Height of Pendulum Calculator

• Height rise from angle: h = L(1 − cosθ)
• Angle from horizontal displacement: θ = asin(x/L)
• Angle from arc length: θ = s/L (radians)
• Height rise from bottom speed: h = v²/(2g)
• Bottom tension (optional): T = m·g + m·v²/L

1. Select an input method that matches what you measured.
2. Enter pendulum length and gravity, then fill the method field.
3. Optionally add mass to compute energy gain and string tension.
4. Choose output units and decimal places for reporting needs.
5. Press Calculate to see results above the form.
6. Use the download buttons to save a CSV or PDF summary.

1) Why height rise matters

Height rise h sets the energy budget of a swing. A rise of 0.10 m stores about 0.981 J per kilogram because g ≈ 9.80665 m/s². That energy becomes speed at the bottom and drives tension in the string.

2) Common measurement inputs

In labs you may know the release angle, a horizontal offset, or arc length along the path. Each describes the same geometry for a simple pendulum of length L measured from pivot to the bob’s center.

3) Geometry link between angle and height

The key relation is h = L(1 − cosθ). With L = 1.00 m and θ = 30°, h ≈ 0.134 m. If you measure horizontal displacement x, the tool uses θ = asin(x/L) before computing height.

4) Speed and energy cross-check

If you measure bottom speed, energy gives h = v²/(2g). For v = 1.50 m/s, h ≈ 0.1147 m. This method is useful when a photogate provides velocity and you want height without reading an angle scale.

5) Angle limits and realism

Real pendulums have drag and a finite bob size, so results are idealized. For many classroom setups, angles below about 10° behave nearly “small‑angle,” while larger swings need care because period and losses change.

6) Period estimation and correction

The calculator shows a small‑angle period T0 = 2π√(L/g) plus a common correction series for larger θ. For L = 1.00 m, T0 is about 2.01 s. At 60°, the corrected period is longer than T0.

7) Units, reporting, and typical uses

Mixed units are a common source of mistakes, so the tool converts length and speed while keeping g in m/s². Engineers often report rise, arc length, and speed together, then export CSV or PDF for lab notes or teaching handouts. For field tests, log temperature and pivot friction, then repeat three trials; average h to reduce random reading error in your report.

1) What does “height rise” mean here?

It is the vertical lift of the bob from the lowest point to the measured release position. It is not the total string length, and it is not the height of the pivot above the floor.

2) Where should I measure pendulum length?

Measure from the pivot point to the center of mass of the bob. If the bob is large, include half the bob diameter so the length matches the circular path radius.

3) Can I use feet or inches?

Yes. Select the same unit for length-based inputs, then choose an output unit you prefer. The calculator converts internally to meters for consistent math and converts back for display.

4) I only know the bottom speed. Is that enough?

Yes. Using h = v²/(2g), the tool estimates rise from speed and gravity. It then back-calculates an equivalent angle using the chosen length so you can compare setups.

5) Why do T0 and corrected period differ?

T0 assumes small angles where sinθ ≈ θ. At larger angles, the pendulum moves more slowly near the ends of its swing, so the real period increases, and the correction approximates that increase.

6) Does mass change the height result?

No. Geometry and gravity set height for a given angle, displacement, or speed. Mass only affects energy in joules and tension in newtons, so those fields appear only when you provide mass.