Kinetic Energy of a Pendulum

General Physics

Calculate a pendulum's kinetic energy using either its velocity or its release height (determined by length and angle).

Examples

Explore some practical examples to understand how the calculator works.

Physics Classroom Example

velocity

A standard lab setup where a pendulum's mass and maximum velocity are known.

Type: undefined

Mass: 2 kg

Length: undefined undefined

Angle: undefined undefined

Grandfather Clock Pendulum

angle

Calculating the energy of a typical longcase clock pendulum released from a small angle.

Type: undefined

Mass: 500 g

Length: 1 m

Angle: 10 deg

High-Velocity Pendulum

velocity

An example involving higher speeds, perhaps from a larger mechanical system.

Type: undefined

Mass: 10 kg

Length: undefined undefined

Angle: undefined undefined

Wrecking Ball Scenario

angle

A simplified model of a wrecking ball, demonstrating the immense energy it carries.

Type: undefined

Mass: 1000 kg

Length: 20 m

Angle: 45 deg

Other Titles
Understanding the Kinetic Energy of a Pendulum: A Comprehensive Guide
Dive deep into the physics behind a swinging pendulum, exploring how its energy transforms and how to calculate it accurately.

What is the Kinetic Energy of a Pendulum?

  • The Role of Energy in a Pendulum's Motion
  • Defining Kinetic and Potential Energy
  • The Principle of Conservation of Energy
The kinetic energy of a pendulum is the energy it possesses due to its motion. As a pendulum swings back and forth, it undergoes a continuous transformation between potential energy (stored energy due to its height) and kinetic energy. At the highest point of its swing, its velocity is momentarily zero, and all its energy is potential. As it swings downwards, this potential energy is converted into kinetic energy. At the very bottom of the swing, the pendulum reaches its maximum speed, and its kinetic energy is at its peak, while its potential energy is at its minimum (usually considered zero).
The Core Formula
The fundamental formula for kinetic energy (KE) is KE = ½ m v², where 'm' is the mass of the object (the pendulum bob) and 'v' is its velocity. This calculator helps you determine this value either by providing the velocity directly or by calculating the velocity from the pendulum's initial height, which is determined by its length and release angle.

Step-by-Step Guide to Using the Calculator

  • Choosing Your Calculation Method
  • Inputting Your Variables Correctly
  • Interpreting the Results
Method 1: Using Mass and Velocity
This is the most direct way to calculate kinetic energy. If you already know how fast the pendulum is moving at its lowest point, select the 'Velocity' method. Input the mass of the pendulum bob and its maximum velocity. The calculator will apply the KE = ½ m v² formula to give you an instant result.
Method 2: Using Mass, Length, and Angle
This method is useful when you don't know the velocity but know the physical setup of the pendulum. Select the 'Angle' method. Input the bob's mass, the length of the pendulum's string or rod, and the angle from which it was released. The calculator first determines the vertical height (h) the pendulum dropped using the formula h = L (1 - cos(θ)). Then, it calculates the maximum velocity using v = √(2 g * h), where g is the acceleration due to gravity. Finally, it computes the kinetic energy with this velocity.

Real-World Applications of Pendulum Energy

  • Timekeeping and Clocks
  • Amusement Park Rides
  • Seismology and Metrology
The principles of pendulum motion and energy are fundamental to many technologies. The most classic application is in pendulum clocks, where the regular swing of the pendulum, governed by its length and gravity, keeps time. The energy to overcome friction is supplied by a weight or spring. In amusement parks, rides like the 'Pirate Ship' are essentially giant pendulums, using the conversion of potential to kinetic energy to create a thrilling experience. In science, pendulums are used in seismometers to measure earthquakes and in metrology to make precise measurements of local gravitational acceleration.

Common Misconceptions and Correct Methods

  • Is Energy Lost in a Swing?
  • The Small Angle Approximation
  • Does Mass Affect the Period?
A common misconception is that a pendulum will swing forever. In reality, energy is always lost to air resistance and friction at the pivot point. This is why clocks need a power source to keep the pendulum going. Another key point is the 'small angle approximation'. For small angles (typically < 15°), the period of a pendulum is nearly independent of the amplitude. Our calculator does not use this approximation for energy calculations, providing exact results based on the given angle. Lastly, while the mass of the pendulum does not affect its period, it is a crucial component in calculating its kinetic energy—more mass means more energy for the same speed.

Mathematical Derivation and Examples

  • From Potential to Kinetic Energy
  • Deriving Velocity from Height
  • Worked Example
Derivation
The principle of conservation of energy states that the maximum potential energy (PEmax) at the peak of the swing equals the maximum kinetic energy (KEmax) at the bottom. The potential energy is given by PE = m g h. The height 'h' is derived from trigonometry: h = L - Lcos(θ) = L(1 - cos(θ)). Therefore, KEmax = PEmax = m g L (1 - cos(θ)). This allows us to find the kinetic energy without ever knowing the velocity.
Example Calculation
Let's use an example from the calculator: a 2 kg mass on a 1.5 m pendulum released from 30°. First, find the height: h = 1.5 (1 - cos(30°)) = 1.5 (1 - 0.866) = 0.201 m. Now, calculate the kinetic energy: KE = m g h = 2 kg 9.81 m/s² 0.201 m ≈ 3.94 Joules.