Engineering calculator

Ohm's law and electrical power

V = I R P = V I = I² R = V² ÷ R

Voltage V (volts), current I (amps), resistance R (ohms) and power P (watts) are tied together by these two laws, so any two of them fix the other two. The default example is a 12 V supply across a 4 Ω resistor: I = 12 ÷ 4 = 3 A, and P = 12 × 3 = 36 W (check: V² ÷ R = 144 ÷ 4 = 36 W). A 36 W load on a quarter-watt resistor would burn it out, which is why the power figure matters as much as the current.

Resistors and capacitors in series and parallel

Resistors of 100 Ω, 220 Ω and 330 Ω add to 650 Ω in series. In parallel the reciprocals add: 1/100 + 1/220 + 1/330 = 0.017576, and 1 ÷ 0.017576 = 56.9 Ω. Capacitors swap the rules, so 10, 22 and 47 µF give 79 µF in parallel and 5.998 µF in series. For two components in the reciprocal case there is a shortcut, product over sum: 10 kΩ and 47 kΩ in parallel are 10 × 47 ÷ 57 = 8.246 kΩ.

Voltage divider

Vout = Vin × R2 ÷ (R1 + R2)

Two resistors in series across a supply split the voltage in proportion to their resistance. With 12 V across R1 = 10 kΩ and R2 = 4.7 kΩ, Vout = 12 × 4.7 ÷ 14.7 = 3.837 V, and 816.3 µA flows through both. The formula assumes nothing is connected to the output. A load RL sits in parallel with R2 and pulls the output down: a 4.7 kΩ load makes the bottom leg 2.35 kΩ and Vout falls to 12 × 2.35 ÷ 12.35 = 2.283 V. Enter a load resistance in the calculator to see this. As a rule, keep the load at least ten times R2, or buffer the output.

Stress, strain and Young's modulus

σ = F ÷ A ε = ΔL ÷ L E = σ ÷ ε ΔL = F L ÷ (A E)

A steel bar with a cross-section of 100 mm² and a length of 2 m carries a 20 kN tensile load and stretches 2 mm. Stress σ = 20,000 N ÷ 0.0001 m² = 200 MPa. Strain ε = 0.002 ÷ 2 = 0.001 (0.1%, with no units). Young's modulus E = 200 MPa ÷ 0.001 = 200 GPa, the textbook value for steel. The “Extension from Young's modulus” option runs the calculation the other way: given E, it predicts the 2 mm stretch. Both hold only in the elastic region, below the yield stress; past it the bar deforms permanently and E no longer applies.

The most common slip is units. Stress in pascals needs the area in square metres, and 1 mm² is 10⁻⁶ m², not 10⁻³. Use the unit selectors rather than converting by hand.

Torque and gear ratio

τ = F r sin θ gear ratio = N_driven ÷ N_driver

A 50 N push at right angles to a 0.3 m spanner gives τ = 50 × 0.3 × sin 90° = 15 N·m, or 11.06 lbf·ft. Push at 30° to the handle and only half the force turns the nut (sin 30° = 0.5), so the torque halves to 7.5 N·m. With a speed in rpm, the calculator also gives the power, P = τ × 2π × rpm ÷ 60.

A 12-tooth driver meshing with a 36-tooth gear gives a 3 : 1 reduction. At 1,500 rpm in, the output turns at 500 rpm, and an input torque of 10 N·m becomes 30 N·m at 100% efficiency. Power is conserved (less friction), so what the gear train gains in torque it loses in speed. For a compound train, multiply the ratios of the individual pairs. The same formula works for belt pulleys, using diameters instead of teeth.

RC time constant

τ = R C V(t) = Vs (1 − e^(−t/τ)) f_c = 1 ÷ (2π R C)

A 10 kΩ resistor charging a 100 µF capacitor has τ = 10,000 × 0.0001 = 1 s. The capacitor voltage rises along the curve in the chart: 63.2% of the supply after one τ, then 86.5%, 95.0%, 98.2% and 99.3% after 2τ to 5τ. Discharging is the mirror image, falling to 36.8% after one τ. The same pair used as a low-pass filter has its −3 dB cut-off at 1 ÷ (2π × 1) = 0.159 Hz.