7.9: Work and Energy for Variable Forces

When an object is acted upon by a variable force, the amount of work done and the change in energy of the object can be more complex to calculate compared to when a constant force is applied. Work is the product of force and displacement, while energy is the capacity of a system to do work. When a constant force is applied to an object, the work done can be calculated as the product of the force and the distance moved in the direction of the force. However, when a variable force is applied, the work done is the area under the force-displacement graph.

The change in energy of an object can also be more complex to calculate when a variable force is applied. In this case, the work-energy principle can be used. This principle states that the work done on an object is equal to its change in kinetic energy plus its change in potential energy. So, the total change in energy of an object can be calculated by adding its change in kinetic and potential energy.

When a variable force is applied to an object, the amount of work done and the change in energy of the object can be calculated using the concepts of calculus and the work-energy principle. Understanding these concepts is important in many fields, including physics and engineering, where the effects of variable forces on objects are often encountered.

The work-energy theorem, which states that the work done on an object is equal to its change in kinetic energy, holds true for varying forces. This means that whether a force is constant or varies over time, the work done on an object can be calculated by integrating the force over the distance traveled, and this work results in a change in the object's kinetic energy.

For example, if an object is pushed with a varying force, the total work done on the object can be found by integrating the force over the distance that the object is pushed. This work results in a change in the object's kinetic energy, which can be calculated using the work-energy theorem. So, the work-energy theorem is a powerful tool that can be used to analyze the motion of objects subject to various forces, including varying forces.

A car moving at variable speed is under the influence of a variable force.

To find the work done by the force in moving the car from point A to B, the one-dimensional varying force with the distance covered by the car is plotted.

The area under the curve is divided  into a number of strips of discrete width.

For each strip, the applied force is approximately constant.

So, the work done by the variable force for the total distance between A and B is given by the sum of the individual work done by the constant force for the small segments.

For better approximation, increase the number of strips such that the width of each strip approaches zero. So, the work done can be given by the integral of the one-dimensional varying force with respect to the distance, and represented as the area under the curve.

Using Newton's second law and applying the chain rule of calculus, the work done can be rewritten in terms of the change in kinetic energy which is the work-energy theorem.