Pendulum Force Calculator

Calculator

Bob mass

Mass unit

Pendulum length

Length unit

Angle from vertical

Angle unit

Gravity

Speed input type

Linear speed

Linear speed unit

Angular speed

Output set

Forces, acceleration, energy, period, and frequency.

Formula used

The angle theta is measured from the vertical line.

Weight force: W = m g
Signed restoring force: F_t = -m g sin(theta)
Radial force demand: F_r = m v² / L = m L omega²
String tension: T = m g cos(theta) + m v² / L
Tangential acceleration: a_t = -g sin(theta)
Radial acceleration: a_r = v² / L
Height: h = L(1 - cos(theta))
Energy: E = m g h + 0.5 m v²
Small angle period: P = 2 pi sqrt(L / g)

How to use this calculator

Enter the bob mass and choose its unit.
Enter the pendulum length from pivot to bob center.
Enter the angle from the vertical line.
Choose linear speed or angular speed.
Enter local gravity, or keep the standard value.
Press the calculate button.
Review the result table above the form.
Use the CSV or PDF button to save results.

Example data table

Mass	Length	Angle	Speed	Gravity	Tension	Restoring force
1 kg	1 m	20 deg	1 m/s	9.80665 m/s^2	10.216 N	3.354 N
0.5 kg	0.8 m	30 deg	0.7 m/s	9.80665 m/s^2	4.552 N	2.452 N
2 lb	3 ft	15 deg	2 ft/s	9.80665 m/s^2	9.139 N	2.303 N

Pendulum Force Guide

Why Pendulum Force Matters

A pendulum is simple to see. Its force picture is richer. Gravity pulls the bob downward. The string pulls the bob toward the pivot. Motion adds a centripetal demand along the string. These actions change with angle and speed. This calculator separates them. It shows tension, restoring force, radial force, acceleration, and energy. The output helps students compare theory with lab data. It also helps builders check small suspended loads. Use it for ideal pendulum studies. It assumes a light string and a point mass.

Reading The Main Forces

The restoring force acts along the arc. It tries to move the bob back toward the lowest point. Its size is m g sin theta. The radial force is different. It points toward the pivot and equals m v squared over length. Tension must support the radial demand and part of weight. At low speed and large angle, tension can be small. If the calculated tension becomes negative, the string cannot stay tight. That means the ideal string model has failed for that condition.

Energy And Motion Notes

Potential energy depends on height above the lowest point. Kinetic energy depends on speed. Their sum is useful when friction is ignored. A pendulum released from rest trades height for speed. This tool accepts measured speed or angular speed. It then converts values to standard units. The small angle period is also shown. It is only a good estimate for modest angles.

Good Practice For Accurate Work

Measure length from pivot to bob center. Use kilograms and meters when possible. Enter the angle from the vertical line. Do not use the angle from the ceiling. Use local gravity if known. For Earth, 9.80665 meters per second squared is common. Keep the bob mass constant during trials. Record angle, speed, and length together. Then export the table for later checking. Compare restoring force signs carefully. The magnitude is usually easier for reports. The signed value is better for direction. Always note that air drag and string mass are not included. Repeat measurements improve confidence. Average similar trials. Watch for loose knots, slipping pivots, and timing errors because they can often change speed and tension estimates greatly.

FAQs

What force does this pendulum calculator find?

It finds weight, restoring force, radial force, string tension, net force, acceleration, and energy. The main pendulum force usually means restoring force or tension, so both are shown.

What angle should I enter?

Enter the angle measured from the vertical resting line. Do not enter the angle from the ceiling or from the horizontal line.

Why is restoring force negative?

The negative sign shows direction. It means the force points back toward the equilibrium point. Use the magnitude when you only need size.

What happens if tension is negative?

A negative tension means the ideal string cannot stay tight. In real motion, the bob would leave the circular path or the string would slacken.

Can I use angular speed?

Yes. Choose angular speed as the input type. The tool converts it to linear speed by using v = omega L.

Does the tool include air resistance?

No. It uses an ideal model. Air drag, string mass, pivot friction, bob size, and elastic stretch are not included.

Which gravity value should I use?

The default 9.80665 m/s^2 works for standard Earth calculations. Use a local measured value for more accurate lab work.

Is the small angle period exact?

No. The displayed period is the small angle estimate. It is best for small swings and becomes less accurate at large angles.