Force Equations Formulas Calculator

Science - Physics

Solving for force.


mass (m)
acceleration (a)
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mass (m)
= 0
= 0
acceleration (a)
= 0
= 0


force (F)

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References - Books:

Tipler, Paul A.. 1995. Physics For Scientists and Engineers. Worth Publishers. 3rd ed.


The force equation is fundamental to the study of physics. It describes the relationship between force, mass, and acceleration. Understanding this equation is critical to understanding many physical concepts, ranging from the motion of objects in free fall to the behavior of subatomic particles in high-energy collisions.
Force is an interaction between two objects which causes a change in object motion. Typically, it causes an object with mass to accelerate or decelerate. Acceleration is a change in velocity. Force can also be described as a pull or push. It is a vector quantity with a direction and magnitude.

Force is directly proportional to both mass and acceleration.
Force is categorized into two types, contact and Action At A Distance. Contact forces require the two object to be in direct contact. Action At A Distance forces do not require contact and are exerted over a distance. Types of force include air resistance, applied, centripetal, elastic, electrical, electromagnetic, friction, gravitational, magnetic, nuclear, normal, pressure, spring and tension.


Sir Isaac Newton first proposed the force equation, also known as Newton's second law of motion, in 1687. The relationship described by Newton's law states that the acceleration of an object is determined by the net force acting on it and is inversely related to the object's mass. The more force applied to an object, the greater its acceleration; the more mass an object has, the less its acceleration will be for the same force.

Force Equation Mathematical Structure:

The mathematical structure of the force equation is quite simple. It is typically written in the form F = ma. F represents the net force acting on an object. m represents the object's mass. a represents its acceleration. Rearranging the equation, we can also write it as a = F/m, which tells us that an object's acceleration equals the net force acting on it divided by its mass.

Force Equation Applications:

The force equation has many applications in physics. For example, it calculates the force required to accelerate an object at a specific rate. If we know the object's mass and the desired acceleration, we can use the force equation to determine how much force we need to exert. Similarly, the force equation is used to study the behavior of objects in free fall. By knowing the force of gravity on an object and its mass, we can use the equation to calculate its acceleration toward the ground.
The force equation is also used to study the behavior of fluids and gases. For example, when air flows over an airplane wing, it creates a force that lifts the plane into the air. The force equation can calculate this lift force by analyzing the air's mass and acceleration around the wing.
Furthermore, the force equation plays a significant role in understanding the behavior of subatomic particles in high-energy collisions. When particles collide at high speeds, enormous forces are involved, and the force equation is used to calculate the resulting acceleration and energy transfer.

Example of using the Force Equation:

In 10 seconds, a car with a mass of 1200 kg goes from being at rest to achieving a velocity of 20 m/s. What force is required to achieve this acceleration?
Start by identifying the given values:
  • Mass (m) = 1200 kg
  • Initial velocity (u) = 0 m/s (rest)
  • Final velocity (v) = 20 m/s
  • Time taken (t) = 10 seconds
Calculate the acceleration (a):
  • We can use the equation a = (v - u) / t. v is the final velocity. u is the initial velocity. t is the time taken.
  • Plugging in the given values, we get a = (20 - 0) / 10 = 2 m/s^2.
Use the force equation (F = ma) to find the force required:
  • We have the mass (m) and the acceleration (a), so we can multiply them together:
  • F = 1200 kg * 2 m/s^2 = 2400 N.
As a result, it can be calculated that the force necessary to accelerate the car, starting from rest and reaching a velocity of 20 m/s within 10 seconds, amounts to 2400 Newtons.

Common Errors when using the force Equation

  • Not considering net force: One common error is not accounting for the net force acting on an object. The force equation distinctly addresses the net force as the combined vector sum of all the forces affecting the object. Neglecting to consider all relevant forces can lead to inaccurate calculations.
  • Confusing mass and weight: Another error occurs when mass and weight are used interchangeably. Mass measures the amount of matter in an object, while weight is the force of gravity acting on an object's mass. The force equation requires mass, not weight, so it is essential to distinguish between the two.
  • Incorrect units: Misusing or misinterpreting units can result in errors. Ensuring all quantities in the force equation are expressed in the appropriate units is crucial. For example, mass should be in kilograms (kg), acceleration in meters per second squared (m/s^2), and force in Newtons (N).
  • Not considering the direction of forces: The force equation includes a vector quantity, meaning that both magnitude and direction are essential. You need to consider the direction of forces to avoid incorrect results. Assigning appropriate signs (+/-) to forces based on their direction is essential.
  • Failing to account for external factors: The force equation assumes an idealized scenario without external factors such as air resistance or friction. Neglecting to account for these factors can introduce inaccuracies in calculations. In real-world situations, it is vital to consider and appropriately modify the force equation to account for these external influences.

Examples of Force:

  • Air Resistance: Air resistance, also known as drag, is a force that acts against the motion of an object as it moves through the air. It becomes more significant as the speed of the object increases.
  • Applied force: A force exerted on an object by external agents through physical contact, such as pushing or pulling.
  • Buoyant force: This is experienced by objects immersed in a fluid (liquid or gas). A fluid exerts a positive force and acts in an upward direction. The buoyant force experienced by an object is equivalent in magnitude to the weight of the fluid that the object displaces.
  • Centrifugal force: This force refers to the apparent outward force felt by an object when it moves in a curved path. It is often referred to as a pseudo-force, as it only arises from the perspective of the rotating frame of reference.
  • Centripetal force: This force acts towards the center of a curved path and keeps an object moving in a circular or curved trajectory. It is necessary to maintain the object's constant speed and change the direction of its velocity.
  • Drag force: This force opposes an object's motion through a fluid (liquid or gas) and is influenced by factors such as the object's shape, size, and speed.
  • Elastic force: This force is exerted by a material when it is stretched or compressed. It is characterized by Hooke's Law, which states that the force is directly proportional to the displacement from the equilibrium position.
  • Electrostatic force: This force is exerted between charged objects. Similar charges repel one another, whereas opposite charges attract each other. It plays a crucial role in static electricity and the behavior of electrically charged particles.
  • Frictional force: The force that opposes the motion or attempted motion between objects in contact. It can be kinetic (during relative motion) or static (when there is no relative motion).
  • Gravitational force: The force of attraction between two objects as a result of their masses. It is responsible for objects falling towards the Earth and has a relationship with mass and distance.
  • Magnetic force: This force is exerted on moving charges or current-carrying conductors in a magnetic field. It acts perpendicular to both the magnetic field and the velocity or current.
  • Normal force: The force a surface exerts to support an object resting on it. It acts perpendicular to the surface and prevents objects from sinking into it.
  • Nuclear force: This force functions within atomic nuclei and is accountable for the cohesive binding of protons and neutrons. It is one of the fundamental forces in nature and is tremendously strong at small distances.
  • Shear force: This force acts parallel to the surface of an object, and it is capable of causing deformation or sliding of the object along the surface.
  • Spring Force: The spring force is the force exerted by a spring that is either stretched or compressed as it attempts to return to its original length at equilibrium. It abides by Hooke's Law, which states that the force is directly proportional to the displacement from the equilibrium position.
  • Tension Force: A tension force is transmitted through a cable or any flexible connector due to pulling forces acting on it. It is always directed along the length of the connector.
  • Torsion force: This force is exerted on an object when it is twisted or rotated around a central axis. It opposes the twisting motion and depends on the material's elasticity and the magnitude of the twist.

Common fields Using the Force Equation

  • Physics: Physics is the primary field that extensively uses the force equation. It is foundational to understanding various aspects of motion, mechanics, and the behavior of objects in different physical systems.
  • Engineering: Engineering disciplines, such as civil, mechanical, aerospace, and electrical engineering, rely on the force equation to design and analyze structures, machinery, control systems, and more. It helps engineers calculate the forces required for specific applications and ensure that designs meet safety standards.
  • Astronomy: In astrophysics and astronomy, the force equation is crucial for understanding celestial mechanics, the motion of heavenly bodies, and the gravitational forces between them. It enables calculations related to orbits, planetary systems, and the behavior of objects in space.
  • Biomechanics: Biomechanics investigates the forces acting on biological systems, such as the human body. Understanding the force equation helps biomechanists analyze movement patterns, assess impacts on the body, design prosthetics, and optimize athletic performance.
  • Automotive engineering: Professionals in the automotive industry use the force equation to design and optimize vehicle systems. It aids in determining factors like engine power, braking force, suspension design, and overall vehicle performance and safety.
  • Sports science: Sports scientists use the force equation to study biomechanics in sports, assess forces applied during movements, and understand the impact of these forces on athletes' performance, injuries, and equipment design.
  • Fluid mechanics: Professionals in chemical engineering, environmental science, and hydraulic engineering employ the force equation to study fluids' behavior, calculate fluid flow rates, design pipelines, and assess the forces on structures submerged in liquids.
  • Material science and engineering: The force equation is essential in material science and engineering for understanding how materials respond to external forces. It assists in calculating stresses and strains and determining the structural integrity and behavior of materials under different loads.
  • Robotics: Robotics engineers utilize the force equation to design robotic systems, calculate joint forces and torques, and ensure the stability and safety of robot movements and interactions with the environment.
  • Particle physics: The force equation is crucial in particle physics, specifically high-energy particle accelerators and collisions. It helps physicists study interactions between subatomic particles, calculate the forces involved, and understand particle behavior under extreme conditions.
The force equation has broad applications, and many other fields beyond those listed above may also utilize it in their respective domains.

Force Frequently Asked Questions (FAQs)

  • What does the force equation (F = ma) represent? The force equation describes the relationship between force, mass, and acceleration. According to this concept, the force acting upon an object is equivalent to the product of its mass and its acceleration.
  • How does the force equation relate to Newton's second law of motion? The force equation is Newton's second law of motion. The spring force follows Hooke's Law, stating that it is directly proportional to the displacement from equilibrium. It also relates force, mass, and acceleration by saying that acceleration is directly proportional to the net force and inversely proportional to mass.
  • Can you explain how the force equation is used in everyday life? One example is when we push a car. The force exerted by our muscles causes the vehicle to accelerate because the force equation tells us that the acceleration is directly proportional to the force applied.
  • How does the force equation indicate the effect of weight on acceleration? The weight of an object is simply the force of gravity acting on it. According to the force equation, objects with greater mass will have less acceleration for the same force. Therefore, a heavier object will have a smaller acceleration than a lighter object when the same force is applied.
  • What happens to the acceleration if the mass remains constant but the force increases? According to the force equation, if the mass of an object remains constant and the force acting on it increases, the acceleration will also increase. This means that the object will experience a greater change in velocity over a given time.
  • What happens to the acceleration if the force remains constant but the mass increases? If the force acting on an object remains constant and the mass increases, the acceleration will decrease. This is due to the force equation, which states that the acceleration of an object is inversely proportional to its mass. Therefore, the object becomes more resistant to acceleration as the mass increases.
  • How does the force equation explain the motion of free-falling objects? Gravity is the sole force that acts on an object during free fall. The force equation calculates the object's acceleration due to gravity based on its mass. Objects with a greater mass will experience a greater force of gravity and, consequently, a greater acceleration.
  • Can the force equation be applied to objects moving in a circular path? Yes, the force equation can be used for objects moving in a circular path. Centripetal force constantly pulls the object toward the circle's center in a circular motion. The force equation can be used to calculate this centripetal force based on the object's mass and the radius of the circular path.
  • Does the force equation apply to situations where multiple forces act on an object? Yes, the force equation applies to situations where various forces act on an object. In such cases, the net force is the vector sum of all the individual forces. We can determine the resulting acceleration by applying the force equation to the net force.
  • Are there any limitations to the force equation? While the force equation is a powerful tool, it has limitations. It assumes that the mass of an object remains constant and that forces are applied linearly. Additionally, it does not account for factors such as air resistance or friction, which can affect the object's acceleration. In complex situations, additional equations and concepts may need to be applied to describe objects' motion accurately.

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