A ladder 6 m long has a mass of 18 kg and its center of gravity is 2.4 m from the bottom. The ladder is placed against a vertical wall so that it makes an angle of 60° with the ground. How far up the ladder can a 72-kg man climb before the ladder is on the verge of slipping? The angle of friction at all contact surfaces is 15°.
Question
A ladder 6 m long has a mass of 18 kg and its center of gravity is 2.4 m from the bottom. The ladder is placed against a vertical wall so that it makes an angle of 60° with the ground. How far up the ladder can a 72-kg man climb before the ladder is on the verge of slipping? The angle of friction at all contact surfaces is 15°.
Solution
To solve this problem, we need to use the principles of static equilibrium and the concept of friction.
Step 1: Identify the forces acting on the ladder There are three forces acting on the ladder:
- The weight of the ladder (W1) acting downwards from its center of gravity
- The weight of the man (W2) acting downwards from the point where he is on the ladder
- The frictional force (F) acting horizontally at the point where the ladder touches the ground
Step 2: Calculate the weights W1 = mass of ladder * gravity = 18 kg * 9.8 m/s^2 = 176.4 N W2 = mass of man * gravity = 72 kg * 9.8 m/s^2 = 705.6 N
Step 3: Set up the equations for static equilibrium For the ladder to be in equilibrium, the sum of the moments about any point must be zero. Let's choose the point where the ladder touches the ground.
The moment due to W1 is W1 * distance from the ground to the center of gravity of the ladder = 176.4 N * 2.4 m = 423.36 Nm
The moment due to W2 is W2 * distance from the ground to the man (x) = 705.6 N * x
The moment due to F is F * length of the ladder = F * 6 m
Step 4: Set up the equation for the frictional force The frictional force is given by F = μN, where μ is the coefficient of friction and N is the normal force. The normal force is the component of the weight perpendicular to the ladder, which is (W1 + W2) * cos(60°). The coefficient of friction is tan(15°).
Step 5: Solve the equations Setting the sum of the moments equal to zero gives:
423.36 Nm + 705.6 N * x = F * 6 m
Substituting F = μN gives:
423.36 Nm + 705.6 N * x = tan(15°) * (W1 + W2) * cos(60°) * 6 m
Solving this equation for x gives the maximum distance the man can climb before the ladder is on the verge of slipping.
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