of ladder from wall = 4 m
top of ladder at 4 sqrt 3 = 6.93 m
Take moments about top point of ladder
total weight = 10 (15+60) = 750 N
Friction force = .45*750 = 338 N
Clockwise:
750 N * 4 m = 3000
Counterclockwise:
x is horizontal distance painter to wall
338*6.93 + 150*2 + 600(x)
so
600 x = 3000 - 2342 - 300 = 358
x = .596 m
along ladder to top = 2x = 1.19 m
so along ladder from bottom = 8 -1.19 = 6.8 m from floor along ladder to painter
height of painter = (6.8)/2 sqrt 3 = 5.9 m