# Linear Motion Under a Variable Force

## Definition and Key Concepts

• Linear motion refers to the motion of an object along a straight line.
• Variable force is a force that changes in magnitude, direction, or both, as the position or velocity of the object changes.
• Hence, when an object is moving under the influence of a variable force along a straight line, it’s said to be experiencing linear motion under a variable force.

## Newton’s Second Law

• Newton’s Second Law of Motion states that the acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
• F = ma, where F is the net force acting on the object, m is the mass of the object, and a is its acceleration.
• When forces are variable, calculus often must be used to apply Newton’s Second Law.

## Variable Forces

• The most common variable force in mechanics is the spring force. Springs obey Hooke’s Law which states that the force a spring exerts on an object is proportional to the displacement of the spring’s end from its equilibrium position. Mathematically, F = -kx, where k is the spring constant and x is the displacement from equilibrium.
• Another frequent variable force is the drag force which is experienced by an object moving through a fluid. Its magnitude usually depends on the speed of the object.

## Energy in Linear Motion Under Variable Force

• An object moving under a variable force also has kinetic energy (due to its motion) and potential energy (due to its position).
• The work done by a variable force can be calculated as the area under a force-position graph.
• As the object moves, potential energy is transformed into kinetic energy, or vice versa, but the total mechanical energy (the sum of kinetic and potential energy) is conserved, assuming a closed system with no non-conservative forces.

## Practical Applications

• Linear motion under variable force is a fundamental concept underlying many real-world applications, from the design of car suspension systems (springs) to the calculation of terminal velocity in fluid dynamics (drag force).

## Problem-Solving Involving Linear Motion Under Variable Force

• In problems involving variable forces, often the first step is to calculate or determine how the force varies with position or velocity.
• Then, you can use Newton’s Second Law to develop a differential equation describing the motion.
• The work-energy theorem, which relates the work done by all the forces acting on an object to the change in its kinetic energy, can also be a powerful tool in solving problems involving variable force.

Understanding the concept of linear motion under a variable force is a key aspect of further mechanics and provides the foundation for more complex topics like oscillatory motion and waves. Knowledge of this topic is also essential for engaging in physical modelling in fields such as engineering and physics.