Kinetic Energy of a Pendulum Calculator

Kinetic Energy of a Pendulum Calculator gives the KE of a swinging bob at the bottom or at any angle from release height, angle or speed, with potential energy, tension and period.

Kinetic Energy of a Pendulum Calculator Formulas

The Kinetic Energy of a Pendulum Calculator finds the energy of motion of a swinging bob at the bottom of its swing or at any angle on the way. It needs the bob mass, the pendulum length, gravity and one fact about how big the swing is.

A pendulum has kinetic energy whenever the bob is moving. The energy is highest at the lowest point, where the bob is fastest, and it is zero at the two ends of the swing, where the bob stops for an instant.

Three formulas give the same answer from different starting facts. The first uses the height the bob was raised before release. OpenStax University Physics derives it from conservation of mechanical energy.

$$KE_{max} = m g h$$

The second is the kinetic energy of a pendulum in terms of theta. It gives the energy at any angle θ for a swing that reaches θmax.

$$KE = m g L (\cos\theta – \cos\theta_{max})$$

The third is the general kinetic energy formula. It applies when the speed of the bob is already known, for example from a light gate at the bottom of the swing.

$$KE = \frac{1}{2} m v^2$$

In all three m is the bob mass in kg, g is gravity in m/s², L is the pendulum length in meters and v is the bob speed in m/s. The result is in joules, or in ft·lbf when Result Units is set to US.

Four Ways to Describe the Swing

The “Swing Given By” list sets which fact the Kinetic Energy of a Pendulum Calculator starts from. The others are worked out and shown in the results, so one entry describes the whole swing.

Every row uses the default 2 kg bob on a 1.5 m pendulum at standard gravity, 9.80665 m/s². Mass accepts kg, g, lb and oz, and lengths accept m, cm, mm, ft and in.

The last option runs the problem backward. Enter the kinetic energy the bob should have at the bottom, in J, mJ, kJ or ft·lbf, and the headline becomes the release height. For 10 J that is 50.99 cm, which means a release at 48.69°.

The length field has two extra units for a pendulum that was timed, not measured. Choose “Small-angle period (s)” or “Small-angle frequency (Hz)” and the length comes from the small-angle period formula. A 2 s period gives a length of 99.36 cm. A wide swing on that pendulum takes longer, so the period card reads 2.148 s for a 0.5 m release.

Worked Example: 2 kg Bob Raised 50 cm

The Kinetic Energy of a Pendulum Calculator opens with a 2 kg bob on a 1.5 m string, lifted 0.5 m above its lowest point and let go. The Angle Now field is 0, which asks for the bottom of the swing.

The total energy is m × g × h. That is 2 × 9.80665 × 0.5, which comes to 9.807 J. At the bottom all of it is kinetic, so the maximum kinetic energy is 9.807 J.

The speed follows from that energy. The square root of 2 × 9.807 ÷ 2 is 3.132 m/s, and the bob’s momentum there is 6.263 kg·m/s. Lifting the bob 0.5 m on a 1.5 m string means it was released at 48.19°.

A textbook case makes a good check. OpenStax releases a bob on a 1.0 m string from 30° and finds 1.62 m/s at the bottom with g = 9.8. The same entries here give 1.621 m/s, and a 2 kg bob carries 2.628 J.

Kinetic Energy in Terms of Theta

Enter any angle in the Angle Now field of the Kinetic Energy of a Pendulum Calculator and the headline changes to the kinetic energy at that point. The hero line also states what share of the total energy is kinetic there.

Take a 45° release on the default pendulum and ask for 20°. The kinetic energy is 2 × 9.80665 × 1.5 × (cos 20° − cos 45°), which is 6.843 J. That is 79.41% of the 8.617 J total, and the bob is moving at 2.616 m/s.

The “Energy Through the Swing” panel lays the whole trade out in five steps. These are its rows for the default 0.5 m release.

Half the angle is not half the energy. At 24.09° the bob already holds 7.243 J, about 74% of the total, because most of the height is lost in the first part of the fall. The first card shows this figure as KE at Half the Angle.

The Angle Now field also takes a time. Switch its unit to “Time since release (s)” and the calculator finds where the bob is at that moment. On the default swing, 0.3 s after release the bob is at 36.05° with 4.173 J, or 42.55% of the total, and it moves at 2.043 m/s.

That position comes from the exact large-swing solution. The small-angle cosine would put the bob at 34.69° instead, a gap that grows with the release angle.

The same formula drops into a spreadsheet. In Excel, with mass in A1, length in A2, the angle now in A3 and the release angle in A4, type =A1*9.80665*A2*(COS(RADIANS(A3))-COS(RADIANS(A4))). Excel works in radians, which is why RADIANS is needed.

Potential Energy of a Pendulum

Potential energy is the other half of the trade. The potential energy of a pendulum formula measures height from the lowest point of the swing, so it is zero at the bottom.

$$PE = m g L (1 – \cos\theta)$$

At the release point the potential energy equals the total, 9.807 J in the default case. The Kinetic Energy of a Pendulum Calculator lists it as Potential Energy Now whenever the angle is above zero. As the bob falls it turns into kinetic energy, and at every angle the two add up to the same figure. With no friction or air drag that total never changes.

That fixes the range of values the kinetic energy can take. It runs from 0 J at the ends of the swing to m × g × h at the bottom, and it is never negative. For energy problems without a string, the Kinetic Energy Calculator and the Potential Energy Calculator handle each side alone.

Speed, String Tension and Force

The speed of the bob depends only on how far it has dropped. Mass cancels out of the energy balance, so a heavy bob and a light one reach the bottom at the same speed. The heavy one simply carries more kinetic energy.

The string does two jobs. It holds up part of the weight, and it pulls the bob inward to keep it on a circle. The Kinetic Energy of a Pendulum Calculator adds the two.

$$T = m g \cos\theta + \frac{m v^2}{L}$$

For the default swing the tension is 13.08 N at release and 32.69 N at the bottom. The bob weighs 19.61 N, so the string pulls 1.667 times the weight at the lowest point. That peak is the figure to use when choosing a cord.

The force that drives the swing is the part of the weight along the arc, m × g × sin θ. At the 48.19° release it is 14.62 N, and it falls to zero at the bottom. The Tension Calculator covers ropes that are not swinging, and the Centripetal Force Calculator isolates the inward pull.

A release above 90° turns the third card into a warning. A string cannot push, so it goes slack and the bob leaves the circle. The energy figures then apply only to a bob on a rigid rod.

Period, Frequency and Large Swings

The fourth card of the Kinetic Energy of a Pendulum Calculator times the swing. The usual formula of a simple pendulum gives the period for small angles, and its reciprocal is the natural frequency.

$$T = 2\pi \sqrt{\frac{L}{g}}$$

A 1.5 m pendulum has a small-angle period of 2.457 s and a natural frequency of 0.407 Hz. OpenStax treats the formula as good below about 15°, where sin θ and θ agree within 1%.

Bigger swings take longer. The headline “Period of This Swing” uses the exact solution, and for the 48.19° default it is 2.571 s, 4.6% more than the small-angle figure. The “Other Release Angles” panel shows the trend on the same pendulum.

Doubling the release angle from 45° to 90° more than triples the kinetic energy. For ω of a pendulum the Angular Frequency Calculator has a simple pendulum mode, and the Frequency Calculator converts a timed period into hertz.

Pendulum Laws, Types and History

The physics behind a pendulum is a steady swap between height and speed under gravity, timed by the length of the string. The questions below come up alongside pendulum energy problems.

What are the four laws of a simple pendulum?

All four come from the period formula. The period does not depend on the size of a small swing, which is the law of isochronism. It grows with the square root of the length. It falls with the square root of gravity. It does not depend on the mass of the bob.

What are the types of pendulums?

The two main types are the simple pendulum, a small bob on a light string, and the physical or compound pendulum, a rigid body swinging about a pivot. OpenStax adds the torsional pendulum, which twists on a wire, as a third. The Kinetic Energy of a Pendulum Calculator covers the simple pendulum only.

What was Galileo’s pendulum theory?

Galileo found that a pendulum takes the same time for wide and narrow swings. The Galileo Project at Rice University dates his serious study to 1602. In the 1650s Huygens showed the rule is only approximate, which matches the period table above.

What is the most famous pendulum?

It is the Foucault pendulum hung in the Panthéon in Paris in 1851, which showed that the Earth rotates. It had a 28 kg iron ball on a wire about 67 m long. Enter that length in the Kinetic Energy of a Pendulum Calculator and the small-angle period reads 16.42 s.

Who lived first, Galileo or Newton?

Galileo came first. He was born in 1564 and died in January 1642. Newton was born on December 25, 1642, by the calendar then used in England, so the two lives did not overlap.

How do you explain a pendulum to a child?

Use a playground swing. At the top of each swing the rider stops for a moment and is highest. At the bottom the rider is lowest and fastest. Height turns into speed on the way down and back into height on the way up.

Other Kinetic Energy Formulas

Kinetic energy has no chemical formula, because it is a quantity and not a substance. The formula for an object moving in a line is ½ m v². OpenStax also writes it as p² ÷ (2 m), where p is momentum.

Two more forms are common. A spinning body has rotational kinetic energy ½ I ω², the quantity behind the Flywheel Energy Storage Calculator. Near the speed of light Einstein’s form applies, KE = (γ − 1) m c².

The total kinetic energy of a system is the sum over its parts. Everyday examples include a moving car, a thrown ball, a falling raindrop, a spinning fan and a swinging pendulum. OpenStax gives an 80 kg runner at 10 m/s as about 4.0 kJ.

For gases the question shifts to temperature. At one temperature the molecules of every gas have the same average kinetic energy, so no molecule is the highest. Lighter molecules simply move faster to carry it.

More tools for swinging and spinning motion are on the Rotational Motion Calculators hub, alongside the angular speed and circular motion tools.