Energy
Energy is one of the most fundamental aspects of physics. We often hear the term colloquially, but fail to understand what it truly represents. The concept of energy is extremely useful, especially for more advanced physics. For the purpose of classical mechanics, we mainly see energy as mainly split into kinetic energy and potential energy.
Work: Before we dive into energy, let's look at work. Work is the dot product of an applied force on an object and the object's displacement vector. Let's take this apart. This simply means that to find the value of work we need to multiply the force and displacement vectors. However, this only works when they are in line with each other. If there is an angle between the two vectors, we only want to multiply the part of the displacement vector which goes in the same direction as the applied force. This gives us the formula of F * x * cos(θ). Note that at 180 degrees, the cosine term gives -1, meaning when Force and displacement are in opposite directions, work done is negative. For example, as the box on the right slides down the ramp a length of d, the Work done by gravity will be represented by mg *d * cos(θ). It will also be helpful to note that there is a term called power. Power is the change of work with respect to time.
Kinetic Energy and the Work-Energy Theorem: This brings us on to kinetic energy. Kinetic energy describes the state of an object in motion. It's formula is given by 0.5 * m * v² where m is mass, and v is the object's velocity. The SI unit to represent energy is Joules. So if a 20 kg man moves at 5 m/s, we can say that the his energy is 0.5 * 20 * 5² or 250 Joules. There is a very intimate relationship between work and energy. The Work-Energy Theorem states that the net work done by the forces acting on an object is equal to the change of kinetic energy on an object. This can be very helpful in calculations when relating the change of velocity in an object to the Force applied and distance moved.
Potential Energy: The other type of energy we focus on in physics is called potential energy. This is used for types of forces that are called conservative forces. If we can move an object through some path and return it to its initial position, and the total work done is 0, the force is a conservative force. Some examples are gravity and spring force. We can calculate potential energy use formulas that are dependent on the situation. For the force of gravity near the earth, potential energy is defined by a function of height represented as U(y) = mgy. The potential energy for springs is represented with U(x) = 0.5 * k * x². The change of potential energy also creates work, similar to kinetic energy. Lastly, a term we define as mechanical energy is the sum of kinetic and potential energy in a system. As long as only conservative forces act, the mechanical energy will stay the same. This is the law of conservation of mechanical energy.
