Mechanical energy stores

In a nutshell

Energy can be described as being in different stores. Energy cannot be created or destroyed, but it can be transferred, dissipated or stored in different ways. Gravitational potential energy is the energy stored when moving an object to height. Kinetic energy is a form of energy that arises due to the movement of an object. Elastic potential energy is the energy stored as a result of applying a force to deform an elastic object.

Equations

Word equation	Symbol equation
$gravitational \space potential \space energy = mass \times gravitational \space field \space strength \times height$	$E_p = m \times g \times h$
$kinetic \space energy = \frac{1}{2} \times mass \times velocity^2$	$E_k = \frac{1}{2}\times m \times v^2$
$elastic \space potential \space energy = \frac{1}{2} \times spring \space constant \times extension^2$	$E_e = \frac{1}{2}\times k \times e^2$

Variable definitions

Quantity Name	Symbol	Unit Name	Unit
$gravitational \space potential \space energy$	$E_p$	$joule$	$J$
$kinetic \space energy$	$E_k$	$joule$	$J$
$elastic \space potential \space energy$	$E_e$	$joule$	$J$
$mass$	$m$	$kilogram$	$kg$
$gravitational \space field \space \space strength$	$g$	$newton \space per \space kilogram$	$N/kg$
$velocity$	$v$	$metre \space per \space second$	$m/s$
$height$	$h$	$metre$	$m$
$spring \space constant$	$k$	$newton \space per \space metre$	$N/m$
$extension$	$e$	$metre$	$m$

Gravitational potential energy

When an object is at any height above the surface of the Earth, it will possess energy in its gravitational potential energy store. Gravitational potential energy increases with increasing height above the Earth's surface. The following equation is used to calculate gravitational potential energy:

$gravitational \space potential \space energy = mass \times gravitational \space field \space strength \space \times height$

$E_p = m \times g \times h$

Example

Calculate the energy in the gravitational potential energy store when a $300\ g$ apple is lifted $2.1\ m$ above the ground.

First, write out the known quantities and make sure they are in the correct form:

$m = 300g = \frac{300}{1000}= 0.3 \space kg \\ \ \\ g = 10 \space N/kg \\ \ \\ h = 2.1 \space m$ 

Next, write down the equation you need to use:

$E_p = m \times g \times h$

Then, substitute the values into the equation:

$E_p = 0.3 \times 10 \times 2.1$ 

The apple has $\underline{6.3\ J}$ of energy in its gravitational potential energy store after being lifted.

Kinetic energy

When an object is moving, it will have energy in its kinetic energy store. Kinetic energy is a form of energy that arises due to the movement of an object. It is dependent on both the speed and mass of the object. The following equation is used to calculate kinetic energy:

$kinetic \space energy = \frac{1}{2} \times mass \times velocity^2 \\ \ \\ E_k = \frac{1}{2} \times m \times v^2$

Example

Calculate the energy in the kinetic energy store of an $85\ kg$ rugby player running across a playing field at a constant speed of $5\ m/s$ .

First, write out the known quantities and make sure they are in the correct form:

$m = 85 \space kg \\ \ \\ v = 5 \space m/s$ 

Next, write down the equation you need to use:

$E_k = \frac{1}{2} \times m \times v^2$

Then, substitute the values into the equation:

$E_k = \frac{1}{2} \times 85 \times (5)^2$

Don't forget to include your units:

$1062.5 \space J$ 

The rugby player has $\underline{1062.5\ J}$ of energy in their kinetic energy store whilst running across the playing field.

Elastic potential energy

Elastic potential energy is the energy stored as a result of applying a force to deform an elastic object, such as a spring. This energy is stored until the spring is released.

Note: A deformation could involve compressing, stretching or twisting.

The following equation is used to calculate elastic potential energy:

$elastic \space potential \space energy = \frac{1}{2} \times spring \space constant \times extension^2 \newline$

$E_e = \frac{1}{2}\times k \times e^2$

Example

A mass is attached to the bottom of a hanging spring which has a spring constant of $250\ N/m$ . It stretches from $9.0\ cm$ to $12.5\ cm$ . Calculate the elastic potential energy stored by the stretched spring.

First, write out the known quantities and make sure they are in the correct form:

$initial \space length = 9.0 \space cm = 0.09 \space m \\ \ \\ final \space length = 12.5 \space cm = 0.125 \space m \\ \ \\ spring \space constant = 250 \space N/m$ 

Next, find the overall extension of the spring. This can be done by working out the difference between the initial and final length of the spring:

$extension = final \space length - initial \space length$

$extension = 0.125 - 0.09 = 0.035 \space m$

Next, write down the equation you need to use:

$E_e = \frac{1}{2} \times k \times e^2$

Then, substitute the values into the equation:

$E_e = \frac{1}{2}\times 250 \times (0.035)^2$

Don't forget to include your units:

$0.15 \space J$

The stretched spring has $\underline{0.15\ J}$  of energy in its elastic potential energy store.

Conservation of energy

The conservation of energy principle states that in a closed system, the total energy in that system must remain constant. When energy is converted from one form to another, the total energy before the change must be equal to the total of all energies after the change.

It is therefore possible to make calculations when energy is transferred between two stores. For example, the following relation can be stated for a falling object (ignoring air resistance):

$energy \space lost \space from \space the \space gravitational \space potential \space energy \space store = energy \space gained \space in \space the \space kinetic \space energy \space store$