Question-127708

Question Number 127708 by ajfour last updated on 01/Jan/21

Commented by ajfour last updated on 01/Jan/21

A b a l l i s d r o p p e d f r o m a t a l l

t o w e r, h e i g h t h, a t t h e e q u a t o r .

F i n d h o w f a r f r o m t h e f o o t o f

t h e b u i l d i n g, d o e s t h e b a l l l a n d .

Commented by mr W last updated on 01/Jan/21

Commented by mr W last updated on 01/Jan/21

t h e b a l l m o v e s i n h o r i z o n t a l d i r e c t i o n

w i t h t h e s a m e v e l o c i t y a s t h e t o p

o f t h e t o w e r w h i c h i s l a r g e r t h a n

t h e s p e e d o f t h e f o o t o f t h e t o w e r .

B C = {A A}^{'} > {B B}^{'} .

t h a t m e a n s t h e b a l l s f a l l s o n g r o u n d

i n f r o n t o f t h e f o o t o f t h e t o w e r,

n o t b e h i n d i t .

Commented by mr W last updated on 02/Jan/21

u = (R + h) ω

h = \frac{1}{2} {g t}^{2}

t = \sqrt{\frac{2 h}{g}}

B C = {A A}^{'} = u t = (R + h) ω \sqrt{\frac{2 h}{g}}

{B B}^{'} = R ω \sqrt{\frac{2 h}{g}}

s = B C - {B B}^{'} = h ω \sqrt{\frac{2 h}{g}}

e x a m p l e :

ω = \frac{2 π}{24 \times 60 \times 60}

h = 100 m

s = 100 \times \frac{2 π}{24 \times 60 \times 60} \times \sqrt{\frac{2 \times 100}{10}}

\approx 0.0325 m = 32.5 m m

Commented by ajfour last updated on 01/Jan/21

t h a n k s S i r, I t o o g o t a n a n s w e r

c l o s e t o y o u r s!

Commented by mr W last updated on 01/Jan/21

Commented by mr W last updated on 01/Jan/21

i n m y c o n s i d e r a t i o n a b o v e t h e

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

i n h o r i z o n t a l d i r e c t i o n g_{x} i s n o t

t a k e n i n t o a c c o u n t (t h e d o t t e d l i n e

i n f o l l o w i n g d i a g r a m) . w h e n t h i s i s

c o n s i d e r e d {w e}^{'} l l g e t t h e c o r r e c t

f o r m u l a

s = \frac{2}{3} h ω \sqrt{\frac{2 h}{g}}

Commented by mr W last updated on 01/Jan/21

Commented by ajfour last updated on 01/Jan/21

Commented by ajfour last updated on 01/Jan/21

m V r = m u (R + h)

V = \frac{u (R + h)}{r}

\frac{d x}{d t} = \frac{u (R + h)}{(R + h) - \frac{{g t}^{2}}{2}}

\int d x = - \frac{2 u (R + h)}{g} \int_{0}^{T} \frac{d t}{t^{2} - \frac{2 (R + h)}{g}}

s = \frac{2 u (R + h)}{g} \times \frac{\sqrt{g}}{2 \sqrt{2 (R + h)}} \ln ∣ \frac{\sqrt{\frac{2 (R + h)}{g}} + t}{\sqrt{\frac{2 (R + h)}{g}} - t} ∣_{0}^{T}

s = u \sqrt{\frac{R + h}{2 g}} \ln (1 + \frac{2 T}{λ})

\approx u \sqrt{\frac{R + h}{2 g}} (2 \sqrt{\frac{2 h}{g}}) \sqrt{\frac{g}{2 (R + h)}}

\approx ω (R + h) \sqrt{\frac{2 h}{g}}

= (\frac{2 π}{1 d a y}) (R + h) \sqrt{\frac{2 h}{g}}

s_{t o w e r} = \sqrt{\frac{2 h}{g}} (\frac{2 π}{1 d a y}) R

△ = (\frac{2 π}{86400}) h \sqrt{\frac{2 h}{g}}

f o r h = 100 m

△ \approx \frac{π (\sqrt{5})}{216} \times 100 c m = 3.25 c m .

Commented by mr W last updated on 01/Jan/21

y o u r r e s u l t Δ = ω h \sqrt{\frac{2 h}{g}} i s t h e s a m e

a s w h a t i d i d a t t h e b e g i n i n g . b u t i t

i s n o t c o r r e c t, b e c a u s e t h e a c c e l e r a t i o n

g i s a l w a y s t o w a r d t h e c e n t e r o f t h e

e a r t h a n d h a s t h e r e f o r e a c o m p o n e n t

i n x - d i r e c t i o n . t h i s c o m p o n e n t g_{x}

r e d u c e s t h e h o r i z o n t a l v e l o c i t y a n d

t h e r e f o r e t h e h o r i z o n t a l

d i s p l a c e m e n t . t h e c o r r e c t a n s w e r i s

Δ = \frac{2}{3} ω h \sqrt{\frac{2 h}{g}} .

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