The-Figure-shows-a-system-consisting-of-i-a-ring-of-outer-radius-3R-rolling-clockwise-without-slipping-on-a-horizontal-surface-with-angular-speed-and-ii-an-inner-disc-of-radius-2R-rotating-anti-

Question Number 20891 by Tinkutara last updated on 06/Sep/17

The Figure shows a system consisting

of (i) a ring of outer radius 3 R rolling

clockwise without slipping on a

horizontal surface with angular speed

ω and (i i) an inner disc of radius 2 R

rotating anti - clockwise with angular

speed ω / 2 . The ring and disc are

separated by frictionless ball bearing .

The system is in the x - z plane . The

point P on the inner disc is at a distance

R from the origin, where O P makes an

angle 30 ° with the horizontal . Then

with respect to the horizontal surface

(a) The point O has a linear velocity

3 R ω \overset{\land}{i}

(b) The point P has a linear velocity

\frac{11}{4} R ω \overset{\land}{i} + \frac{\sqrt{3}}{4} R ω \overset{\land}{k}

Commented by Tinkutara last updated on 06/Sep/17

Commented by ajfour last updated on 06/Sep/17

Commented by Tinkutara last updated on 06/Sep/17

But since radius at O is 0, so linear

velocity = r ω = 0 (ω) = 0 ? Why not ?

Commented by ajfour last updated on 06/Sep/17

l e t v e l o c i t y o f p o i n t O b e v_{0} .

a s t h e b o t t o m m o s t p o i n t d o e s n o t

s l i p i t s v e l o c i t y = v_{0} - ω (3 R) = 0

\Rightarrow {\vec{v}}_{0} = 3 ω R \hat{i}

v e l o c i t y o f p o i n t P b e v_{P}; t h e n

{\vec{v}}_{P} = {\vec{v}}_{0} + \frac{ω R}{2} (- \sin 30 ° \hat{i} + \cos 30 ° \hat{k})

= (3 ω R - \frac{ω R}{4}) \hat{i} + \frac{ω R}{2} \times \frac{\sqrt{3}}{2} \hat{k}

= \frac{11 ω R}{4} \hat{i} + \frac{ω R \sqrt{3}}{4} \hat{k} .

b o t h a r e t r u e .

Commented by ajfour last updated on 06/Sep/17

t h a t ? w o u l d b e t h e c a s e i f y o u

h a n g t h e w h e e l i n a i r, l i k e a p u l l e y .

Commented by ajfour last updated on 06/Sep/17

r o l l i n g i s r o t a t i n g p l u s s l i d i n g . .

Commented by Tinkutara last updated on 06/Sep/17

Then why v_{O} \neq (2 R) (\frac{ω}{2}) = R ω ? Why

outer radius is taken only ?

Commented by ajfour last updated on 06/Sep/17

o w i n g t o r o t a t i o n t h e s p e e d i s

z e r o, b u t d u e t o t r a n s l a t i o n i t i s

3 ω R .

Commented by Tinkutara last updated on 06/Sep/17

Thank you very much Sir!

Commented by ajfour last updated on 06/Sep/17

o n l y t h e o u t e r r a d i u s i s u s e d f o r

v_{O} b e c a u s e i t i s t h e o u t e r r i n g

w h i c h i s i n c o n t a c t w i t h g r o u n d .

s o w e h a v e a r e l a t i o n o n l y o f t h e

a n g u l a r v e l o c i t y o f r i n g a n d

l i n e a r v e l o c i t y o f i t s c e n t e r .

v_{O} = 3 ω R . T h e d i s c c a m h a v e a n y

a n g u l a r v e l o c i t y (h e r e ω / 2) . .

Commented by Tinkutara last updated on 06/Sep/17

Thanks . Now I understand .