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Question Number 24772 by Tinkutara last updated on 25/Nov/17

The ratio of acceleration of points A,  B and C is [assume all surfaces are  smooth, pulley and strings are light]

$$\mathrm{The}\:\mathrm{ratio}\:\mathrm{of}\:\mathrm{acceleration}\:\mathrm{of}\:\mathrm{points}\:{A}, \\ $$$${B}\:\mathrm{and}\:{C}\:\mathrm{is}\:\left[\mathrm{assume}\:\mathrm{all}\:\mathrm{surfaces}\:\mathrm{are}\right. \\ $$$$\left.\mathrm{smooth},\:\mathrm{pulley}\:\mathrm{and}\:\mathrm{strings}\:\mathrm{are}\:\mathrm{light}\right] \\ $$

Commented by Tinkutara last updated on 25/Nov/17

Commented by mrW1 last updated on 26/Nov/17

I think B and C can move independently  from A, therefore there is no certain  ratio between a_A  and a_B .

$${I}\:{think}\:{B}\:{and}\:{C}\:{can}\:{move}\:{independently} \\ $$$${from}\:{A},\:{therefore}\:{there}\:{is}\:{no}\:{certain} \\ $$$${ratio}\:{between}\:{a}_{{A}} \:{and}\:{a}_{{B}} . \\ $$

Commented by Tinkutara last updated on 26/Nov/17

Answer is wrong.  Why a_B =a_C =((9F)/m)?

$$\mathrm{Answer}\:\mathrm{is}\:\mathrm{wrong}. \\ $$$$\mathrm{Why}\:{a}_{{B}} ={a}_{{C}} =\frac{\mathrm{9}{F}}{{m}}? \\ $$

Commented by mrW1 last updated on 26/Nov/17

a_A =((3F)/m)  a_F =3a_A =((9F)/m) (acceleration of the end of string)  a_B =a_C =0

$${a}_{{A}} =\frac{\mathrm{3}{F}}{{m}} \\ $$$${a}_{{F}} =\mathrm{3}{a}_{{A}} =\frac{\mathrm{9}{F}}{{m}}\:\left({acceleration}\:{of}\:{the}\:{end}\:{of}\:{string}\right) \\ $$$${a}_{{B}} ={a}_{{C}} =\mathrm{0} \\ $$

Commented by Tinkutara last updated on 26/Nov/17

But answer is still wrong as ratio is  integer.

$${But}\:{answer}\:{is}\:{still}\:{wrong}\:{as}\:{ratio}\:{is} \\ $$$${integer}. \\ $$

Commented by mrW1 last updated on 26/Nov/17

This is I think the right answer.

$${This}\:{is}\:{I}\:{think}\:{the}\:{right}\:{answer}. \\ $$

Commented by ajfour last updated on 26/Nov/17

Is B a rope point or centre of  pulley ?

$${Is}\:{B}\:{a}\:{rope}\:{point}\:{or}\:{centre}\:{of} \\ $$$${pulley}\:? \\ $$

Commented by mrW1 last updated on 26/Nov/17

B (as well as C) is center of pulley.

$${B}\:\left({as}\:{well}\:{as}\:{C}\right)\:{is}\:{center}\:{of}\:{pulley}. \\ $$

Commented by ajfour last updated on 26/Nov/17

3x+2l+3y=constant  3(a_C −a_A )−3a_B +a_F =0    ....(i)  Fv_F =mv_A a_A   ⇒Fa_F =ma_A ^2 +0  Further    3F=ma_A   a_A =((3F)/m)   ⇒   a_F =3a_A  = ((9F)/m) .  Now from (i):  a_C =a_B  = (whatever, depends on  mass of small pulleys).

$$\mathrm{3}{x}+\mathrm{2}{l}+\mathrm{3}{y}={constant} \\ $$$$\mathrm{3}\left({a}_{{C}} −{a}_{{A}} \right)−\mathrm{3}{a}_{{B}} +{a}_{{F}} =\mathrm{0}\:\:\:\:....\left({i}\right) \\ $$$${Fv}_{{F}} ={mv}_{{A}} {a}_{{A}} \\ $$$$\Rightarrow{Fa}_{{F}} ={ma}_{{A}} ^{\mathrm{2}} +\mathrm{0} \\ $$$${Further}\:\:\:\:\mathrm{3}{F}={ma}_{{A}} \\ $$$${a}_{{A}} =\frac{\mathrm{3}{F}}{{m}}\: \\ $$$$\Rightarrow\:\:\:{a}_{{F}} =\mathrm{3}{a}_{{A}} \:=\:\frac{\mathrm{9}{F}}{{m}}\:. \\ $$$${Now}\:{from}\:\left({i}\right): \\ $$$${a}_{{C}} ={a}_{{B}} \:=\:\left({whatever},\:{depends}\:{on}\right. \\ $$$$\left.{mass}\:{of}\:{small}\:{pulleys}\right). \\ $$

Commented by Tinkutara last updated on 26/Nov/17

Answer is 3:1:1

$${Answer}\:{is}\:\mathrm{3}:\mathrm{1}:\mathrm{1} \\ $$

Commented by jota+ last updated on 26/Nov/17

La resultante sobre el sistema B  y C is zero. 2F−2F=0  ⇒ a_B =a_C =0

$${La}\:{resultante}\:{sobre}\:{el}\:{sistema}\:{B} \\ $$$${y}\:{C}\:{is}\:{zero}.\:\mathrm{2}{F}−\mathrm{2}{F}=\mathrm{0} \\ $$$$\Rightarrow\:{a}_{{B}} ={a}_{{C}} =\mathrm{0} \\ $$

Commented by mrW1 last updated on 26/Nov/17

Answer 3:1:1 is wrong in each case,  that wurde mean a_A = 3a_B =3a_C .  But this is wrong even if B and C  are the rope points. If B and C are  the points on the rope, then the  answer should be 1:3:3 with  a_A =((3F)/m)  a_B =a_C =3a_A =((9F)/m)  This was my original answer.    If B and C are the center of pulley,  then a_A :a_B :a_C =uncertain.

$${Answer}\:\mathrm{3}:\mathrm{1}:\mathrm{1}\:{is}\:{wrong}\:{in}\:{each}\:{case}, \\ $$$${that}\:{wurde}\:{mean}\:{a}_{{A}} =\:\mathrm{3}{a}_{{B}} =\mathrm{3}{a}_{{C}} . \\ $$$${But}\:{this}\:{is}\:{wrong}\:{even}\:{if}\:{B}\:{and}\:{C} \\ $$$${are}\:{the}\:{rope}\:{points}.\:{If}\:{B}\:{and}\:{C}\:{are} \\ $$$${the}\:{points}\:{on}\:{the}\:{rope},\:{then}\:{the} \\ $$$${answer}\:{should}\:{be}\:\mathrm{1}:\mathrm{3}:\mathrm{3}\:{with} \\ $$$${a}_{{A}} =\frac{\mathrm{3}{F}}{{m}} \\ $$$${a}_{{B}} ={a}_{{C}} =\mathrm{3}{a}_{{A}} =\frac{\mathrm{9}{F}}{{m}} \\ $$$${This}\:{was}\:{my}\:{original}\:{answer}. \\ $$$$ \\ $$$${If}\:{B}\:{and}\:{C}\:{are}\:{the}\:{center}\:{of}\:{pulley}, \\ $$$${then}\:{a}_{{A}} :{a}_{{B}} :{a}_{{C}} ={uncertain}. \\ $$

Commented by Tinkutara last updated on 26/Nov/17

Thanks Sir! I will confirm.

$${Thanks}\:{Sir}!\:{I}\:{will}\:{confirm}. \\ $$

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