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Calculation templates

Welcome to the Physics library, here we have calculations and content for general physics concepts all the way through to topic areas like fluid mechanics and Astrophysics!

Statics

Dynamics

Thermodynamics and Fluid Dynamics

Acoustics and Optics

Relativity and Quantum Mechanics

Astrophysics

Physics Calculations

Free-To-Use Physics Calculations: Make Complex Physics Problems Easy with CalcTree. Get Accurate Results in No Time! Start using now! 

Simple Harmonic Motion Calculator

Frequency of a Simple Harmonic Motion Calculator

Time Period of a Simple Harmonic Motion Calculator

Velocity of a Simple Harmonic Motion Calculator

Acceleration of a Simple Harmonic Motion Calculator

Acceleration of a Simple Harmonic Motion Calculator (related to the displacement)

Damped Harmonic Motion Energy Loss Calculator

Displacement of Harmonic Motion Calculator

Impulse and Momentum Calculator

Rotational Kinetic Energy Calculator

Work Calculator

Freefall Height Calculator

Freefall Velocity Calculator

Potential Energy Calculator

Kinetic Energy Calculator

Force Due To Acceleration

Pendulum Calculator

Centripetal Force Calculator

Longitudinal Vibration of Rods Calculator

Dunkerley’s Method for Beam Vibration Calculator

Acceleration vs. Velocity equations

Angular Velocity Calculator

Angular Acceleration Calculator

Average Angular Acceleration Calculator

Total Harmonic Distortion (THD) Calculator

View and execute free dynamics calculations.

Physics - Dynamics

This page provides insight into determining the potential energy of a body and provides a function to help you calculate it. Calculation variables include mass, acceleration due to gravity, and vertical distance from the ground.

Potential energy is stored in an object due to its position or arrangement with other objects. When moving, potential energy can be transformed into kinetic energy (energy of motion). An object's potential energy depends on its mass, gravity, and height above a reference point.

If you liked this, check out our other articles and resources!

This page provides insight into determining a body's potential energy and provides tools to help you calculate it.

According to Newtonian physics, freefall can be classified as the motion of an object when no factor except gravity is exerting a force on it. The height of a given object in freefall can be calculated by obtaining the initial velocity, time taken to fall and the acceleration due to gravity. 

Note: This is the same equation as height in projectile motion! 

Freefall can be classified as the motion of an object when no factor except gravity is exerting a force on it. Calculate freefall height here!

According to Newtonian physics, freefall can be classified as the motion of an object when no factor except gravity is exerting a force on it. The velocity of a given object in freefall can be calculated by obtaining the initial velocity, time taken to fall and the acceleration due to gravity. 

See how this equation relates to projectile motion and the 

Freefall can be classified as the motion of an object when no factor except gravity is exerting a force on it. Calculate freefall velocity here.

This page provides calculators for determining displacement, velocity and acceleration and provides some functions to help you calculate it! Read more about the 

Rearranging the equations above to determine:

In kinematics, the basic motion concept of an object, such as velocity, position, and acceleration performed at regular intervals of time, is referred to as equations of motion. 

 govern the motion of an object in 1, 2, and 3 dimensions. Hence, the equations of motion are: 

This page provides equations for determining displacement, velocity and acceleration and provides some functions to help you calculate it!

Acceleration vs. Velocity equations and calculator

Rotational kinetic energy is the energy an object possesses due to its rotation around an axis. It is a scalar quantity and can be calculated by considering the object's moment of inertia and angular velocity. 

The moment of inertia measures an object's resistance to rotational motion and is calculated based on the distribution of mass around the axis of rotation. The greater the moment of inertia, the greater the rotational kinetic energy of an object rotating at a given angular velocity. This is depicted in Figure 1.

The rotational kinetic energy can be described in the following equation:

Rotational kinetic energy is the energy an object possesses due to its rotation around an axis. Calculate it here!

This page provides insight into determining the angular velocity of a body and provides a function to help you calculate it using linear velocity and the radius.

Angular velocity measures the rate of change of an object's angular position over time. It is a vector quantity usually expressed in radians per second (rad/s). 

Angular velocity (ω) is the speed at which an object rotates around a fixed axis. The magnitude of angular velocity represents the speed of rotation, and the direction of the angular velocity vector indicates the axis of rotation.

The equations for angular velocity are as follows: 

This equation is useful for algebraic manipulation involving derivatives and integrals of angular velocity (i.e. Angular Acceleration, Angular Position).

This page provides insight into determining the angular velocity of a body and provides a calculator to evaluate it.

This page provides insight into determining 

 and some functions to help you calculate it!

Work is the amount of energy transferred by a force acting on an object as it is displaced. It is a function of work and distance.

One can calculate the normal force of an object and the spring force of an object. The spring force of an object can be calculated using our 

This page provides insight into determining work and provides functions to help you calculate it!

This page provides insight into determining the impulse and momentum and provides a function to help you calculate it using the 'velocity change' and 'time interval' approaches.

Figure 1 illustrates the magnitude of an impulse experienced by a particle when its momentum changes from the initial state to the final state. The diagram depicts the velocity of a particle before, during and after a collision.

Impulse and momentum are fundamental concepts in physics, and they play a crucial role in our understanding of motion and energy transfer. Impulse is the change in momentum, while momentum is the product of an object's mass and velocity.

As impulse is the change in momentum, it can be expressed in the following equation:

Impulse can also be calculated by using the following equation: 

This is useful when you know the total force acting on a system and the specific time interval for which you measure impulse. 

Determining the impulse and momentum, and provides a function to help you calculate it using the velocity change and impulse length.

This page provides some insight into determining force and provides a force calculator to evaluate it for a given mass and acceleration.

, the push or pull on an object with mass causes it to change its velocity. Force is an external agent capable of changing a body's state of rest or motion. 

It has a magnitude and a direction. The SI units of force are Newton (N) and the imperial units of force are Pounds (lb). 

Calculate force due to acceleration here.

Force Due To Acceleration Calculator

This page provides insight into determining the kinetic energy of a body and provides a function to help you calculate it using mass and velocity.

Kinetic energy is the energy possessed by an object due to its motion. It is a scalar quantity with only magnitude and no direction. Kinetic energy is important in various physical phenomena, such as collisions and the motion of objects under the influence of forces.

Kinetic energy can be expressed mathematically as: 

This page provides insight into determining the frequency of a pendulum and provides a function to help you calculate it using the pendulum's period.

Pendulums are used in many scientific instruments, such as accelerometers, seismometers, and gravimeters, to measure the acceleration of gravity in geophysical surveys. They're even used in clocks, so it's important to understand the maths behind them.

The period of the pendulum is the time it takes for a pendulum to complete one cycle, to move left and right.

One can calculate different things, like the frequency, as seen in the equation below:

We can also calculate the period of the pendulum.

From this, we can also calculate the length of the pendulum.

Learn about pendulum properties, and calculate their period, frequency and length.

Pendulum Properties Calculator

This template calculates the displacement, velocity and acceleration of an object undergoing simple harmonic motion (SHM), a type of periodic motion where the restoring force is directly proportional to the displacement. Simple harmonic motion is commonly observed in systems like pendulums and springs. This calculator is essential for understanding the fundamental principles of oscillatory motion in physics and engineering.

Simple Harmonic Motion (SHM) is a fundamental concept in mechanics and physics, serving as a model for various motions such as spring oscillations and pendulum swings.

This tool is particularly useful in scenarios such as:

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This template calculates the displacement, velocity and acceleration of an object undergoing simple harmonic motion (SHM).

This tool calculates the frequency of an object undergoing simple harmonic motion (SHM), a type of periodic motion where the restoring force is directly proportional to the displacement. 

This tool calculates the frequency of an object undergoing simple harmonic motion (SHM).

This tool calculates the period of an object undergoing simple harmonic motion (SHM), a type of periodic motion where the restoring force is directly proportional to the displacement. 

This calculator is particularly beneficial when:

This tool calculates the time period of an object undergoing simple harmonic motion (SHM).

This template is designed to calculate the energy loss in a damped harmonic motion system, which is where a spring or pendulum experiences a decrease in amplitude over time due to a damping force.

The formula for energy loss in a damped harmonic motion system involves the initial energy, damping coefficient, mass and time, highlighting how these factors influence the energy dissipation in the system.

Harmonic motion is a fundamental concept in mechanics and physics, serving as a model for various motions such as spring oscillations and pendulum swings. A damped harmonic motion system is where a spring or pendulum experiences a decrease in amplitude over time due to a damping force, such as internal friction or air resistance. 

The energy loss in the system is given as:

The energy of a damped harmonic oscillator decreases exponentially over time, meaning the rate of energy loss is time-dependent. The longer the system oscillates, the more energy it loses. This is quantified using the exponential decay factor 

This template is particularly useful in scenarios such as:

Calculate the energy loss in a damped harmonic motion system, which is where the amplitude of a spring or pendulum decreases over time.

Use this template to find the Centripetal Force & Centripetal Acceleration acting on an object while in circular motion. You can also find related missing parameters such as mass, tangential velocity and angular velocity of the object and the radius of the circular path.

❗Important: you only need to input three parameters for the Output table to fully populate

Table

Table (1)

The centripetal force acts on an object moving in a circular path, directing it towards the centre of the circle. It is responsible for keeping an object in circular motion and preventing it from moving in a straight line. 

Centripetal force can arise from various sources, such as tension in a string, gravitational attraction or friction.

You can use three different formulas to calculate centripetal force depending on the parameters given or yet to be found.

You can also use other bridging formulas to calculate parameters that aren't directly a part of each individual formula. Here are a few bridging relationships.

We can rearrange the equations above to create many combinations of formulas for all our parameters!

Use this template to find the Centripetal Force and Centripetal Acceleration acting on an object while in circular motion. 

Centripetal Force and Centripetal Acceleration Calculator

This tool calculates the instantaneous velocity of an object undergoing simple harmonic motion (SHM), a type of periodic motion where the restoring force is directly proportional to the displacement. 

matplotImage

This is the derivative of the instantaneous displacement, 

 in simple harmonic motion which is given by:

The velocity is a sinusoidal waveform meaning the object's velocity changes dynamically as it oscillates. That is. the velocity varies depending on the object's position in its oscillatory path, reaching its maximum velocity at the equilibrium point and diminishing to zero at the maximum displacement points. This characteristic velocity pattern directly results from the harmonic restoring force acting on the object.

This calculator has applications such as:

This tool calculates the instantaneous velocity of an object undergoing simple harmonic motion (SHM).

This template is used to calculate the natural frequency of a rod undergoing longitudinal vibration. Such vibrations are a fundamental aspect in the fields of mechanical engineering, structural analysis and materials science.

Analogous to a spring system which is often used as an idealised model of how a beam bends, a vibrating rod is an idealised model of longitudinal vibrations of many different objects. A vibrating rod can also be considered as an elongated elastic band or string. It is used to understand longitudinal vibrations in diverse engineering contexts including the shaking behaviour of a building, the sound of instruments and the toughness of materials.

Longitudinal vibration is when particles within our idealised “rod” move back and forth in a direction parallel to the rod's length. Longitudinal vibration is a form of harmonic motion, which means this back and forth movement is a periodic oscillation of particles. This oscillatory motion leads to alternating regions of compression and elongation along the rod, a phenomenon critical for the analysis and design of mechanical systems to prevent resonant frequencies that could cause structural failure.

The fundamental frequency, also known as the natural frequency, is the frequency at which our rod naturally vibrates in. It is the simplest form of vibration where the whole rod stretches or compresses along its length. It is important to predict the natural frequency of your object under consideration in order to avoid external vibrations at the same frequency, which would lead to resonance and therefore potential failure. 

The fundamental frequency has an inverse relationship to the rod’s length, which means an increase in rod length leads to a reduction of the natural frequency.

The fundamental frequency is found in the rod’s first "mode", a mode is a specific pattern of vibration. The second mode shape has an additional node in the centre showing the two halves of the rod moving in opposite direction to each other, leading to more complex stress distributions in the rod. 

As a general pattern the frequency of mode 

As the mode number increases, so does the number of nodes and therefore dividing the rod into more segments that vibrate out of phase with adjacent segments, leading to increasingly complex patterns of vibration.

his template is particularly useful in scenarios such as:

This template is used to calculate the natural frequency of a rod undergoing longitudinal vibration.

This template uses Dunkerley's method to approximate the natural frequency of a system by considering the natural frequencies of its component parts. 

Dunkerley’s method is an approximation technique that considers the effects of added mass on the natural frequency of a system. It provides a simple yet effective way to estimate the natural frequency of beams and structures.

This template uses Dunkerley's method to approximate the natural frequency of a system by considering the natural frequencies of its components.

This calculator calculates the instantaneous acceleration of an object undergoing simple harmonic motion (SHM), as a function of it's displacement from the equilibrium position.

 of an object undergoing simple harmonic motion at time 

 is the first derivative of the object's displacement and the acceleration 

 is the second derivate of the object's displacement (or the first derivative of the object's velocity). We can re-write the acceleration 

This calculator calculates the instantaneous acceleration of an object undergoing simple harmonic motion, as a function of it's displacement.

Acceleration as a function of displacement of a Simple Harmonic Motion Calculator 

This calculator finds the angular acceleration of a spinning object, which is the rate of change of it's angular velocity with respect to time. The angular acceleration is commonly calculated when a torque is applied to an object. 

Angular acceleration of a spinning object is the rate of change of it's angular velocity with respect to time. It is commonly calculated when a torque is applied to an object. 

 is a twisting force that causes an object to rotate. Torque is a vector quantity that requires both magnitude and direction to define it. The magnitude is given by 

 is the radius from the axis of rotation. The direction is determined using the right-hand rule as per the below figure: curl the four fingers of your right hand in the direction of rotation and your thumb is pointing in the direction of 

When the torque of an object is constant its angular acceleration is also constant. The constant angular acceleration, 

 can be calculated by the ratio of the total torque of the object, 

The figure below shows the direction of the angular acceleration and torque due to an applied force, creating a counterclockwise angular acceleration about the rotational axis and a positive torque.

This calculator finds the angular acceleration of a spinning object, which is the rate of change of it's angular velocity with respect to time. 

This calculator calculates the instantaneous acceleration of an object undergoing simple harmonic motion (SHM), a type of periodic motion where the restoring force is directly proportional to the displacement. 

 and a second derivative of the displacement, 

 in simple harmonic motion which are given by:

The acceleration is a cosine waveform meaning the object's acceleration changes dynamically as it oscillates. We can write 

 meaning the acceleration acts in opposite direction to the displacement and is proportional to it.

This calculator calculates the instantaneous acceleration of an object undergoing simple harmonic motion (SHM).

This calculator determines the average angular acceleration over a specific period based on a specified initial and final angular velocity. Angular acceleration tells us the rate at which the angular velocity of a spinning object changes over time. 

 is a foundational concept in rotational motion that explains how quickly an object's angular velocity, 

 changes with time, due to external forces. Angular velocity denotes the speed at which an object rotates around a fixed axis. When an object rotates, it experiences changes in its angular velocity, either speeding up or slowing down. Angular acceleration measures the rate of this change and provides critical insights into the dynamics of rotational motion.

 angular acceleration means the object's rotation is accelerating, while negative 

 angular acceleration means it's decelerating. In the figure below, the left image shows positive angular acceleration while the right image shows negative angular acceleration.

Mathematically, we can define angular acceleration as the ratio of the change in angular velocity to the change in time. This is expressed as: 

Angular acceleration is crucial in real-world applications such as engineering and physics, where it helps analyse the motion of rotating machinery, celestial bodies and gyroscopes. Angular acceleration provides a quantitative measure of how an object's angular velocity changes, enabling scientists and engineers to understand and predict the behaviour of rotating objects in various contexts.

A lathe is one of many applications in the real world that may involve calculations of angular acceleration. The image below shows a lathe at high speed to bore a hole.

This calculator determines the average angular acceleration over a specific period based on a specified initial and final angular velocity.

This calculator computes the Total Harmonic Distortion

(THD) of a signal in audio and electric power systems and outputs a graph of the actual distorted waveform. THD quantifies the deviation of a signal's harmonic content from a pure signal by comparing the voltages of the signal's harmonic components to the voltage of the signal's fundamental frequency.

is not an input to the THD equation, it is an input to the sinusoidal function for the signal waveforms. 

The waveform generated can either be from an audio signal or an electric power system, depending on the application. 

matplotImage (1)

Total Harmonic Distortion (THD) measures the deviation of a signal from its intended pure form due to harmonics, which are multiples of the fundamental frequency. Harmonics can originate from the signal's interaction with equipment or external noise, affecting signal quality. While a signal may possess an infinite number of harmonics, the first three are primarily analyzed because they significantly impact audio fidelity and power efficiency. The focus on these harmonics is practical, as higher-order harmonics have diminishing effects on overall signal quality.

THD is critical in assessing the purity of audio and electrical power signals, where a lower THD suggests a signal that closely resembles the original sound or intended electrical frequency, leading to improved audio reproduction and power system performance.

Typically, the THD value can be interpreted as:

See the equations used in this calculator in the toggles below.

Discover more about the different parameters in the toggles below.

Waveform analysis of THD is applied in the real-world, for example, to create an Active Harmonic Filter (AHF). An active filter setup demonstrating corrective action on a non-linear load is shown in the below image. The AHF produces a compensating current 

, which, when combined with the distorted load current 

Additionally, the significance of THD includes:

This calculator computes the Total Harmonic Distortion (THD) of a signal in audio and electric power systems.

This calculator determines the displacement of a harmonic wave at a given point and time. Harmonic waves are a foundational concept in physics that describes the propagation of waves with a sinusoidal shape. 

Harmonic waves are fundamental to our understanding of many physical phenomena, from the simple oscillations of a pendulum to the complex propagation of electromagnetic waves. These waves are distinguished by their sinusoidal pattern, exhibiting smooth and continuous oscillations that repeat at regular intervals. 

 of a point on a harmonic wave at a given time 

 is mathematically represented by the equation:

Displacement in a harmonic wave follows a sinusoidal pattern, peaking at the amplitude during oscillation crests or troughs and hitting zero at equilibrium. This cycle, dictated by the harmonic wave equation, is key for understanding wave dynamics, showing energy transmission and interaction with the medium. 

Harmonic waves are crucial in various fields, including acoustics, optics, and electromagnetic theory, offering insights into the behavior of waves and their interaction with different media. This calculator has applications such as:

This calculator determines the displacement of a harmonic wave at a given point and time. Harmonic waves are a foundational concept in physics.