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 $(t_o,v_o)(t,v)$

$a=\frac{v-v_o}{t-t_o}=\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}$

From above

$\mathrm{\Delta }v=a\mathrm{\Delta }t$

$\mathrm{\Delta }t=\frac{v-v_o}{a}$

Speed of linear motion

$v=v_o+a\mathrm{\Delta }t$

$v_o=v-a\mathrm{\Delta }t$

Distance of linear motion

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

Fromm above

$v^2=v_o^2+2as$

Linear Motion with acceleration not equal zero

$a=\frac{v-v_o}{t-t_o}=\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}$

$v=v_o+a\mathrm{\Delta }t$

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

$a=\frac{v-v_o}{t-t_o}=\frac{v-v_o}{t}$

$v=v_o+at$

$s=t(v_o+\frac{at}{2})$

$a=\frac{v-0}{t-0}=\frac{v}{t}$

$v=at$

$s=\frac{1}{2}vt=\frac{1}{2}a{t}^2$

Linear Motion with acceleration equal zero

$a=\frac{\mathrm{\Delta }v}{\mathrm{\Delta }t}=\frac{v-v_o}{t-t_o}=0$

$v=v_o$

$s=v_{o}t$

Linear Motion with acceleration equal constant

$a=g$

$v=gt$

$s=g{t}^2$

Curve Motion

Continous motion

Average acceleration for curve motion

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

Average distance for curve motion

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

As $\mathrm{\Delta }t\to 0$

Instantanous acceleration for curve motion

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

Instantanous distance for curve motion

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

Bounded continous motion

Distance for curve motion from A to B

$s_{ab}=\underset{a}{\overset{b}{\int }}f(x)dx=F(b)-F(a)$

$s_{ab}=-\underset{b}{\overset{a}{\int }}f(x)dx=F(a)-F(b)$

$s_{avg}=\frac{1}{b-a}\underset{a}{\overset{b}{\int }}f(x)dx$

$s_{rms}=\sqrt{s_{avg}}$

Circular Motion

Full circle

$s=2\pi$

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

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

Arc of circle

$\alpha =\frac{\omega -{\omega }_o}{t-t_o}=\frac{\mathrm{\Delta }\omega }{\mathrm{\Delta }t}$

$\omega ={\omega }_o+a\mathrm{\Delta }t$

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

$s=r\theta$

$v=r\omega$

$a=r\alpha$

Oscillation

Horizontal Oscillation

Oscillation of spring in the horizontal direction

$F_a=-F_x$

$mg=-kx$

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

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

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

Vertical Oscillation

Oscillation of spring in the vertical direction

$-F_g=F_y$

$-mg=ky$

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

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

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

Inclined Oscillation

Oscillation of pendulum in the titlted direction

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

$-mg=l\theta$

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

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

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

Wave

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

Wave equation and wave function

Wave form

Wave equation

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

Wave function

$f(t)=Asin\omega t$

$\omega =\sqrt{\beta }$

Formula of motions

Moment

v < C

Tính Chất	Ký Hiệu	Công thức
Time	$t$	$t$
Speed	$v$	$v$
Acceleration	$a$	$\frac{v}{t}$
Distance	$s$	$vt$
Force	$F$	$ma=m\frac{v}{t}=\frac{p}{t}$
Work	$W$	$Fs=F\frac{p}{t}s=pv$
Energy	$E$	$\frac{W}{t}=\frac{pv}{t}=pa$
Moment	$p$	$mv=Ft$

v = C

Tính Chất	Ký Hiệu	Công thức
Time	$t$	$t$
Speed	$v$	$C=\lambda f$
Acceleration	$a$	$\frac{C}{t}$
Distance	$s$	$Ct$
Force	$F$	$ma=\frac{p}{t}=\frac{\frac{h}{\lambda }}{t}=\frac{hf}{\lambda }$
Work	$W$	$Fs=pC=hf$
Energy	$E$	$\frac{W}{t}=\frac{pC}{t}$
Moment	$p$	$\frac{h}{\lambda }$

v > C

Tính Chất	Ký Hiệu	Công thức
Time	$t$	$t$
Speed	$v$	$v$
Acceleration	$a$	$\frac{v}{t}$
Distance	$s$	$vt$
Force	$F$	$ma=m\frac{v}{t}=\beta \frac{p}{t}$
Work	$W$	$Fs=F\frac{p}{t}s=\beta pv$
Energy	$E$	$\frac{W}{t}=\frac{pv}{t}=\beta pa$
Moment	$p$	$m\beta p$

Linear motion

Inclined Linear motion

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

Vertical Linear motion

Tính Chất	Ký Hiệu	Công thức
Time	$t$	$t$
Acceleration	$a$	$-g=-\frac{MG}{h^2}$
Speed	$v$	$-gt$
Distance	$s$	$-g{t}^2$
Force	$F$	$-mg=-m\frac{MG}{h^2}$
Work	$W$	$-mgh=-m\frac{MG}{h}$
Energy	$E$	$-mg\frac{h}{t}==-m\frac{MG}{ht}$

Horizontal Linear motion

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

Curve motion

v(t) =

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	$F$	$ma=m\frac{d}{dt}v(t)$
Work	$W$	$Fs=F\int v(t)dt$
Energy	$E$	$\frac{W}{t}=\frac{F}{t}\int v(t)dt$

s(t) =

Tính Chất	Ký hiệu	Công thức
Time	$t$	$t$
Distance	$s$	$s(t)$
Speed	$v$	$\frac{d}{dt}s(t)$
Acceleration	$a$	$\frac{d^2}{d{t}^2}s(t)$
Force	$F$	$ma=m\frac{d^2}{d{t}^2}s(t)$
Work	$W$	$Fs=Fvt=pv=p\frac{d}{dt}s(t)$
Energy	$E$	$\frac{W}{t}=Fv=pa=p\frac{d^2}{d{t}^2}s(t)$

Circular motion

Full circle motion

Tính Chất	Ký Hiệu	Công thức
Time	$t$	$t$
Distance	$s$	$2\pi$
Speed	$v$	$\frac{2\pi }{t}=2\pi f=\omega$
Acceleration	$a$	$\frac{\omega }{t}$
Force	$F$	$ma=m\frac{\omega }{t}$
Work	$W$	$Fs=Fvt=pv=p\omega$
Energy	$E$	$\frac{W}{t}=Fv=pa=p\frac{\omega }{t}$

Circular Arc motion

Tính Chất	Ký Hiệu	Công thức
Time	$t$	$t$
Distance	$s$	$r\theta$
Speed	$v$	$r\omega$
Acceleration	$a$	$r\alpha$
Force	$F$	$ma=mr\alpha$
Work	$W$	$Fs=Fvt=pv=pr\omega$
Energy	$E$	$\frac{W}{t}=Fv=pa=p\frac{r\omega }{t}$

Oscillation motion

Horizontal Osicllation

$F_a=-F_x$

$ma=-kx$

$a=-\beta x=\frac{d^2}{d{t}^2}x$

$x=Asin\omega t$

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

Vertical Osicllation

$-F_g=F_y$

$-mg=ky$

$g=-\beta y=\frac{d^2}{d{t}^2}y$

$y=Asin\omega t$

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

Inclined Osicllation

$-F_g=F_{\theta }$

$-mg=l\theta$

$g=-\beta \theta =\frac{d^2}{d{t}^2}\theta$

$\theta =Asin\omega t$

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

Wave motion

Wave function

$f(t)=Asin\omega t$

Wave equation

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

$\omega =\sqrt{\beta }$

Tính Chất	Ký Hiệu	Công thức
Time	$t$	$t$
Distance	$s$	$\lambda$
Speed	$v$	$\lambda f=\omega$
Acceleration	$a$	$\frac{\omega }{t}$
Force	$F$	$ma=m\frac{\omega }{t}$
Work	$W$	$Fs=Fvt=pv=p\omega$
Energy	$E$	$\frac{W}{t}=Fv=pa=p\frac{\omega }{t}$

Force and Motion

From Newton laws

Force equals zero . Matter is stationary

Force is not zero . Matter is in motion

Sum of the forces equal zero . Matter is in equilibrium

Force of momentum travels horizontaly

$F_p=m\frac{V}{t}=\frac{p}{t}$

$p=mv=Ft$

$m=\frac{p}{v}=\frac{Ft}{v}$

$v=\frac{p}{m}=\frac{Ft}{m}$

$t=\frac{mv}{F}=\frac{p}{F}$

Force of momentum travels verticaly

O

|

V

$F_g=mg=m\frac{MG}{h^2}$

$g=\frac{MG}{h^2}$

$h=\sqrt{\frac{MG}{g}}$

Equilibrium for hang in the air

F_p > F_g Matter stays above

F_p < F_g Matter stays below

F_p = F_g Matter stays hang in the air

$F_p=F_g$

$m\frac{v}{t}=mg$

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

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

$a=g=\frac{MG}{h^2}$

$h=\sqrt{\frac{MG}{a}}$