Uniform Beam Loads, Capacitor Energy, and Gear Ratios Explained
Three more engineering formulas — uniform beam deflection, stored capacitor energy, and gear ratio speed conversion — worked through with real numbers.
These three formulas span structural, electrical, and mechanical engineering, but each reduces a physical system to one clean calculable number.
Uniform beam load: distributed weight, not a single point
For a simply supported beam under a uniformly distributed load (rather than a single point load), maximum deflection is δ = 5wL⁴/(384EI). A beam carrying a uniform 3,000 N/m load over a 5-meter span, with elastic modulus 200 GPa and moment of inertia 0.0002 m⁴: δ = (5 × 3,000 × 5⁴)/(384 × 200×10⁹ × 0.0002) ≈ 0.00061 meters (about 0.61mm) of deflection at the center.
Why the uniform-load formula differs from the point-load formula
A point load concentrates all the weight at one spot (using PL³/48EI), while a uniform load spreads the same total weight evenly across the span (using 5wL⁴/384EI) — the different constant (5/384 vs. 1/48) and the fact that w is load-per-unit-length rather than a single force reflect this fundamentally different loading pattern, even though both describe deflection in the same beam type.
Capacitor energy: voltage squared
Stored Energy = ½CV². A 2,200 microfarad (0.0022F) capacitor charged to 9V: Energy = 0.5 × 0.0022 × 9² = 0.5 × 0.0022 × 81 ≈ 0.089 joules. Because voltage is squared, a capacitor charged to double the voltage stores 4 times the energy — the same squared relationship seen in kinetic energy elsewhere in physics.
Gear ratio: teeth count determines speed change
Gear Ratio = Driven Gear Teeth / Driving Gear Teeth. Output RPM = Input RPM / Gear Ratio. A 15-tooth driving gear turning a 45-tooth driven gear gives a ratio of 45/15 = 3:1 — meaning for every 3 turns of the driving gear, the driven gear turns once. At 1,200 RPM input: output speed = 1,200/3 = 400 RPM, one-third the input speed, trading speed for increased torque at the output shaft.
Common mistakes to avoid
- Applying the point-load beam deflection formula to a uniform-load scenario (or vice versa) — the constant and the load variable's meaning both change between the two cases
- Forgetting that capacitor energy scales with voltage squared, not linearly — doubling voltage is far more consequential for stored energy than doubling capacitance
- Mixing up which gear is "driving" vs. "driven" in a gear ratio calculation, which inverts the resulting speed relationship
Calculate your own scenarios with the uniform load beam deflection calculator, capacitor energy calculator, and gear ratio calculator.