Question-217565

Question Number 217565 by mr W last updated on 16/Mar/25

Commented by mr W last updated on 16/Mar/25

t h e s t r i n g s a r e m a s s l e s s a n d t h e r e

i s n o f r i c t i o n b e t w e e n s t r i n g s a n d

p u l l e y s .

f i n d t h e a c c e l e r a t i o n s o f a l l o b j e c t s .

Answered by ajfour last updated on 16/Mar/25

A c c . o f 5 k g b l o c k b e A .

r e l a t i v e t o 4 k g p u l l e y l e t a c c . o f 3 k g b e a_{1} ↓

a n d t h a t a c c . o f 4 k g b l o c k i t s e l f b e a ↓ r e l a t i v e

t o 5 k g b l o c k .

3 g - T_{1} = 3 (A + a + a_{1}) \dots (i)

T_{1} - 2 g = 2 (A + a - a_{1}) \dots (i i)

T_{2} - g = A + a \dots . (i i i)

2 T_{2} - T_{1} = 4 (a - A) \dots (i v)

200 - 2 T_{1} = 5 A \dots (v)

\dots \dots \dots \dots \dots \dots

4 (A + a + g) - 2 T_{1} = 8 (a - A)

200 - 2 T_{1} = 5 A

\Rightarrow 200 - 4 (A + a + g) = 13 A - 8 a . . (1)

a d d i n g r (i) & (i i)

g = 5 A + 5 a + a_{1} \dots . (2)

u d i n g (i i) i n (v)

200 - 4 g - 4 (A + a - a_{1}) = 5 A

\Rightarrow 200 - 4 g = 9 A + 4 a - 4 a_{1}

n o w u s i n g (2)

200 - 4 g = 9 A + 4 a - 4 (g - 5 A - 5 a)

\Rightarrow 200 = 29 A + 24 a

f r o m (1)

{200 - 4 g = 17 A - 4 a} \times 6

1200 - 24 g = 102 A - 24 a

200 = 29 A + 24 a

a d d i n g a b o v e t w o

1400 - 24 g = 131 A

A = \frac{1400 - 24 g}{131}

r e s t c a n b e f o u n d n o w .

Commented by mr W last updated on 17/Mar/25

t h a n k s s i r!

y o u g o t A = \frac{1160}{131} \approx 8.85 m / s^{2}

i g o t a c c . o f 5 k g p u l l e y

A_{1} = \frac{5590}{421} \approx 13.28 m / s^{2}

Commented by ajfour last updated on 18/Mar/25

I l l c h e c k s i r . T h a n k s f o r s o l v i n g t h e

p r o j e c t i l e o n e t o o .

Answered by mr W last updated on 17/Mar/25

Commented by mr W last updated on 17/Mar/25

δ_{1} r e l a t i v e a c c . o f s t r i n g t o p u l l e y 1

δ_{2} r e l a t i v e a c c . o f s t r i n g t o p u l l e y 2

a_{1} = A_{1} + δ_{1}

A_{2} = A_{1} - δ_{1}

a_{2} = A_{2} + δ_{2} = A_{1} - δ_{1} + δ_{2}

a_{3} = A_{2} - δ_{2} = A_{1} - δ_{1} - δ_{2}

F_{2} = m_{2} (g + a_{2}) = m_{2} (g + A_{1} - δ_{1} + δ_{2})

F_{2} = m_{3} (g + a_{3}) = m_{3} (g + A_{1} - δ_{1} - δ_{2})

F_{1} = m_{1} (g + a_{1}) = m_{1} (g + A_{1} + δ_{1})

F_{1} - 2 F_{2} = M_{2} (g + A_{2}) = M_{2} (g + A_{1} - δ_{1})

F - 2 F_{1} = M_{1} (g + A_{1})

F - 2 m_{1} (g + A_{1} + δ_{1}) = M_{1} (g + A_{1})

(M_{1} + 2 m_{1}) A_{1} + 2 m_{1} δ_{1} = F - (M_{1} + 2 m_{1}) g

7 A_{1} + 2 δ_{1} = 130 \dots (i)

m_{1} (g + A_{1} + δ_{1}) - 2 m_{2} (g + A_{1} - δ_{1} + δ_{2}) = M_{2} (g + A_{1} - δ_{1})

(m_{1} - 2 m_{2} - M_{2}) A_{1} + (M_{2} + m_{1} + 2 m_{2}) δ_{1} - 2 m_{2} δ_{2} = (M_{2} - m_{1} + 2 m_{2}) g

- 7 A_{1} + 9 δ_{1} - 4 δ_{2} = 70 \dots (i i)

m_{2} (g + A_{1} - δ_{1} + δ_{2}) = m_{3} (g + A_{1} - δ_{1} - δ_{2})

(m_{2} - m_{3}) A_{1} + (- m_{2} + m_{3}) δ_{1} + (m_{2} + m_{3}) δ_{2} = (m_{3} - m_{2}) g

- A_{1} + δ_{1} + 5 δ_{2} = 10 \dots (i i i)

δ_{1} = \frac{7800}{421}

δ_{2} = \frac{400}{421}

A_{1} = \frac{7800}{421} + 5 \times \frac{400}{421} - 10 = \frac{5590}{421}

A_{2} = \frac{5590}{421} - \frac{7800}{421} = - \frac{2210}{421}

a_{1} = \frac{5590}{421} + \frac{7800}{421} = \frac{13390}{421}

a_{2} = - \frac{2210}{421} + \frac{400}{421} = - \frac{1810}{421}

a_{3} = - \frac{2210}{421} - \frac{400}{421} = - \frac{2610}{421}

Commented by Tawa11 last updated on 18/Mar/25

Sir, you assumed the cylinder are rotating here .

But Ajfour did not assume the cylinder is rotating .

Commented by mr W last updated on 18/Mar/25

T h e p u l l e y s {d o n}^{'} t r o t a t e a n d i a l s o

{d i d n}^{'} t a s s u m e t h a t t h e y r o t a t e .

Commented by Tawa11 last updated on 18/Mar/25

Thanks sir .

Commented by mr W last updated on 18/Mar/25

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

a n d p u l l e y, t h e s t r i n g c a n a l s o

m o v e r e l a t i v e l y t o t h e p u l l e y w i t h o u t

t h a t t h e p u l l e y r o t a t e s .

i n t h e w o r k i n g s a b o v e δ_{1} a n d δ_{2} a r e

t h e r e l a t i v e m o t i o n s o f t h e s t r i n g s

a l o n g t h e p u l l e y s . t h e r e a r e n o t t h e

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

Commented by mr W last updated on 18/Mar/25

Commented by Tawa11 last updated on 18/Mar/25

Ohh, thanks sir .

I appreciate your time .

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