Rotational Motionocw.nctu.edu.tw/course/physics/physics_lecturenotes/ch07.pdf · 2018-01-09 · Rotational Motion 轉動與 ... comparison provides some insight into how much the

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交通大學李威儀

Rotational Motion

轉動與圓周運動

http://blog.twpaddy.net/2005/08/blog-post_31.html

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QUIZ 姓名學號

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(1) An object at rest in space explodes into three equal

parts. The velocities of two of them are, respectively, 2 i

and -4 j. Find the resulting velocity of the third part.

QUIZ 姓名學號

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(2) Please compare the ( real ) gravitational force to the

( fictitious ) centrifugal force on the surface of the earth. This

comparison provides some insight into how much the earth

rotation affects what we observe.

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Fc =





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Fc =

w =





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Fc =

w =

Fc = ( assuming r = 108 m )





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Fc =

w =

Fc = ( assuming r = 108 m )

Fg =

Fg / Fc =





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簡介及假設

三種運動形式 :

translational motion, 平移運動 ( 移動 )

rotation, 旋轉運動 ( 轉動 )

vibration, 振動

假設這一章討論的轉動物体 :

是不會變形的硬体沒有振動

其轉軸相對於轉動物体是固定的

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旋轉的量測 ( Measurement of Rotation )

degrees ( o )

radians ( 無單位 )

在物理應用上對角度的使用單位以 radians 為主

旋轉一圈 ( 1 revolution ) = 360o = 2 p ( rad )

s = r q ( in radians )

s

r

q

s = x 2pr qo ( in degrees )

360o

q ( in radians ) = s / r

( 圓弧長 / 圓弧半徑 )

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旋轉運動 ( Rotational Motion )

旋轉物体上各點的平移位移量 ( s ) 不一樣

但卻有相同的角度變化或角位移 ( q )

wav = Dq / Dt = ( qf – qi ) / ( tf – ti ) ( rad/s )

q = s / r

w (瞬時角速度) = lim Dq / Dt = d q / dt Dt 0

定義角速度為向量 , 其方向遵守右手定則

假如轉軸的方向在慣性座標中是固定的 :

Choose z as the direction of the rotation

axis

counter-clockwise rotation “ + ” w

clockwise rotation “ – “ w

w

x

y

z

r

s q

vT

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• freq. f ( rev / sec ) = 1/T , T : period ( sec )

• for constant w, w = 2p / T = 2p f

Ex. If f = 1 rev/sec w = 2p rad/sec

• w = d q / dt = d ( s/r ) / dt = 1/r ( ds / dt ) = 1/r • v = 1/r • vT

vT = w r , if w is constant , r vT

tangent to the circle of motion

constantly changing direction

• angular acceleration

aav = Dw / Dt

vT = d s / dt : tangential velocity

a = d w / dt ( rad / s2 ) , 如轉軸的方向是固定的 a = d w / dt

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轉動方程式 ( Equations of Rotational Motion )

w – wo = at

q – qo = wot + ½ at2

w2 = wo2+ 2a ( q – qo )

一維運動公式 ( One-Dimensional Equation of Kinematics )

Linear ( a = constant )

v = vo + at

x = xo + vot + ½ at2

v2 = vo2 + 2a ( x – xo )

Rotation ( a = constant )

w = wo + at

q = qo + wot + ½ at2

w2 = wo2 + 2a ( q – qo )

x q

v w

a a

• for constant a :

a = dw / dt dw = a dt dw = a dt = a dt

w

wo

t

0

t

0

• dq = w dt = ( wo + at ) dt

q

qo

t

0

t

0

相對於平移運動的運動方程式

t = (w – wo) / a

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向心加速度 ( Radial Acceleration )

ar = v2 / r = r w2 ( m /s2 ) 方向為指向旋轉軸心

Dr / r = Dv / v

( if v1 = v2 = v , at = 0 )

Dv = ( v / r ) | Dr |

as Dt 0 , Dr = v Dt

Dv / Dt = v2 / r

a = lim Dv / Dt Dt 0

( v = r w )

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向心加速度 ( ar )

切線加速度 ( at )

ar = - w2 r

net linear acceleration : a = ar + at , a = ( ar2 + at

2 ) 1/2

at = r a

a = 0 , 等速率圓周運動

a 0

at

ar

r

vt

對於非等速率的圓周運動, 還必須考慮切線加速度

ar ar

ar

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向心力 ( Centripetal Force ) 與離心力 ( Centrifugal Force )

FR = m ar = m ( vt2 / r ) = m r w2 : 向心力

向心力的來源有各種可能性, 舉例來說 :

可以是繩子的張力, 萬有引力, 或摩擦力

影片

fs : static friction

( the tires are momentarily at rest with respect to the road )

FSM07VD1.MOV

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Ex. A small coin is placed at the rim of a turn table, with R = 10 cm

and v = 30 rev/min. Find the minimum coefficient of static friction

for the coin to stay on.

Fy = N – mg = 0

Fx = f = mv2 / r

f = m N = m mg = mv2 / r

m = v2 / rg , v = ( 30 x 2pr ) / 60 (sec/min) = pr

m = ( p2r ) / g = 0.15

N

mg

f a

y

x

摩擦力向心力

F = N + W + f = m a

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Fapp

f fs,max

= msN

(stick-slip)

smooth fk

at rest smooth sliding

vmax = 0 on flat and frictionless road

http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html

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Where is the needed

centripetal force ?

Not easy to figure out !

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對於非等速率的圓周運動, 還必須考慮切線方向上所受的力

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什麼是離心力 ?

離心力是在一個非慣性的旋轉座標中觀察到的一種假想力

舉例而言 :

由於旋轉中的物体是處於加速度 ( 向心加速度 ) 運動中的物体 , 因此若

要以旋轉中的座標為觀察座標 , 它將是一個非慣性觀察座標 , 在此非慣

性座標中觀察實際上以等速度運動的物体 , 會被以為受外力作用而處

在加速運動中 , 離心力的錯覺就是因此而出現的

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The observer sees the extension of the spring and

concludes there is a force applied to the block.

http://memo.cgu.edu.tw/yun-

ju/cguweb/phychiu/H103CircularMotion/centrifugal.htm

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影片

FSM08VD2.MOV

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究竟是什麼力讓你妳驚聲尖叫？

是離心力？

是什麼力讓飛行員在疾昇時昏厥？

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是什麼力讓它翻車的？

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是什麼力讓它翻車的？

摩擦力 !

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除了離心力之外 , 還有一個假想力產生柯式力 Coriolis Force


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萬有引力與軌道運動 ( Gravitation and Orbital Motion )

最重要的圓周運動之一就是行星或衛星運行的軌道運動

軌道運動中的向心力是由萬有引力提供的

牛頓則是第一個提出萬有引力這個觀念的人

F12 = F21 = F = , G = 6.67 x 10-11 N m2 / kg2 G m1 m2

r2

簡單的說萬有引力就是兩個質量 ( m1, m2 )之間的相互吸引力 , 其大小為 :

center of mass

CM

CM

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Ex. 一個 100 kg 的男生和一個 100 kg 的女生相距 ( 66.7 )1/2公尺

他們之間萬有引力的大小為 F = ( 6.67 x 10-11 x 100 x 100 ) / 66.7

男女異性之間的吸引力應該不是萬有引力 !

F = ma a = 10-8 / 100 = 10-10 ( m/s2 )

r

F12 F21 m1

m2

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• 拋体運動 ( projectile motion ) 與軌道運動 ( orbital motion )

牛頓也是第一個將萬有引力、拋体運動與衛星軌道運動聯想在一起的人

只要有足夠大小的起始速度軌道運動

假設圓心物体的質量作軌道運動的衛星物体的質量

圓心物体靜止不動

( G M m ) / r2 = m v2 / r

vorb = ( GM / r )1/2 – 與衛星質量無關

– r vorb

( Person in the satellite feels

“weightless” because the person

is free falling along with the

satellite and experiences no force

of support from the satellite. )

r is

measured

from the

center of

the earth


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- Kepler’s 3rd Law ( 西元 1618 年發現 )

k depends only on M

對太陽系的行星而言, k =

若T 與 r 為已知 M 可以計算得知

例如 : 知道月亮繞行地球的 T 及 r Mearth 可知

T ( period ) = 2pr / vorb = r3 2 p

G M 週期

T2 = r3 = k r3 4p2

GM

4p2

G Msun

r

M

Johannes Kepler, 1571 – 1630

Probably the first science fiction writer

“Solemnium” ( 1618 )

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同步軌道通信衛星

T ( period ) = 2pr / vorb = r3 = 24 hours 2 p

G M 週期


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地球的轉動

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地球自轉

Fc = mv2/r = m w2 r

w = 2p (rad) / 24 (hrs) = 7.27 x 10-5 (rad/s)

rotational angular speed of the earth is very small

Fc = m x ( 3.37 x 10-2 m/s2 )

Fg = mg = m x ( 9.8 )

Fg / Fc = 9.8 / ( 3.37 x 10-2 ) = 290

Comparing the ( real ) gravitational force to the ( fictitious )

centrifugal force provides some insight into how much the

earth rotation affects what we observe.

FSX08VD1.MOV

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Typhoon Winnie ( 1997/8/13 )

in Northern Hemisphere

Storm in Southern

Hemisphere near New

Zealand ( 1973/12 )


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Long exposure photo

showing earth’s rotation

in the Star Trails.

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昏了 , 昏了

已經轉昏了 …..

Documents

Rotational Motionocw.nctu.edu.tw/course/physics/physics_lecturenotes/ch07.pdf · 2018-01-09 · Rotational Motion 轉動 與 ... comparison provides some insight into how much the

Rotational Motionocw.nctu.edu.tw/course/physics/physics_lecturenotes/ch07.pdf · 2018-01-09 · Rotational Motion 轉動與 ... comparison provides some insight into how much the