PHYS 218 Chapter 6

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Work

Work of a force over an object moving from $r_{1}$ to $r_{2}$

$W=\int _{r_{1}}^{r_{2}}F\cdot \ dl$

• $F$ is force
• $\cdot$ is the dot product
• $dl$ is the displacement vector</math>

If $F$ is constant (in magnitude, direction, and time), then $W=F\cdot (r_{2}-r_{1})$

Units

Work is measured in Joules [J] (J = N × m = ${\tfrac {kg\times m^{2}}{s^{2}}}$ )

Example

A train moves 100 meters along a straight track with a force of 1000 N applied at an angle of 60°

Since force is constant, $W=F\cdot \Delta r=1000\cos {60^{\circ }}\cdot 100=50000\ {\mbox{J}}$

Work by Net Forces

Given forces $F_{1}$, $F_{2}$, and $F_{3}$,

${\begin{aligned}W&=\int _{r_{1}}^{r_{2}}\sum {F}\cdot dl&({\mbox{where}}\ dl\ {\mbox{is change in displacement}})\\&=\int _{r_{1}}^{r_{2}}ma\cdot dl\\&=\int _{r_{1}}^{r_{2}}m{\frac {dv}{dt}}\cdot dl\\&=\int _{r_{1}}^{r_{2}}m\,dv\cdot {\frac {dl}{dt}}\\&=\int _{r_{1}}^{r_{2}}mv\ dv\\&={\frac {m{v_{2}}^{2}}{2}}-{\frac {m{v_{1}}^{2}}{2}}\end{aligned}}$

Net work is related to change in velocities.

Kinetic Energy

$K={\frac {mv^{2}}{2}}$ (J)

Work Energy Theorem

Work of all external forces (J) = change in kinetic energy (J)
$W=K_{2}-K_{1}\,\!$ (J)

Example

Object:

• mass = 2 kg
• initial velocity = 10 m/s
• presence of gravity

What is work of gravitational force

1. object from bottom to top
2. object from top to bottom
3. object during complete journey

1. ${\begin{aligned}W&=K(t_{T})-K(t_{0})&=0-{\tfrac {1}{2}}\times 2\times 100&=-100\end{aligned}}$

2. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W=100 - 0}

3. Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W=100-100 = 0}

Example 2

Object:

• mass = 10 kg
• initial velocity = 2 m/s
• Force applied = 110 N

Find velocity when displacement is at 20 m.

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} W &= K_2-K_1 \\ &= \frac{m{v_2}^2}{2}-\frac{m{v_1}^2}{2} \\ &= 20 \\ 2200 &= \frac{m{v_2}^2}{2}-20\\ &= \ldots \end{align}}

Springs

An uncompressed spring's length is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle l_0} or natural length

In small increments, the force that the spring exerts on the block is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -k(x-l_0)} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k} is the spring constant [N/m]

• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x>l_0 \rightarrow F<0}
• Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x<l_0 \rightarrow F>0}

Example

An object of mass Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is pushing against a spring. When let go, the spring pushes the object to a known velocity and returns to its natural length

Find the work done by the spring

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} W &= \int_{r_1=l_0-5}^{r_2=l_0} -k(x-l_0)\ dx \\ &= -k\int_{l_0-5}^{l_0} x-l_0 \ dx \\ &= -k\left(\frac{{l_0}^2}{2}-\frac{(l_0-5)^2}{2}-l_0(l_0-(l_0-5))\right) \\ &= \ldots \frac{k\ \Delta x^2}{2} = \frac{25k}{2} & \mbox{energy stored in spring when compressed} \end{align}}

Power

Rate of change of Work over time measured in Joules per second or watts [ J/s = W ]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align}P &= \frac{dW}{dt} \\ &= F \cdot (v_2-v_1) & \mbox{if force is constant}\end{align}}

For example, a 100 Watt light bulb is converting 100 J of work into light each second.

Note: Energy through power: kilowatt hour (energy in units of joules)