# Work (12) (ELCA)

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• Work — the application of a force that results in the movement of an object in the direction (or opposite direction) of the force.

What is required for work?

• Movement
1. Must be parallel movement to the force exerted.
2. When the force applied is the opposite to the direction of motion, the force is negative.

## Important Background InformationEdit

### VectorsEdit

$a = \frac{v_f - v_i}{\Delta t}$

• $a$ – acceleration
• $v_f$ – Final Velocity
• $v_i$ – Initial Velocity
• $\Delta t$ – Change in Time (s)

$x = \frac{1}{2} (v_i + v_f) \Delta t$

• $x$ – distance (m)
• $v_i$ – Initial Velocity
• $v_f$ – Final Velocity
• $\Delta t$ – Change in Time (s)

### ForceEdit

$F = ma$

• $F$ – Force (N)
• $m$ – Mass (kg)
• $a$ – acceleration $\left ( \frac{m}{s^2} \right )$

$F_g = mg\sin \Theta$

• $F_g$ - Force of gravity
• $m$ - mass (kg)
• $g$ - Gravity $\left ( \frac{m}{s^2} \right )$
• $\Theta$ - Angle of Object / Slope (º)

$F_\text{net} = F_g - F_f$

• $F_\text{net}$ - Net Force (N)
• $F_g$ - Force of Gravity $\left ( \frac{m}{s^2} \right )$
• $F_f$ - Force of Friction (N)

$F_\text{net} = mg\sin \Theta - F_f$

• $F_\text{net}$ - Net Force (N)
• $m$ - Mass (kg)
• $g$ - Gravity $\left ( \frac{m}{s^2} \right )$
• $\Theta$ - Angle (º)
• $F_f$ - Force of Friction (N)

## WorkEdit

Work (W) = Energy Applied (E)

$W = Fd$

• $W$ = Work (J)
• $F$ = Force (N)
• $d$ = Distance (m)

Work and Motion Relation
Direction of Force Change
Force is in the direction of motion Positive
Force opposes motion Negative
Force is 90º to motion No Work
Object does not move No Work

## Kinetic EnergyEdit

### Solve for Kinetic EnergyEdit

$\frac{1}{2}mv^2=E_K$

• $m$ = mass (kg)
• $v$ = velocity $\left ( \frac{m}{s} \right )$
• $E_K$ = Kinetic Energy (J)

${{\sqrt\frac{2E_K}{m}}} = v$

• $m$ = mass (kg)
• $v$ = velocity $\left ( \frac{m}{s} \right )$
• $E_K$ = Kinetic Energy (J)

$\frac{2E_K}{v^2} = m$

• $m$ = mass (kg)
• $v$ = velocity $\left ( \frac{m}{s} \right )$
• $E_K$ = Kinetic Energy (J)

### Net WorkEdit

$W_\text{net} = \Delta E_K = E_\text{Kf} - E_\text{Ki}$

• $W_\text{net}$ = Net Work (J)
• $\Delta E_K$ = change in Kinetic Energy (J)
• $E_\text{Kf}$ = Final Kinetic Energy (J)
• $E_\text{Ki}$ = Initial Kinetic Energy (J)

$W_\text{net}=\frac{1}{2}mv_f^2-\frac{1}{2}mv_i^2$

• $W_\text{net}$ = Net Work (J)
• $m$ = Mass (kg)
• $v_f$ = Final Velocity $\left ( \frac{m}{s} \right )$
• $v_i$ = Initial Velocity $\left ( \frac{m}{s} \right )$

## Potential EnergyEdit

• Potential Energy ($U$) – stored energy that can be used at some later time.

### Gravitational Potential EnergyEdit

• Gravitational Potential Energy ($U_\text{g}$) – potential energy due to position above the Earth's surface.

$U_\text{g} = mgh$

• $U_\text{g}$ = Potential Gravitational Energy (J)
• $m$ = mass (kg)
• $g$ = gravity $\left ( \frac{m}{s} \right )$
• $h$ = height (m)

$\frac{U_\text{g}}{mg} = h$

• $U_\text{g}$ = Potential Gravitational Energy (J)
• $m$ = mass (kg)
• $g$ = gravity $\left ( \frac{m}{s} \right )$
• $h$ = height (m)

$\frac{U_\text{g}}{gh} = m$

• $U$ = Potential Gravitational Energy (J)
• $m$ = mass (kg)
• $g$ = gravity $\left ( \frac{m}{s} \right )$
• $h$ = height (m)

### Elastic Potential EnergyEdit

• Elastic Energy($U_\text{e}$) – potential energy due to elasticity.

$U_\text{e} = \frac{1}{2}Kx^2$

• $U_\text{e}$ = Elastic Energy (J)
• $K$ = Spring Constant $\left ( \frac{N}{m} \right )$
• $x$ = Stretched Length (m)

${{\sqrt\frac{2U_\text{e}}{K}}} = x$

• $U_\text{e}$ = Elastic Energy (J)
• $K$ = Spring Constant $\left ( \frac{N}{m} \right )$
• $x$ = Stretched Length (m)

$\frac{2U_\text{e}}{x^2} = K$

• $U_\text{e}$ = Elastic Energy (J)
• $K$ = Spring Constant $\left ( \frac{N}{m} \right )$
• $x$ = Stretched Length (m)

## Mechanical EnergyEdit

• Mechanical Energy - The sum of all the potential and kinetic energies in a system.

$E_\text{mi} = E_\text{mf}$

• $E_\text{mi}$ - Initial Mechanical Energy
• $E_\text{mf}$ - Finial Mechanical Energy

$U_\text{gi} + U_\text{ei} + E_\text{Ki} = U_\text{gf} + U_\text{ef} + E_\text{Kf}$

• $U_\text{gi}$ - Initial Potential Gravitational Energy
• $U_\text{ei}$ - Initial Potential Elastic Energy
• $E_\text{Ki}$ - Initial Kinetic Energy
• $U_\text{gf}$ - Final Potential Gravitational Energy
• $U_\text{ef}$ - Finial Potential Elastic Energy
• $U_\text{Kf}$ - Final Kinetic Energy

## PowerEdit

• Power – the rate at which work is performed or energy is transmitted, or the amount of energy required or expended for a given unit of time.

$P = \frac{W}{t}\,$

• $P$ – Power (W)
• $W$ – Work (J)
• $t$ – Time (s)

$P = Fv$

• $P$ – Power (W)
• $F$ – Force (N)
• $v$ – Velocity $\left ( \frac{m}{s} \right )$
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