A 10.9-kg monkey climbs a uniform ladder with weight w = 1.24 102 N and length L = 2.70 m as shown in the figure below. The ladder rests against the wall and makes an angle of θ = 60.0° with the ground. The upper and lower ends of the ladder rest on frictionless surfaces. The lower end is connected to the wall by a horizontal rope that is frayed and can support a maximum tension of only 80.0 N.
To solve this problem, we can use the principles of equilibrium to determine the maximum angle at which the monkey can climb the ladder without causing it to slip or the rope to break.
First, let's consider the forces acting on the ladder. There are three forces: the weight of the ladder (W_ladder), the weight of the monkey (W_monkey), and the tension in the rope (T).
The weight of the ladder and the monkey can be calculated using the equations:
W_ladder = mass_ladder * g
W_monkey = mass_monkey * g
Where:
mass_ladder = 10.9 kg (mass of the ladder)
mass_monkey = unknown (mass of the monkey)
g = 9.8 m/s^2 (acceleration due to gravity)
Next, we need to analyze the forces acting on the ladder in the x and y directions. In the x direction, there is only one force, the tension force (T), acting to the right.
In the y direction, there are two forces: the weight of the monkey (W_monkey) acting downwards, and the vertical component of the tension force (T_y) acting upwards.
Using trigonometry, we can find the relationship between the tension force and its vertical and horizontal components:
T_y = T * sin(theta)
Now, let's analyze the forces acting on the monkey. There are two forces: the weight of the monkey (W_monkey) acting downwards, and the vertical component of the tension force (T_y) acting upwards.
To ensure that the ladder doesn't slip, the friction force between the ladder and the ground must be greater than or equal to zero. The friction force can be calculated using the equation:
Friction force = coefficient of friction * normal force
Since we know that the ladder rests on a frictionless surface, the friction force is zero. Therefore, the normal force is also zero.
Now, we can determine the maximum angle at which the monkey can climb the ladder without causing the ladder to slip or the rope to break.
To find this angle, we need to set up an equation that equates the vertical component of the tension force to the weight of the monkey:
T_y = W_monkey
Substituting the values we know:
T * sin(theta) = W_monkey
Now, we can solve for theta:
theta = arcsin(W_monkey / T)
Substituting the known values:
theta = arcsin((mass_monkey * g) / T)
We also know that the tension in the rope cannot exceed 80.0 N. Therefore, we can set an inequality to find the maximum mass of the monkey:
mass_monkey * g ≤ T
Solving for the maximum mass of the monkey:
mass_monkey ≤ T / g
Substituting the values we know:
mass_monkey ≤ 80.0 N / 9.8 m/s^2
Finally, we can substitute the maximum mass of the monkey into the equation for theta to find the maximum angle at which the monkey can climb the ladder without causing it to slip or the rope to break.