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 comp…
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.
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Q1: How do you calculate work done by a variable force?
Work done by a variable force equals the area under the force-displacement graph. To find this, divide the graph into small strips where force is approximately constant, then sum the work for each strip. For better accuracy, increase the number of strips until their width approaches zero, converting the sum into an integral of force with respect to distance.
Q2: What is the relationship between work and kinetic energy?
The work-energy theorem states that work done on an object equals its change in kinetic energy. Using Newton's second law and calculus, the work done by a variable force can be rewritten in terms of kinetic energy change. This principle holds true whether the force is constant or varies, making it a powerful tool for analyzing object motion.
Q3: Why is graphical representation useful for variable forces?
A force-displacement graph visually represents how force changes over distance. The area under this curve directly represents total work done. By dividing the area into discrete strips, each with approximately constant force, you can sum individual work contributions. This graphical method makes complex variable force calculations manageable and intuitive.
Q4: How does the work-energy principle apply to variable forces?
The work-energy principle states that work done on an object equals its change in kinetic energy plus change in potential energy. For variable forces, you integrate force over distance to find total work. This work results in changes to the object's kinetic and potential energy, allowing you to calculate total energy change systematically.
Q5: What role does calculus play in analyzing variable forces?
Calculus enables precise calculation of work done by variable forces through integration. Instead of approximating with discrete strips, integration converts the sum into a continuous mathematical expression. This approach yields exact work values by integrating force with respect to distance, providing the foundation for the work-energy theorem with varying forces.
Q6: How does approximation improve when analyzing variable force work?
Approximation improves by increasing the number of strips in the force-displacement graph. As strip width decreases toward zero, the assumption that force is constant within each strip becomes more accurate. This refinement process transforms discrete summation into integration, yielding increasingly precise work calculations until reaching the exact mathematical limit.
Q7: Why is the work-energy theorem important in physics and engineering?
The work-energy theorem provides a universal method for analyzing motion under any force condition, constant or variable. It connects force, displacement, and energy change through a single principle applicable across physics and engineering. This makes it invaluable for solving real-world problems involving complex variable forces on moving objects.