### Key equations (Biomechanics)

#### Key equations

Biomechanics, meaning “life mechanics”, makes use of several key equations which are listed below. For examples see the dynamics section of the Mechanics page.

##### Force

A force ($F \ \mathrm{N}$) is a push or pull acting on a body. The action of a force causes a body's mass ($m \ \mathrm{kg}$) to accelerate with acceleration $a \ \mathrm{ms^{-2}}$, so we have $$F = ma.$$

##### Weight

Weight ($W \ \mathrm{N}$) is the amount of gravitational force exerted on a body. The acceleration due to gravity is defined as $g = 9.81 \mathrm{ms^{-2}}$. We have that $$W = mg,$$ where $m$ is the body's mass in $\mathrm{kg}$.

##### Example
###### Weight

If an individual has a mass of $73 \mathrm{kg}$ what is the individual's weight?

###### Solution

We have that $m = 73 \mathrm{kg}$. We can use the formulas \begin{align} W & = mg, \\ 1 \mathrm{kg} & = 2.2 \mathrm{lb}, \end{align} in order to find the weight of the individual. We have that $g = 9.81 \mathrm{ms^{-2} }$, therefore \begin{align} W & = mg, \\ & = 73 \mathrm{kg} \times 9.81 \mathrm{ms^{-2} }, \\ & = 716.13 \mathrm{N}. \end{align} We can also multiply by the conversion factor $2.2 \mathrm{lb/kg}$ to convert to weight in pounds $$73 \mathrm{kg} \times 2.2 \mathrm{lb/kg} = 160.6 \mathrm{lb}.$$

##### Pressure

Pressure ($P \ \mathrm{N/cm^{-2} }$) is defined as force ($F \ \mathrm{N}$) over a given area ($A \ \mathrm{cm^{-2} }$), therefore $$P = \frac{F}{A}.$$

##### Density

Density combines the mass of a body ($m \mathrm{kg}$) with the body volume (in $\mathrm{m^{-3} }$), the symbol for density is $\rho$ and we have that $$\rho = \frac{m}{\text{volume} }.$$

##### Torque

The rotary effect created by a force ($F \ \mathrm{N}$) is known as the torque ($T \ \mathrm{Nm}$). Torque is the product of the force and the perpendicular distance ($d \ \mathrm{m}$) from the forces line of action to the axis of rotation, $$T = Fd \ \mathrm{N m}.$$ This is also known as the moment of force.

##### Impulse

The product of the force ($F \ \mathrm{N}$) and the amount of time is it applied ($t \ \mathrm{s}$) is known as the impulse ($J \ \mathrm{N s}$), $$J = Ft \ \mathrm{Ns}.$$

#### The SUVAT equations

The equations of motion, also known as SUVAT equations, are used when acceleration, $a$, is constant. They are known as SUVAT equations because they contain the following variables: $s$ - distance, $u$ - initial velocity, $v$ - velocity at time $t$, $a$ - acceleration and $t$ - time. However, each SUVAT equation does not contain all variables so for answering some questions it might be necessary to use one or more of them. The equations are as follows:

\begin{align*} v &= u + at,\\ s & = \left(\frac{u+v}{2}\right)t, \\ v^2 &= u^2 + 2as,\\ s &= ut + \frac{1}{2}at^2,\\ s & = vt - \frac{1}{2}at^2. \end{align*}

The quantities $s$, $u$, $v$ and $a$ are all vector quantities so therefore their sign represents the direction of motion.

You need to ensure that the measurements are in base SI units before substituting them into the formulae. These are:

\begin{align*} \text{time } t &= \text{seconds } \mathrm{s},\\ \text{displacement } s &= \text{metres } \mathrm{m}, \\ \text{velocity } v \text{ or } u & = \text{metres per second } \mathrm{ms^{-1} },\\ \text{acceleration }a &=\text{metres per second per second } \mathrm{ms^{-2} }. \end{align*}

For worked examples involving the SUVAT equations see the Mechanics page, specifically equations of motion.

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