7.1 - Kinetic Energy

The kinetic energy $K$ associated with the motion of a particle of mass $m$ and speed $v$, where $v$ is well below the speed of light, is:

The SI unit of kinetic energy is the joule (J), defined as:

7.2 - Work and Kinetic Energy

Work $W$ is energy transferred to or from an object via a force acting on the object. Energy transferred to the object is positive work, and from the object, negative work.

When two or more forces act on an object, their net work is the sum of the individual works done by the forces, which is also equal to the work that would be done on the object by the net force $\vec F_{net}$ of those forces.

For a particle, a change $\Delta K$ in the kinetic energy equals the net work $W$ done on the particle:

In which $K_i$ is the initial kinetic energy of the particle and $K_f$ is the kinetic energy after the work is done. This is known as the work-kinetic energy theorem. This formula can also be rearranged as the following:

The work done on a particle by a constant force $\vec F$ during displacement $\vec d$ is:

in which $\phi$ is the constant angle between the directions of $\vec F$ and $\vec d$.

Only the component of $\vec F$ that is along the displacement $\vec d$ can do work on the object.

7.3 - Work Done by the Gravitational Force

The work $W_g$ done by the gravitational force $F_g$ on a particle-like object of mass $m$ as the object moves through a displacement $\vec d$ is given by:

in which $\phi$ is the angle between $\vec F_g$ and $\vec d$.

The work $W_a$ done by an applied force as a particle-like object is either lifted or lowered is related to the work $W_g$ done by the gravitational force and the change $\Delta K$ in the object's kinetic energy by:

If $K_f = K_i$, that is, the object's kinetic energy did not change, then the equation reduces to:

or in other words, the applied force transfers as much energy to the object as the gravitational force transfers from it.

7.4 - Work Done by a Spring Force

The force $\vec F_s$ from a spring is:

where $\vec d$ is the displacement of the spring's free end from its position when the spring is in its relaxed state (not compressed or extended), and $k$ is the spring constant (a measure of the spring's stiffness). If an $x$ axis lies along the spring, with the origin at the location of the spring's free end when the spring is in its relaxed state, we can write:

A spring force is thus a variable force: It varies with the displacement of the spring's free end. (Yay calculus!)

If an object is attached to the spring's free end, the work $W_s$ done on the object by the spring force when the object is moved from an initial position $x_i$ to a final position $x_f$ is:

If $x_i = 0$ and $x_f = x$, the equation could be written as:

Written as an integral makes this easier to work with:

7.5 - Work Done by a General Variable Force

When the force $\vec F$ on a particle-like object depends on the position of the object, the work done by $\vec F$ on the object while the object moves from an initial position $r_i$ with coordinates ($x_i$, $y_i$, $z_i$) to a final position $r_f$ with coordinates ($x_f$, $y_f$, $z_f$) must be found by integrating the force.

If we assume that components only depend on their respective coordinates, then the work is:

If $\vec F$ only has an $x$ component, then this reduces to:

7.6 - Power

The power due to a force is the rate at which that force does work on an object. If the force does work $W$ during a time interval $\Delta t$, the average power due to the force over that time interval is:

Instantaneous power is the instantaneous rate of doing work:

In other words, Power is the derivative of Work.

For a force $\vec F$ at an angle $\phi$ to the direction of travel of the instantaneous velocity $\vec v$, the instantaneous power is: