Enter the mass and the angle between the pendulum and the vertical into the calculator to determine the pendulum’s restoring (tangential) force due to gravity.
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Pendulum Force Formula and Reference
For a simple pendulum, the force that pulls the bob back toward equilibrium is the tangential component of gravity. When the angle is measured from the vertical, the restoring force depends on the bob mass, local gravity, and deflection angle. If your use case is string tension rather than restoring force, see the comparison table below.
F = m*g*sin(\theta)
Rearranged equations
If you are solving for a missing input, these are the most useful rearrangements.
| Unknown | Formula |
|---|---|
| Force | F = m*g*sin(\theta) |
| Mass | m = \frac{F}{g*sin(\theta)} |
| Gravity | g = \frac{F}{m*sin(\theta)} |
| Angle | \theta = \sin^{-1}\left(\frac{F}{m*g}\right) |
Variable guide
| Variable | Meaning | Practical note |
|---|---|---|
| Force | Tangential restoring force along the swing path | This is the component that drives the bob back toward the centerline. |
| Mass | Mass of the pendulum bob and any attached moving hardware | Use the total moving mass. |
| Gravity | Local gravitational acceleration | On Earth, 9.81 m/s² is commonly used. |
| Angle | Deflection from the vertical | At 0°, the pendulum hangs straight down and restoring force is zero. |
Angle quick reference
| Angle from vertical | Relative force factor | Interpretation |
|---|---|---|
| 0° | 0.000 | No restoring force |
| 15° | 0.259 | Small restoring force |
| 30° | 0.500 | Moderate force |
| 45° | 0.707 | Strong mid-range force |
| 60° | 0.866 | Large restoring force |
| 90° | 1.000 | Maximum tangential component of weight |
Restoring force vs. string tension
These are related but not identical. The restoring force acts tangent to the arc, while tension acts along the string. This distinction matters when checking pivot load, cable load, or support reactions.
| Quantity | Use it when you need | Formula |
|---|---|---|
| Restoring force | The force causing the pendulum to swing back toward equilibrium | F_t = m*g*sin(\theta) |
| Along-string weight component | The portion of weight aligned with the string | F_r = m*g*cos(\theta) |
| Tension at the release point | String load when the bob is momentarily at rest at angle θ | T = m*g*cos(\theta) |
| Tension while moving | String load during motion when speed is not zero | T = m*g*cos(\theta) + \frac{m*v^2}{L} |
Small-angle behavior
For small deflections, the pendulum behaves almost linearly. This is why simple pendulums are often modeled as simple harmonic motion for modest swing angles.
F \approx m*g*\theta
Use that approximation only when the angle is in radians and the swing angle is small, typically under about 15°.
Example
A 3 kg pendulum bob displaced 30° from vertical on Earth has a restoring force of:
F = 3*9.81*sin(30^\circ) = 14.72
The restoring force is 14.72 N toward the equilibrium position.
Common input mistakes
- Using angle from the horizontal instead of angle from the vertical.
- Entering bob mass only and forgetting hooks, brackets, or attached moving parts.
- Confusing restoring force with string tension or pivot load.
- Using a local gravity value that does not match the intended environment.
- Applying the small-angle approximation to large swing angles.
When this calculator is most useful
- Checking the tangential force acting on a pendulum bob at a given angle.
- Estimating how strongly gravity pulls a pendulum back toward center.
- Comparing restoring force across different masses or gravity values.
- Solving for a missing angle, force, mass, or gravity input from known values.
