## Gravitational Potential Energy

In classical mechanics, the **gravitational potential energy** (U) is energy an object possesses because of its position in a **gravitational field**.** Gravitational potential** (V; the gravitational energy per unit mass) at a location is equal to the work (energy transferred) **per unit mass** that would be needed to move the object from a fixed reference location to the location of the object. The most common use of gravitational potential energy is for an object near the surface of the Earth where the gravitational acceleration can be assumed to be constant at about **9.8 m/s ^{2}**.

**U = mgh**

## Conservation of Mechanical Energy

First the principle of the **Conservation of Mechanical Energy** was stated:

**The total mechanical energy** (defined as the sum of its potential and kinetic energies) of a particle being acted on by only conservative forces** is constant**.

See also: Conservation of Mechanical Energy

**An isolated system** is one in which **no external force** causes energy changes. If only **conservative forces** act on an object and **U** is the** potential energy** function for the total conservative force, then

**E**_{mech}** = U + K**

**The potential energy, U**, depends on the position of an object subjected to a conservative force.

It is defined as the object’s ability to do work and is increased as the object is moved in the opposite direction of the direction of the force.

**The potential energy** associated with a system consisting of Earth and a nearby particle is **gravitational potential energy**.

**The kinetic energy, K**, depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them.

** K = ½ mv ^{2}**

The above mentioned definition (**E**_{mech}** = U + K**) assumes that the system is **free of friction** and other **non-conservative forces**. The difference between a conservative and a non-conservative force is that when a conservative force moves an object from one point to another, the work done by the conservative force is independent of the path.

In any real situation, **frictional forces** and other non-conservative forces are present, but in many cases their effects on the system are so small that the principle of **conservation of mechanical energy** can be used as a fair approximation. For example the frictional force is a non-conservative force, because it acts to reduce the mechanical energy in a system.

Note that non-conservative forces do not always reduce the mechanical energy. A non-conservative force changes the mechanical energy, there are forces that increase the total mechanical energy, like the force provided by a motor or engine, is also a non-conservative force.

## Block sliding down a frictionless incline slope

The 1 kg block starts out a height H (let say 1 m) above the ground, with **potential energy** **mgH** and** kinetic energy** that is equal to 0. It slides to the ground (without friction) and arrives with no potential energy and kinetic energy **K = ½ mv ^{2}**. Calculate the velocity of the block on the ground and its kinetic energy.

*E*

_{mech}

*= U + K = const**=> ½ mv*^{2}* = mgH*

*=> v = √2gH = 4.43 m/s*

*=> K*_{2}* = ½ x 1 kg x (4.43 m/s)*^{2}* = 19.62 kg.m*^{2}*.s*^{-2}* = 19.62 J*

## Pendulum

Assume a **pendulum** (ball of mass m suspended on a string of length **L** that we have pulled up so that the ball is a height **H < L** above its lowest point on the arc of its stretched string motion. The pendulum is subjected to the **conservative gravitational force** where frictional forces like air drag and friction at the pivot are negligible.

We release it from rest. **How fast is it going at the bottom?**

The pendulum reaches **greatest kinetic energy** and **least potential energy** when in the **vertical position**, because it will have the greatest speed and be nearest the Earth at this point. On the other hand, it will have its **least kinetic energy** and **greatest potential energy** at the **extreme positions** of its swing, because it has zero speed and is farthest from Earth at these points.

If the amplitude is limited to small swings, the period *T* of a simple pendulum, the time taken for a complete cycle, is:

where ** L** is the length of the pendulum and

**is the local acceleration of gravity. For small swings the period of swing is approximately the same for different size swings. That is,**

*g***the period is independent of amplitude**.