An 9.0 m, 210 N uniform ladder rests against a smooth wall. The coefficient of static friction between the ladder and the ground is 0.75, and the ladder makes a 50.0° angle with the ground. How far up the ladder can an 890 N person climb before the ladder begins to slip?
All of the weight of ladder and person are balanced by the normal force F at the lower contact point, since the upper contact point is frictionless.
F = 1100 N
At the slipping condition, the friction force at the bottom is the maximum allowed by static friction, which would be 1100*0.75 = 825 N. It would be horizontal and pointed to the wall.
Now apply a moment balance using the three forces F, ladder weight and person weight. For simplicity, apply it at the upper (wall) contact point so you won't have to solve for the force there. You should be able to solve for the unknown position of the man.
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To determine how far up the ladder an 890 N person can climb before the ladder begins to slip, we need to find the maximum static friction force that can be exerted between the ladder and the ground.
Let's break down the problem step by step:
Step 1: Calculate the vertical and horizontal components of the ladder's weight.
The ladder's weight (W) has two components: the vertical component (Wv) and the horizontal component (Wh). We can calculate these components using trigonometry:
Wv = W * cos(angle)
Wh = W * sin(angle)
Given:
Weight of the ladder (W) = 210 N
Angle with the ground (angle) = 50.0°
Wv = 210 N * cos(50.0°)
Wv = 210 N * 0.6428
Wv = 135 N
Wh = 210 N * sin(50.0°)
Wh = 210 N * 0.7660
Wh = 161 N
Step 2: Calculate the normal force exerted by the ground on the ladder.
The normal force (N) is the force exerted by a surface perpendicular to it. In this case, the ground exerts a normal force against the ladder. The normal force equals the ladder's vertical component weight (Wv).
N = Wv
N = 135 N
Step 3: Determine the maximum static friction force.
The maximum static friction force (Ffs) can be calculated using the formula:
Ffs = coefficient of static friction (μs) * Normal force (N)
Given:
Coefficient of static friction (μs) = 0.75
Normal force (N) = 135 N
Ffs = 0.75 * 135 N
Ffs = 101.25 N
Step 4: Calculate the distance up the ladder using the maximum static friction force.
To find the distance (d) that an 890 N person can climb before the ladder begins to slip, we need to consider the static friction force is equal to the maximum achievable static friction force (Ffs).
Ffs = m * g, where m is the mass of the person and g is the acceleration due to gravity (9.8 m/s^2).
Rearranging the formula to solve for mass:
m = Ffs / g
Given:
Ffs = 101.25 N
g = 9.8 m/s^2
m = 101.25 N / 9.8 m/s^2
m = 10.33 kg
Now, using the weight of the person (890 N) as the mass (m) in the formula:
Ffs = m * g
Ffs = 890 N = m * 9.8 m/s^2
m = 890 N / 9.8 m/s^2
m = 90.82 kg
The mass of the person is approximately 90.82 kg.
Finally, we can calculate the distance (d) using the formula:
d = Ffs / Wh
Given:
Ffs = 101.25 N
Wh = 161 N
d = 101.25 N / 161 N
d ≈ 0.628 m
Therefore, an 890 N person can climb approximately 0.628 m up the ladder before it begins to slip.