PHYS 218 Chapter 6

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Work

Work of a force over an object moving from 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 r_1} to 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 r_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=\int_{r_1}^{r_2} F \cdot\ dl}

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

If 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 F} is constant (in magnitude, direction, and time), then 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=F \cdot (r_2-r_1)}

Units

Work is measured in Joules [J] (J = 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 \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, 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=F\cdot \Delta r = 1000\cos{60^\circ} \cdot 100 = 50000\ \mbox{J}}

Work by Net Forces

Given forces 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 F_1} , 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 F_2} , and 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 F_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 \begin{align} 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{align}}

Net work is related to change in velocities.

Kinetic Energy

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

Work Energy Theorem

Work of all external forces (J) = change in kinetic energy (J)
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=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. 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(t_T)-K(t_0) &= 0-\tfrac{1}{2} \times 2 \times 100 &= -100 \end{align}}

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)