Physics for engineer/Motion

Motion is cause by a force interacts with matter to make matter move from one place to another place . Strength to kick a ball to move from place A to place B

Chracteristics of motion

Velocity

Speed indicates the speed of motion to travel a distance over time

$v={\frac {s}{t}}$

Acceleration

Acceleration indicates the change of speed of motion

$a={\frac {v}{t}}$

Distance

Distance indicates the length of travel path of motion

$s=vt$

Force

A physcial quantity intaeracts with matter to perform a task

$F=ma$

Work

Work indicates the capability of a force to perform a task

$W=Fs=Fvt$

Energy

Energy indicates the capability of a force to perform a task over time

$E={\frac {W}{t}}={\frac {Fs}{t}}=Fv$

Formula

Tính Chất	Ký Hiệu	Công thức
Time	$t$	$t$
Acceleration	$a$	${\frac {v}{t}}$
Speed	$v$	$v$
Distance	$s$	$vt$
Force	$F$	$ma$
Work	$W$	$Fs=Fvt$
Energy	$E$	${\frac {W}{t}}=F{\frac {s}{t}}=Fv$

Types of Motion

Linear Motion

Linear Motion represents motion that follows straight line without changing its direction

Accelerationn of linear motion passes through 2 points $(t_{o},v_{o})(t,v)$

$a={\frac {v-v_{o}}{t-t_{o}}}={\frac {\Delta v}{\Delta t}}$

From above

$\Delta v=a\Delta t$

$\Delta t={\frac {v-v_{o}}{a}}$

Speed of linear motion

$v=v_{o}+a\Delta t$

$v_{o}=v-a\Delta t$

Distance of linear motion

$s=v_{o}\Delta t+{\frac {1}{2}}\Delta v\Delta t=\Delta t(v_{o}+{\frac {\Delta v}{2}})=\Delta t(v_{o}+{\frac {a\Delta t}{2}})=\Delta t(v-{\frac {a\Delta t}{2}})=({\frac {v-v_{o}}{a}})({\frac {2v_{o}+v-v_{o}}{2}})={\frac {v^{2}-v_{o}^{2}}{2a}}$

Fromm above

$v^{2}=v_{o}^{2}+2as$

Linear Motion with acceleration not equal zero

$a={\frac {v-v_{o}}{t-t_{o}}}={\frac {\Delta v}{\Delta t}}$

$v=v_{o}+a\Delta t$

$s=v_{o}\Delta t+{\frac {1}{2}}\Delta v\Delta t=\Delta t(v_{o}+{\frac {\Delta v}{2}})=\Delta t(v_{o}+{\frac {a\Delta t}{2}})=\Delta t(v-{\frac {a\Delta t}{2}})={\frac {v^{2}-v_{o}^{2}}{2a}}$

Chracteristics of Inclined Linear motion are listed below

Tính Chất	Ký Hiệu	Công thức
Time	$t$	$t$
Acceleration	$a$	${\frac {\Delta v}{\Delta t}}={\frac {v-v_{o}}{t-t_{o}}}$
Speed	$v$	$v_{o}+a\Delta t$
Distance	$s$	$(v_{o}+a\Delta t)\Delta t$
Force	$F$	$m{\frac {\Delta v}{\Delta t}}$
Work	$W$	$F(v_{o}+a\Delta t)\Delta t$
Energy	$E$	$F(v_{o}+a\Delta t)$

Linear Motion with acceleration equal zero

$a={\frac {\Delta v}{\Delta t}}={\frac {v-v_{o}}{t-t_{o}}}=0$

$v=v_{o}$

$s=v_{o}t$

Chracteristics of Horizontal Linear motion are listed below

Tính Chất	Ký Hiệu	Công thức
Time	$t$	$t$
Acceleration	$a$	${\frac {\Delta v}{\Delta t}}={\frac {v-v_{o}}{t-t_{o}}}$
Speed	$v$	$v_{o}+a\Delta t$
Distance	$s$	$(v_{o}+a\Delta t)\Delta t$
Force	$F$	$m{\frac {\Delta v}{\Delta t}}$
Work	$W$	$F(v_{o}+a\Delta t)\Delta t$
Energy	$E$	$F(v_{o}+a\Delta t)$

Linear Motion with acceleration equal constant

$a=g$

$v=gt$

$s=gt^{2}$

Chracteristics of Vertical Linear motion are listed below

Tính Chất	Ký Hiệu	Công thức
Time	$t$	$t$
Acceleration	$a$	${\frac {\Delta v}{\Delta t}}={\frac {v-v_{o}}{t-t_{o}}}$
Speed	$v$	$v_{o}+a\Delta t$
Distance	$s$	$(v_{o}+a\Delta t)\Delta t$
Force	$F$	$m{\frac {\Delta v}{\Delta t}}$
Work	$W$	$F(v_{o}+a\Delta t)\Delta t$
Energy	$E$	$F(v_{o}+a\Delta t)$

Curve Motion

Continous motion

Average acceleration for curve motion

$a={\frac {\Delta v(t)}{\Delta t}}={\frac {v(t+\Delta t)-v(t)}{(t+\Delta t)-t}}$

Average distance for curve motion

$s=\Delta t[v(t)+{\frac {\Delta v(t)}{2}}]$

As $\Delta t\rightarrow 0$

Acceleration for curve motion

$a(t)=\lim _{\Delta t\to 0}\sum {\frac {\Delta v(t)}{\Delta t}}={\frac {d}{dt}}v(t)=v^{'}(t)$

Distance for curve motion

$s(t)=\lim _{\Delta t\rightarrow 0}\sum [v(t)+{\frac {\Delta v(t)}{2}}]\Delta t=\int v(t)dt=V(t)+C$

Non-continous motion

Distance for curve motion from A to B

$s=\int \limits _{a}^{b}f(x)dx=F(b)-F(a)$

Characteristics of Curve motion are listed below

Tính Chất	Ký hiệu	Công thức
Distance	$s$	$\int v(t)dt$
Time	$t$	$t$
Speed	$v$	$v(t)$
Acceleration	$a$	${\frac {d}{dt}}v(t)$
Force	$F$	$m{\frac {d}{dt}}v(t)$
Work	$W$	$F\int v(t)dt$
Energy	$E$	${\frac {F}{t}}\int v(t)dt$

Example

Chuyển động		v	a	s
Chuyển động cong		$v(t)$	${\frac {d}{dt}}v(t)dt$	$\int v(t)dt$
Chuyển động thẳng ngang		$v$	$0$	$vt+c$
Chuyển động thẳng dọc		$t$	$1$	${\frac {t^{2}}{2}}+c$
Chuyển động thẳng nghiêng		$at$	$a$	${\frac {at^{2}}{2}}+c$
Chuyển động thẳng nghiêng		$at+v$	$a$	${\frac {at^{2}}{2}}+vt+c$

Circular Motion

Full circle

$s=2\pi$

$v={\frac {2\pi }{t}}=2\pi f=\omega$

$a={\frac {\omega }{t}}$

Arc of circle

$\alpha ={\frac {\omega -\omega _{o}}{t-t_{o}}}={\frac {\Delta \omega }{\Delta t}}$

$\omega =\omega _{o}+a\Delta t$

$\theta =\omega _{o}\Delta t+{\frac {1}{2}}\Delta \omega \Delta t=\Delta t(\omega _{o}+{\frac {\Delta \omega }{2}})=\Delta t(\omega _{o}+{\frac {\alpha \Delta t}{2}})=\Delta t(\omega -{\frac {\alpha \Delta t}{2}})={\frac {\omega ^{2}-\omega _{o}^{2}}{2\alpha }}$

$s=r\theta$

$v=r\omega$

$a=r\alpha$

Oscillation

Horizontal Oscillation

Oscillation of spring in the horizontal direction

$F_{a}=-F_{x}$

$mg=-kx$

$a=-{\frac {k}{m}}x=-\beta x={\frac {d^{2}}{dt^{2}}}x$

$x=A\sin(\omega t)$

$\omega ={\sqrt {\beta }}={\sqrt {\frac {k}{m}}}$

Vertical Oscillation

Oscillation of spring in the vertical direction

$-F_{g}=F_{y}$

$-mg=ky$

$g=-{\frac {k}{m}}y=-\beta y={\frac {d^{2}}{dt^{2}}}y$

$y=A\sin(\omega t)$

$\omega ={\sqrt {\beta }}={\sqrt {\frac {k}{m}}}$

Inclined Oscillation

Oscillation of pendulum in the titlted direction

$-F_{\theta }=F_{\theta }$

$-mg=l\theta$

$g=-{\frac {l}{m}}\theta =-\beta \theta ={\frac {d^{2}}{dt^{2}}}\theta$

$\theta =A\sin(\omega t)$

$\omega ={\sqrt {\beta }}={\sqrt {\frac {l}{m}}}$

Wave

From above, oscillation generates sin wave that process wave equation and wave function

Wave equation and wave function

Wave form

Wave equation

$f^{''}(t)=-\beta f(t)$

Wave function

$f(t)=Asin\omega t$

$\omega ={\sqrt {\beta }}$