You may already know a lot about work and energy, such as how to calculate work done on or by an object, the types of energy, and how energy is conserved. But did you know that all of these physics concepts are closely related? In this article, we’ll explain the work-energy theorem and how to apply the equation in order to explore and analyze systems, including when work is done by friction.
What We Review
Reviewing Work and How to Calculate It
What is Work?
In Physics, we define work as the change of energy in a system caused by a force applied over a displacement. We say that work is done when a force acting on an object causes it to move. You can read more about work here.
The Formula for Calculating Work
We can calculate work using the following formula:
| Work Formula W=F \cdot d |
..where:
- W is work
- F is force
- d is displacement
Work-Energy Theorem
What is the Work-Energy theorem?
The work-energy theorem states that the work done on a system is equal to its change in kinetic energy. We can understand where this theorem comes from if we break down work and energy.
When work is done on an object, there must be a force causing the object to move. As the force is applied, the object accelerates or changes its velocity. For example, when a kickboxer kicks a heavy bag that is initially at rest, he is doing work on the bag, and the force he exerts changes the bag’s velocity. We also know that kinetic energy is the energy of motion and depends on velocity. When an object’s velocity changes, its kinetic energy also changes. Since doing work on an object changes its motion, the object’s kinetic energy also changes. When the kickboxer does work on the heavy bag, he is changing the bag’s kinetic energy by increasing its velocity.
How Work and Energy are Related
The Work-Energy Theorem Equation
The work-energy theorem is expressed by the following equation:
| Work-Energy Theorem W=\Delta E_{k} |
…where:
- W is work
- \Delta E_{k} is the change in kinetic energy
Check out this video from GPB Education for more details about the work-energy theorem.
Applying the Work-Energy Theorem
How to Use the Equation to Solve Problems
The work-energy theorem provides us with a tool we can use to analyze a variety of systems. When solving problems, you will often combine the work-energy theorem with the equations you’ve learned before for work and kinetic energy. The information given will lead you to how to approach the given problem.
Say you are given the force applied on an object and its resulting displacement. You want to determine the object’s change in kinetic energy. First, you’ll apply the equation for work as the product of the force and displacement. Once you calculate work, the work-energy theorem tells us that this work is equal to the change in kinetic energy.
However, if you are given the object’s mass and initial and final velocities, you’ll want to start by first calculating the object’s initial and final kinetic energy. As a reminder, the equation for kinetic energy is:
| Kinetic Energy Formula E_{k}=\frac{1}{2}mv^2 |
…where:
- m is mass
- v is velocity
Once you determine the object’s initial and final kinetic energy, you can calculate the change in kinetic energy by subtracting the initial kinetic energy from the final kinetic energy. Finally, by applying the work-energy theorem you’ll know that the work done on the object is equal to the change in kinetic energy you calculated.
Common Mistakes and Misconceptions about Work and Energy
One of the most common mistakes occurs when calculating the change in kinetic energy. It is important to separately calculate the initial and final kinetic energies, then subtract them to determine the change in kinetic energy. Some students assume that you could save a step by first calculating the change in velocity and then calculating the kinetic energy only once. However, this will result in an incorrect answer because the velocity is squared in the formula for kinetic energy.
Other common misconceptions are often involved with work. In order for work to be done on an object, the force must cause a displacement. If the object doesn’t move, no work is done and the kinetic energy does not change. In more complex systems that may involve multiple forces, it is also important to pay careful attention to only the net force that causes the displacement and does work on the object. For example, imagine you push a box horizontally across a table. In addition to your applied force, there are other forces acting on the box such as the normal force from the table and the force of gravity. However, these forces are not causing the horizontal displacement of the box. Only the net force you apply by pushing the box across the table should be included in any calculations of work or change in kinetic energy.
Example of the Work-Energy Theorem
How to Calculate the Work Done by Friction
Suppose a car has a mass of 1{,}250\text{ kg} with an initial velocity of 20\text{ m/s}. While applying the brakes, the car slows to a final velocity of 10\text{ m/s}. How much work is done by friction as the brakes are applied?
We know that according to the work-energy theorem, the work done by friction will be equal to the change in the kinetic energy of the car. Begin by calculating the initial kinetic energy:
E_{ki}=\frac{1}{2}mv_{i}^2
E_{ki}=\frac{1}{2}(1{,}250\text{ kg})(20\text{ m/s})^2
E_{ki}=250{,}000\text{ J}
Then, find the final kinetic energy. The final kinetic energy, E_{kf}, is:
E_{kf}=\frac{1}{2}mv_{f}^2
E_{kf}=\frac{1}{2}(1{,}250\text{ kg})(10\text{ m/s})^2
E_{kf}=62{,}500\text{ J}
Now, the change in kinetic energy is the difference:
\Delta E_{k}=E_{kf}-E_{ki}=62{,}500\text{ J}-250{,}000\text{ J}
\Delta E_{k}=-187{,}500\text{ J}
Therefore, the work done by friction is -187{,}500\text{ J}. The work done by friction is negative because the force of friction opposes the car’s displacement and the car is losing kinetic energy.
Conclusion
The work-energy theorem allows us to combine our understanding of work and kinetic energy. When work is done on an object, the force exerted on the object causes a displacement. Since kinetic energy is the energy of motion, the force is also changing the object’s kinetic energy as it causes movement. Knowing that work is equal to the change in kinetic energy will allow you to analyze and solve many problems.
