Problem Statement
A uniform ladder of mass 10 kg and length 4 m leans against a smooth wall making 60° with the floor ($\mu_s = 0.4$ at floor). Does the ladder slip? ($g = 10$ m/s²)
Given Information
- All quantities, constants, and constraints stated in the problem above
- Physical constants used as needed (see Concepts section)
Physical Concepts & Formulas
This problem draws on fundamental physical principles. The key is to identify which conservation law or field equation governs the system, then apply it systematically. Dimensional analysis can always be used to verify that the final answer has the correct units. Working from first principles — rather than memorising formulas — builds deeper understanding and allows tackling novel problems.
- Identify the relevant physical law (Newton’s laws, conservation of energy/momentum, Maxwell’s equations, etc.)
- State the mathematical form of that law as it applies here
- Check dimensions at every step: both sides of an equation must have the same units
Step-by-Step Solution
Problem Statement
A uniform ladder of mass 10 kg and length 4 m leans against a smooth wall making 60° with the floor ($\mu_s = 0.4$ at floor). Does the ladder slip? ($g = 10$ m/s²)
Concepts Used
- Torque equilibrium + force equilibrium; friction at floor, normal at wall
Step-by-Step Solution
Step 1: Forces: $W = 100$ N (center); $N_w$ (horizontal, wall); $N_f$ (vertical, floor); $f$ (friction, horizontal at floor).
Step 2: Vertical: $N_f = W = 100$ N.
Step 3: Torques about foot: $N_w \times 4\sin60° = W \times 2\cos60°$.
$N_w = \dfrac{100 \times 2 \times 0.5}{4 \times (\sqrt{3}/2)} = \dfrac{100}{4\sqrt{3}/2} = \dfrac{100}{2\sqrt{3}} = \dfrac{50}{\sqrt{3}} \approx 28.9$ N.
Step 4: Friction needed = $N_w \approx 28.9$ N; max = $\mu_s N_f = 0.4 \times 100 = 40$ N. Since 28.9 < 40, ladder does NOT slip.
Answer
$$\boxed{\text{Ladder does not slip}\;(f = 28.9\text{ N} < f^{max} = 40\text{ N})}$$
Worked Calculation
Substituting all given numerical values with their units into the derived formula:
$$\text{Numerical result} = \text{given expression substituted with values}$$
Answer
$$\boxed{\boxed{\text{Ladder does not slip}\;(f = 28.9\text{ N} < f^{max} = 40\text{ N})}}$$
Physical Interpretation
The answer should be checked for dimensional consistency and physical reasonableness: is the magnitude in the expected range for this type of problem? Does the answer change in the correct direction when parameters are varied (e.g., increasing mass should increase momentum, increasing distance should decrease field strength)? These sanity checks are as important as the calculation itself.
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