Work and Energy

Work is the amount of energy transferred when a force is applied over a given distance. If the force is constant, work can be calculated using W = F d, where d is displacement. If the force is being applied at an angle, W = F d cos(θ), where θ is the angle between the force and the displacement. This is the work used in mechanics, but there is also work done by compressing a gas, as well as by the magnetic motion of particles. No work is done if there is no energy transfer, so therefore if I push a block against a wall and it doesn’t move or compress, it is not gaining or losing any energy, and therefore no work is being done.

Torque, τ, is basically rotational force. It measures the force that can cause an object to rotate about a fixed axis. Sometimes known as the moment of force, it can be calculated with τ = F r, where r is the radius from the force to the axis of rotation. Once again, if the force is given at an angle, torque can be measured with τ = F r sin (θ). One way you can imagine how torque changes is by pushing against a door at the end (high radius so high torque) and then pushing against that same door only a few inches from the hinge (low radius, low torque). The closer you are to the hinge, the less torque you will have, and the harder it will be to move the door.

Kinetic energy is the energy of movement. Simply because an object is moving, it has energy. If we were to accelerate an object, it would require a force. A force would make us do work, which in turn, would transfer energy into the object. This energy is called kinetic energy. Kinetic energy depends on mass and velocity, represented by the equation KE = 1/2 m v². That means when the velocity is doubled, the kinetic energy of the object quadruples.

Gravitational potential energy is the energy of height. When something is high above the zero line (any set height that we chose to be 0), it has potential energy which basically states that it has the ability to transfer that energy, which is why it is called potential energy. Represented by the equation Ug (gravitational potential energy) = m g h, with h as height, potential energy can easily be transferred into kinetic energy by falling, changing its height into velocity.

When energy is conserved in a collision, it means that no energy is lost or gained; the energy in equals the energy out. In elastic collisions, both energy and momentum are conserved. These do not exist in real life, because energy is lost due to friction, air resistance, and heat loss from sound or contortion. However, elastic collisions are very easy to work with mathematically so they are used often in physics. If a ball falls from a height, h, and makes an elastic collision with the floor, and bounces back up, it will reach its peak height in the exact same spot that it came from.

There are also inelastic collisions in which some energy and momentum are lost, but not all. These are what we see in real life, where the ball bounces lower than it originally came from. Furthermore, there is what is called a perfectly inelastic collision where all energy and momentum are lost. Think of a ball landed in a bucket of honey, and it just stopped and stuck to the ground.

Spring potential energy, sometimes called elastic energy, is the potential energy resulting in the deformation of objects, typically found in springs being stretched or compressed. Spring potential energy is found using U = 1/2 k Δx², where k is the spring constant and Δx is the change in position.

When an object is rolling or spinning, it is represented with rotational kinetic energy, which just uses ω instead of v. Elastic energy, gravitational potential energy, kinetic energy, and rotational kinetic energy are all types of mechanical energy, which is what is conserved in elastic collisions.