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Unit 4

4.2 Work and Mechanical Energy

3 min readβ€’june 8, 2020

peter57616

Peter Apps


AP Physics 1 🎑

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Enduring Understanding 3.E πŸ‘¨β€πŸ’»

A force exerted on an object can change the kinetic energy of the object.

Enduring Understanding 4.C πŸ‘¨β€πŸ’»

Interactions with other objects or systems can change the total energy of a system.

Essential Knowledge 3.E.1 πŸ’‘

The change in the kinetic energy of an object depends on the force exerted on the object and on the displacement of the object during the interval that the force is exerted.

Essential Knowledge 4.C.2 πŸ’‘

Mechanical energy (the sum of kinetic and potential energy) is transferred into or out of a system when an external force is exerted on a system such that a component of the forces is parallel to its displacement. The process through which the energy is transferred is called work.

Kinetic Energy

Kinetic Energy (K) is energy an object has due to its motion. It can be calculated using the equation:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%201.34.19%20PM.png?alt=media&token=98d4da7d-a498-4962-9350-1cecadf8c4b3
Where m is the mass of the object and v is the velocity. Notice that since the velocity term is squared, the kinetic energy of an object must always be positive. When work is done on an object to increase its speed, that work increases the object’s kinetic energy. To see how, let’s do a simple derivation.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%201.35.19%20PM.png?alt=media&token=48da3448-1686-4f82-95bc-5c117090340c
We’re going to assume an object of mass m is being pushed by a net force (F) over some displacement (d). Using Newton’s 2nd Law and some kinematics:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%201.36.20%20PM.png?alt=media&token=56c28c6c-f283-4a3b-89ae-288163b13286
we can substitute values in for the net force and acceleration.

Gravitational Potential Energy

Instead of increasing the velocity of the object, we might imagine work being done to change the position of the object. If the force is directed upwards, the object might gain height relative to the Earth (or some other reference point). In this case, the work being done is defined as changing the gravitational potential energy (Ug) of the object.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%201.37.48%20PM.png?alt=media&token=8f35d248-1fd2-4d6e-a26d-e63c41e939ec
The equation above is the one most often used to calculate potential energy while the object is on or near the Earth. In fact, in order to have gravitational potential energy relative to the Earth, the Earth must be part of the system chosen.
However, it’s also possible to describe gravitational potential energy relative to non-Earth objects (or relative to Earth for a satellite or another planet). In this case, we define 0 potential energy to be a point that is an infinite distance away from the Earth. Therefore the closer to the Earth (or other planet) we get the more negative the potential energy becomes.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%201.38.37%20PM.png?alt=media&token=077e3647-e240-4efb-a406-5fe0429d8a6e

Elastic Potential Energy

A third type of energy that can be changed as a result of work is elastic potential energy. This is energy that is stored in a spring, rubber band, or other stretchable material. (For AP 1, this material is ideal, and obeys Hooke’s Law). Elastic Potential Energy for an ideal spring is described by the equation:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%201.39.46%20PM.png?alt=media&token=f2b7a577-05fc-43cf-a9d6-d1207484653c
Where k is the spring constant, and x is the displacement of the spring from its equilibrium position. The derivation of this equation is similar to the gravitational potential energy derivation, except the force is caused by Hooke’s Law and is dependent on x.
You may remember using a plot of Fs vs x to determine the spring constant for a spring. We can use the same plot to derive the energy stored in the spring too! Remember that Work is the area under the curve.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%201.42.01%20PM.png?alt=media&token=f01a05f3-16ba-4654-8d8f-0e1985099f92
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2FScreen%20Shot%202020-04-13%20at%201.42.36%20PM.png?alt=media&token=73e4e4ff-f8e9-4be0-9406-7c7c66b9a9b4

Image courtesy of Khan Academy.

Thermal Energy

This is a catch-all for any energy that isn’t potential or kinetic. We can talk about thermal energy as energy lost in the form of heat or sound. This is typically due to friction being present, but could also be due to heat from a collision. In AP 1, we don’t have a formal equation to define this, but AP 2 will go into more detail defining this.
πŸŽ₯Watch: AP Physics 1 - Unit 4 Streams

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