Kinetic Energy of a Pendulum Calculator – Calculate PE & KE Energy

  • Adam
  • November 27, 2025

Calculate kinetic energy of a pendulum at any point in its swing. Free calculator with energy conservation formulas, step-by-step solutions for maximum KE at bottom, velocity calculations, and PE-KE distribution analysis.

Kinetic Energy of a Pendulum Calculator

A kinetic energy of a pendulum calculator determines the energy of motion at any point in a pendulum's swing using energy conservation principles. The calculator computes kinetic energy based on the pendulum's mass, length, initial angle, and position, helping students and engineers understand energy transformations in simple harmonic motion and oscillating systems.

Calculate Pendulum Kinetic Energy

Default: 9.8 m/s² (Earth's surface)

Result:

What is Kinetic Energy of a Pendulum?

Kinetic energy of a pendulum is the energy of motion possessed by the pendulum bob as it swings through its arc. As the pendulum oscillates, energy continuously transforms between kinetic energy (maximum at the bottom of the swing) and gravitational potential energy (maximum at the highest points). The kinetic energy at any point depends on the bob's velocity at that position and follows the fundamental kinetic energy formula KE = ½mv².

Key Characteristics of Pendulum Kinetic Energy:

  • Maximum at bottom: Kinetic energy reaches its maximum value when the pendulum passes through the lowest point (equilibrium position)
  • Zero at turning points: At the highest points of the swing, velocity is zero, so kinetic energy is zero
  • Energy conservation: In an ideal pendulum with no friction, total mechanical energy (KE + PE) remains constant
  • Velocity dependent: Kinetic energy is proportional to the square of velocity (KE ∝ v²)
  • Independent of path: Only the height difference matters, not the path taken between positions

Kinetic Energy of Pendulum Formula

The kinetic energy of a pendulum can be calculated using different approaches depending on which parameters are known. The formulas connect kinetic energy to velocity, angle, and energy conservation principles.

Basic Kinetic Energy Formula

KE = ½mv²

Where:

  • KE = kinetic energy (Joules, J)
  • m = mass of the pendulum bob (kg)
  • v = velocity of the bob (m/s)

Kinetic Energy at Bottom (Maximum KE)

When a pendulum is released from an initial angle θ₀, all the initial potential energy converts to kinetic energy at the bottom:

KEmax = mgL(1 - cos θ₀)

or equivalently

KEmax = mgh₀

Where:

  • m = mass of pendulum bob (kg)
  • g = gravitational acceleration (9.8 m/s² on Earth)
  • L = length of pendulum (m)
  • θ₀ = initial angle from vertical (radians)
  • h₀ = initial height above bottom = L(1 - cos θ₀)

Velocity at Bottom

vmax = √[2gL(1 - cos θ₀)]

Kinetic Energy at Any Angle

For a pendulum at angle θ after being released from initial angle θ₀:

KE(θ) = mgL(cos θ - cos θ₀)

Energy Conservation in Pendulum

Total Mechanical Energy (constant):

Etotal = KE + PE = constant

At any point:

½mv² + mgh = constant

Conservation equation:

KE1 + PE1 = KE2 + PE2

How to Calculate Kinetic Energy of a Pendulum

Calculating pendulum kinetic energy requires understanding energy conservation and the relationship between position and velocity. Follow these systematic steps:

  1. Identify the reference point: Choose the lowest point of the swing as the zero potential energy reference level.
  2. Determine initial conditions: Identify the initial angle θ₀ from which the pendulum is released (usually from rest, so initial KE = 0).
  3. Find the height difference: Calculate the vertical height difference using h = L(1 - cos θ), where θ is measured from the vertical.
  4. Apply energy conservation: Initial PE converts to KE as the pendulum falls: PEinitial = KEat bottom.
  5. Calculate kinetic energy: Use KE = mgL(1 - cos θ₀) for maximum KE at the bottom, or KE = mgL(cos θ - cos θ₀) at any angle θ.
  6. Find velocity if needed: Use v = √(2gL(1 - cos θ₀)) to calculate velocity at any point.

Calculation Example 1: KE at Bottom

Problem: A pendulum with mass 0.5 kg and length 2 m is released from an angle of 30° from vertical. Find the kinetic energy at the bottom.

Solution:

Given: m = 0.5 kg, L = 2 m, θ₀ = 30° = 0.524 rad, g = 9.8 m/s²

Using formula: KEmax = mgL(1 - cos θ₀)

KEmax = 0.5 × 9.8 × 2 × (1 - cos 30°)

KEmax = 9.8 × (1 - 0.866)

KEmax = 9.8 × 0.134

KEmax = 1.31 Joules

At the bottom, all initial potential energy has converted to kinetic energy.

Calculation Example 2: KE at Intermediate Angle

Problem: Using the same pendulum from Example 1, find the kinetic energy when the pendulum is at 15° from vertical.

Solution:

Given: m = 0.5 kg, L = 2 m, θ₀ = 30°, θ = 15°, g = 9.8 m/s²

Using formula: KE(θ) = mgL(cos θ - cos θ₀)

KE(15°) = 0.5 × 9.8 × 2 × (cos 15° - cos 30°)

KE(15°) = 9.8 × (0.966 - 0.866)

KE(15°) = 9.8 × 0.100

KE(15°) = 0.98 Joules

At this point, the pendulum has both kinetic and potential energy.

Calculation Example 3: Velocity from KE

Problem: For the pendulum with KE = 1.31 J at the bottom and mass 0.5 kg, find the velocity.

Solution:

Given: KE = 1.31 J, m = 0.5 kg

Using formula: KE = ½mv²

Rearranging: v = √(2KE/m)

v = √(2 × 1.31 / 0.5)

v = √(5.24)

v = 2.29 m/s

This is the maximum velocity of the pendulum, occurring at the lowest point.

Energy Transformation in Pendulum Motion

A swinging pendulum demonstrates continuous energy transformation between kinetic and potential energy, providing a clear example of mechanical energy conservation.

Energy at Different Positions

Position Kinetic Energy Potential Energy Velocity
Highest Point (θ₀) Zero (v = 0) Maximum (PE = mgL(1-cos θ₀)) 0 m/s
Middle (θ = θ₀/2) Intermediate Intermediate Intermediate
Bottom (θ = 0) Maximum (KE = mgL(1-cos θ₀)) Zero (reference level) Maximum
Opposite High (−θ₀) Zero (v = 0) Maximum (same as start) 0 m/s

Energy Conservation Principle

In an ideal pendulum without air resistance or friction:

At highest point (release):

Etotal = PEmax + 0 = mgL(1 - cos θ₀)

At lowest point (bottom):

Etotal = 0 + KEmax = mgL(1 - cos θ₀)

At any intermediate angle θ:

Etotal = PE(θ) + KE(θ) = mgL(1 - cos θ₀)

The sum of kinetic and potential energy remains constant throughout the motion.

Factors Affecting Pendulum Kinetic Energy

Several parameters influence the kinetic energy of a pendulum at any given point:

  • Initial angle (θ₀): Larger release angles result in greater maximum kinetic energy because the bob falls through a greater height
  • Pendulum length (L): Longer pendulums have greater maximum KE for the same angle because the vertical drop distance is proportional to length
  • Bob mass (m): Kinetic energy is directly proportional to mass; doubling the mass doubles the KE at any given velocity
  • Gravitational field (g): Stronger gravity results in greater kinetic energy for the same initial conditions
  • Energy losses: Real pendulums lose energy to air resistance and friction at the pivot, reducing maximum KE over time

Applications of Pendulum Energy Calculations

Understanding pendulum kinetic energy is essential in various scientific and engineering applications:

  • Physics Education: Pendulums serve as classic demonstrations of energy conservation and simple harmonic motion principles
  • Timekeeping: Historical pendulum clocks rely on consistent energy exchange and periodic motion for accurate timekeeping
  • Seismology: Seismometers use pendulum principles to detect and measure ground motion during earthquakes
  • Engineering Design: Cranes, wrecking balls, and suspended structures require pendulum energy analysis for safety calculations
  • Amusement Rides: Swing rides and pendulum rides apply energy principles to ensure safe operation within design limits
  • Metrology: Precision gravity measurements use pendulum period analysis to determine local gravitational acceleration
  • Biomechanics: Analysis of human gait and limb motion often models legs as compound pendulums with energy considerations

Frequently Asked Questions

What is the formula for kinetic energy of a pendulum at the bottom?
The kinetic energy at the bottom of a pendulum's swing is KEmax = mgL(1 - cos θ₀), where m is mass, g is gravitational acceleration (9.8 m/s²), L is pendulum length, and θ₀ is the initial release angle from vertical. This formula comes from energy conservation: all the initial gravitational potential energy converts to kinetic energy at the lowest point. For small angles (θ₀ < 15°), the approximation KEmax ≈ ½mgLθ₀² works well.
Where is the kinetic energy maximum in a pendulum?
Kinetic energy is maximum at the lowest point of the pendulum's swing (when θ = 0, at the equilibrium position). At this point, the pendulum bob has its highest velocity and zero potential energy (using the bottom as reference level). The velocity is maximum because all the gravitational potential energy from the release height has converted to kinetic energy. Conversely, kinetic energy is zero at the highest points (turning points) where the bob momentarily stops before reversing direction.
How do you calculate the velocity of a pendulum at the bottom?
The velocity at the bottom is calculated using vmax = √[2gL(1 - cos θ₀)], derived from energy conservation. First, calculate the height difference h = L(1 - cos θ₀). Then use conservation of energy: mgh = ½mv², which simplifies to v = √(2gh). Substituting h gives the formula. For example, if L = 2 m and θ₀ = 30°, then vmax = √[2 × 9.8 × 2 × (1 - 0.866)] = 2.29 m/s.
What is the relationship between potential and kinetic energy in a pendulum?
In an ideal pendulum, potential and kinetic energy continuously transform into each other while maintaining constant total mechanical energy. When PE increases (bob moving up), KE decreases (bob slowing down). When PE decreases (bob falling), KE increases (bob speeding up). At any point: PE + KE = constant = mgL(1 - cos θ₀). The relationship is complementary: PE = Etotal - KE and KE = Etotal - PE. Maximum PE occurs at the highest points where KE = 0, and maximum KE occurs at the bottom where PE = 0.
Does the mass of a pendulum affect its kinetic energy?
Yes, mass directly affects kinetic energy. Since KE = ½mv², doubling the mass doubles the kinetic energy for the same velocity. Similarly, KEmax = mgL(1 - cos θ₀) is directly proportional to mass. However, mass does not affect the maximum velocity or period of the pendulum because both potential and kinetic energy scale with mass proportionally. A heavier bob has more total energy but reaches the same maximum velocity as a lighter bob released from the same angle.
How does the initial angle affect pendulum kinetic energy?
Larger initial angles result in greater maximum kinetic energy because the pendulum bob falls through a greater vertical height. The relationship follows KEmax = mgL(1 - cos θ₀), so KE increases nonlinearly with angle. For example, releasing from 60° gives about four times more kinetic energy than releasing from 30° for the same pendulum. For small angles (< 15°), KEmax ≈ ½mgLθ₀², showing approximately quadratic dependence on angle. Larger angles also increase the maximum velocity proportionally to √(1 - cos θ₀).
What happens to kinetic energy when a pendulum reaches its highest point?
At the highest point of its swing, the pendulum's kinetic energy becomes zero because the velocity is momentarily zero as the bob changes direction. All the mechanical energy is stored as gravitational potential energy at this turning point. The pendulum bob decelerates as it rises, converting kinetic energy to potential energy, until it stops completely at the maximum height. Then it accelerates back down, converting potential energy back to kinetic energy. This is why KE = 0 and PE = maximum at the turning points.
How do you find kinetic energy at an intermediate angle?
For kinetic energy at an intermediate angle θ (between the release angle θ₀ and the bottom), use KE(θ) = mgL(cos θ - cos θ₀). First calculate the height difference from the current position to the release position: Δh = L(cos θ - cos θ₀). Then apply energy conservation: the decrease in PE equals the increase in KE. Since PEinitial = mgL(1 - cos θ₀) and PEcurrent = mgL(1 - cos θ), then KE = PEinitial - PEcurrent = mgL(cos θ - cos θ₀).
What is the total energy of a pendulum?
The total mechanical energy of an ideal pendulum remains constant and equals the initial potential energy: Etotal = mgL(1 - cos θ₀), where θ₀ is the release angle. This total energy continuously redistributes between kinetic and potential forms as the pendulum swings. At the bottom, all energy is kinetic (E = KEmax). At the highest points, all energy is potential (E = PEmax). At intermediate positions, energy is split between both forms. In real pendulums, total energy gradually decreases due to friction and air resistance.
Can pendulum kinetic energy be greater than its initial potential energy?
No, in an isolated system without external forces, the maximum kinetic energy can never exceed the initial potential energy due to conservation of energy. The kinetic energy at the bottom exactly equals the initial potential energy: KEmax = PEinitial. If the pendulum appears to gain energy, external work is being done (such as pushing the swing) or there's measurement error. Energy can only transform between kinetic and potential forms; it cannot be created. If friction is present, maximum KE will actually be less than initial PE due to energy dissipation.

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