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}}

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}}at^{2}

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

Linear Motion with acceleration equal constant

a=g

v=gt

s=gt^{2}

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 $\Delta t\rightarrow 0$

Instantanous 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)

Instantanous 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

Bounded continous motion

Distance for curve motion from A to B

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

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

s_{avg}={\frac {1}{b-a}}\int \limits _{a}^{b}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 {\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 }}

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	$a$	${\frac {\Delta v}{\Delta t}}={\frac {v-v_{o}}{t-t_{o}}}$
Speed	$v$	$v_{o}+a\Delta t$
Distance	$s$	$(v_{o}+a\Delta t)\Delta t$
Force	$F$	$ma=m{\frac {\Delta v}{\Delta t}}$
Work	$W$	$Fs=F(v_{o}+a\Delta t)\Delta t$
Energy	$E$	${\frac {W}{t}}=F(v_{o}+a\Delta t)$

Vertical Linear motion

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

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}}{dt^{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}}{dt^{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}}{dt^{2}}}\theta

\theta =Asin\omega t

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

Wave motion

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