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Physics

Fluid-Mechanics

Easy

Question

A uniform rod of mass m, length L, area of cross-section A and Young’s modulus Y hangs from the ceiling. Its elongation under its own weight will be

zero
$\frac{m g L}{2 A Y}$

The correct answer is: $fraction numerator m g L over denominator 2 A Y end fraction$

Mass of section BC m/L (L - y).
$therefore$ tension at B=T=m / L(L-y) g.
$therefore$ elongation of element dy at B
= d x =( dy ) T / AY = m / L ( L - y ) g d y / AY
Total elongation $equals integral d x equals m g divided by L A Y integral subscript 0 superscript L left parenthesis L minus y right parenthesis d y equals m g L divided by 2 Y A$

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A wire of cross-section area 3.0 mm² and natural length 50 cm is fixed at one end and a mass of 2.1 kg is hung from the other end. Young’s modulus of the material of wire is $1.9 cross times 10 to the power of 11 N divided by m squared$ . The potential energy stored in the wire in steady state will be (g = 10 m/s²)

A wire of cross-section area 3.0 mm² and natural length 50 cm is fixed at one end and a mass of 2.1 kg is hung from the other end. Young’s modulus of the material of wire is $1.9 cross times 10 to the power of 11 N divided by m squared$ . The potential energy stored in the wire in steady state will be (g = 10 m/s²)

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A metal wire of length L and radius r is clamped rigidly at one end. A force F is applied at another end so that its length increases by L. The increase in length of another metal wire of length 2L and radius 2r, when stretched by a force 2F, will be

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PhysicsFluid-Mechanics

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$A t space 40 to the power of 0 space C comma space a space b r a s s space w i r e space o f space 1 space m m space r a d i u s space i s space h u n g space f r o m space t h e space c e i l i n g. space A space s m a l l space m a s s comma space M space i s space h u n g space f r o m space t h e space f r e e space e n d space o f space t h e space w i r e. space W h e n space t h e space w i r e space i s space c o o l e d space d o w n space f r o m space 40 to the power of 0 C space t o space 20 to the power of 0 C space i t space r e g a i n s space i t s space o r i g i n a l space l e n g t h space o f space 0.2 space m. space T h e space v a l u e space o f space M space i s space c l o s e space t o space colon space left parenthesis C o e f f i c i e n t space o f space l i n e a r space e x p a n s i o n space a n d space Y o u n g apostrophe s space m o d u l u s space o f space b r a s s space a r e 10 to the power of negative 5 end exponent divided by C presuperscript ring operator text and end text 10 to the power of 11 N divided by m squared comma space space r e s p e c t i v e l y semicolon space g space equals 10 space space m s to the power of negative 2 end exponent right parenthesis$

$A t space 40 to the power of 0 space C comma space a space b r a s s space w i r e space o f space 1 space m m space r a d i u s space i s space h u n g space f r o m space t h e space c e i l i n g. space A space s m a l l space m a s s comma space M space i s space h u n g space f r o m space t h e space f r e e space e n d space o f space t h e space w i r e. space W h e n space t h e space w i r e space i s space c o o l e d space d o w n space f r o m space 40 to the power of 0 C space t o space 20 to the power of 0 C space i t space r e g a i n s space i t s space o r i g i n a l space l e n g t h space o f space 0.2 space m. space T h e space v a l u e space o f space M space i s space c l o s e space t o space colon space left parenthesis C o e f f i c i e n t space o f space l i n e a r space e x p a n s i o n space a n d space Y o u n g apostrophe s space m o d u l u s space o f space b r a s s space a r e 10 to the power of negative 5 end exponent divided by C presuperscript ring operator text and end text 10 to the power of 11 N divided by m squared comma space space r e s p e c t i v e l y semicolon space g space equals 10 space space m s to the power of negative 2 end exponent right parenthesis$

PhysicsFluid-Mechanics

parallel

Fluid-Mechanics

$T h e space e l a s t i c space l i m i t space o f space b r a s s space i s space 379 space M P a. space W h a t space s h o u l d space b e space t h e space m i n i m u m space d i a m e t e r space o f space a space b r a s s space r o d space i f space i t space i s space t o space s u p p o r t space a space 400 space N space l o a d space w i t h o u t space e x c e e d i n g space i t s space e l a s t i c space l i m i t ?$

$T h e space e l a s t i c space l i m i t space o f space b r a s s space i s space 379 space M P a. space W h a t space s h o u l d space b e space t h e space m i n i m u m space d i a m e t e r space o f space a space b r a s s space r o d space i f space i t space i s space t o space s u p p o r t space a space 400 space N space l o a d space w i t h o u t space e x c e e d i n g space i t s space e l a s t i c space l i m i t ?$

PhysicsFluid-Mechanics

Fluid-Mechanics

$A space c u b i c a l space b l o c k space o f space s i d e space 0.5 space m space f l o a t s space o n space w a t e r space w i t h space 30 space percent sign space o f space i t s space v o l u m e space u n d e r space w a t e r. space W h a t space i s space t h e space m a x i m u m space w e i g h t space t h a t space c a n space b e space p u t space o n space t h e space b l o c k space w i t h o u t space f u l l y space s u b m e r g i n g space i t space u n d e r space w a t e r ? space left square bracket T a k e comma space d e n s i t y space o f space w a t e r equals 10 cubed space k space g divided by m cubed right square bracket$

$A space c u b i c a l space b l o c k space o f space s i d e space 0.5 space m space f l o a t s space o n space w a t e r space w i t h space 30 space percent sign space o f space i t s space v o l u m e space u n d e r space w a t e r. space W h a t space i s space t h e space m a x i m u m space w e i g h t space t h a t space c a n space b e space p u t space o n space t h e space b l o c k space w i t h o u t space f u l l y space s u b m e r g i n g space i t space u n d e r space w a t e r ? space left square bracket T a k e comma space d e n s i t y space o f space w a t e r equals 10 cubed space k space g divided by m cubed right square bracket$

PhysicsFluid-Mechanics

Fluid-Mechanics

$I n space a n space e x p e r i m e n t comma space b r a s s space a n d space s t e e l space w i r e s space o f space l e n g t h space 1 space m space e a c h space w i t h space a r e a s space o f space c r o s s space s e c t i o n space 1 space m m space squared space a r e space u s e d. space T h e space w i r e s space a r e space c o n n e c t e d space i n space s e r i e s space a n d space o n e space e n d space o f space t h e space c o m b i n e d space w i r e space i s space c o n n e c t e d space t o space a space r i g i d space s u p p o r t space a n d space o t h e r space e n d space i s space s u b j e c t e d space t o space e l o n g a t i o n. space T h e space s t r e s s space r e q u i r e d space t o space p r o d u c e space a space n e t space e l o n g a t i o n space o f space 0.2 space m m space i s comma space left square bracket G i v e n comma space t h e space Y o u n g apostrophe s space M o d u l u s space f o r space s t e e l space a n d space b r a s s space a r e comma space r e s p e c t i v e l y comma space 120 cross times 10 to the power of 9 N divided by m squared text and end text open 60 cross times 10 to the power of 9 N divided by m squared close square brackets right square bracket$

$I n space a n space e x p e r i m e n t comma space b r a s s space a n d space s t e e l space w i r e s space o f space l e n g t h space 1 space m space e a c h space w i t h space a r e a s space o f space c r o s s space s e c t i o n space 1 space m m space squared space a r e space u s e d. space T h e space w i r e s space a r e space c o n n e c t e d space i n space s e r i e s space a n d space o n e space e n d space o f space t h e space c o m b i n e d space w i r e space i s space c o n n e c t e d space t o space a space r i g i d space s u p p o r t space a n d space o t h e r space e n d space i s space s u b j e c t e d space t o space e l o n g a t i o n. space T h e space s t r e s s space r e q u i r e d space t o space p r o d u c e space a space n e t space e l o n g a t i o n space o f space 0.2 space m m space i s comma space left square bracket G i v e n comma space t h e space Y o u n g apostrophe s space M o d u l u s space f o r space s t e e l space a n d space b r a s s space a r e comma space r e s p e c t i v e l y comma space 120 cross times 10 to the power of 9 N divided by m squared text and end text open 60 cross times 10 to the power of 9 N divided by m squared close square brackets right square bracket$

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$A space s u b m a r i n e space e x p e r i e n c e s space a space p r e s s u r e space o f space 5.05 cross times 10 to the power of 6 P a space a t space a space d e p t h space o f space d subscript 1 space i n space a space s e a. space W h e n space i t space g o e s space f u r t h e r space t o space a space d e p t h space o f space d subscript 2 comma space i t space e x p e r i e n c e s space a space p r e s s u r e space o f space 8.08 cross times 10 to the power of 6 P a. space T h e n space d subscript 2 minus space d subscript 1 space i s space a p p r o x i m a t e l y space left parenthesis d e n s i t y space o f space w a t e r space equals 10 cubed k g divided by m cubed space a n d space a c c e l e r a t i o n space d u e space t o space g r a v i t y space $ equals 10 space m s to the power of negative 2 end exponent right parenthesis colon$

$A space s u b m a r i n e space e x p e r i e n c e s space a space p r e s s u r e space o f space 5.05 cross times 10 to the power of 6 P a space a t space a space d e p t h space o f space d subscript 1 space i n space a space s e a. space W h e n space i t space g o e s space f u r t h e r space t o space a space d e p t h space o f space d subscript 2 comma space i t space e x p e r i e n c e s space a space p r e s s u r e space o f space 8.08 cross times 10 to the power of 6 P a. space T h e n space d subscript 2 minus space d subscript 1 space i s space a p p r o x i m a t e l y space left parenthesis d e n s i t y space o f space w a t e r space equals 10 cubed k g divided by m cubed space a n d space a c c e l e r a t i o n space d u e space t o space g r a v i t y space $ equals 10 space m s to the power of negative 2 end exponent right parenthesis colon$

PhysicsFluid-Mechanics

Fluid-Mechanics

$W a t e r space f r o m space a space t a p space e m e r g e s space v e r t i c a l l y space d o w n w a r d s space w i t h space a n space i n i t i a l space s p e e d space o f space 1.0 space m s to the power of space minus 1 end exponent. space T h e space c r o s s minus s e c t i o n a l space a r e a space o f space t h e space t a p space i s space 10 to the power of negative 4 end exponent space m squared. space A s s u m e space t h a t space t h e space p r e s s u r e space i s space c o n s tan t space t h r o u g h o u t space t h e space s t r e a m space o f space w a t e r space a n d space t h a t space t h e space f l o w space i s space s t r e a m l i n e d. space T h e space c r o s s minus s e c t i o n a l space a r e a space o f space t h e space s t r e a m comma space 0.15 space m space b e l o w space t h e space t a p space w o u l d space b e colon space left parenthesis T a k e space g space equals 10 space m s to the power of negative 2 end exponent right parenthesis$

$W a t e r space f r o m space a space t a p space e m e r g e s space v e r t i c a l l y space d o w n w a r d s space w i t h space a n space i n i t i a l space s p e e d space o f space 1.0 space m s to the power of space minus 1 end exponent. space T h e space c r o s s minus s e c t i o n a l space a r e a space o f space t h e space t a p space i s space 10 to the power of negative 4 end exponent space m squared. space A s s u m e space t h a t space t h e space p r e s s u r e space i s space c o n s tan t space t h r o u g h o u t space t h e space s t r e a m space o f space w a t e r space a n d space t h a t space t h e space f l o w space i s space s t r e a m l i n e d. space T h e space c r o s s minus s e c t i o n a l space a r e a space o f space t h e space s t r e a m comma space 0.15 space m space b e l o w space t h e space t a p space w o u l d space b e colon space left parenthesis T a k e space g space equals 10 space m s to the power of negative 2 end exponent right parenthesis$

PhysicsFluid-Mechanics

Fluid-Mechanics

$T h e space r a t i o space o f space s u r f a c e space t e n s i o n s space o f space m e r c u r y space a n d space w a t e r space i s space g i v e n space t o space b e space 7.5 space w h i l e space t h e space r a t i o space o f space t h e i r space d e n s i t i e s space i s space 13.6. space T h e i r space c o n t a c t space a n g l e s comma space w i t h space g l a s s comma space a r e space c l o s e space t o space 135 degree space a n d space 0 degree comma space r e s p e c t i v e l y. space I t space i s space o b s e r v e d space t h a t space m e r c u r y space g e t s space d e p r e s s e d space b y space a n space a m o u n t space h space i n space a space c a p i l l a r y space t u b e space o f space r a d i u s space r subscript 1 comma space w h i l e space w a t e r space r i s e s space b y space t h e space s a m e space a m o u n t space h space i n space a space c a p i l l a r y space t u b e space o f space r a d i u s space r subscript 2. space T h e space r a t i o space left parenthesis r subscript 1 divided by r subscript 2 right parenthesis comma space i s space t h e n space c l o s e space t o space colon$

$T h e space r a t i o space o f space s u r f a c e space t e n s i o n s space o f space m e r c u r y space a n d space w a t e r space i s space g i v e n space t o space b e space 7.5 space w h i l e space t h e space r a t i o space o f space t h e i r space d e n s i t i e s space i s space 13.6. space T h e i r space c o n t a c t space a n g l e s comma space w i t h space g l a s s comma space a r e space c l o s e space t o space 135 degree space a n d space 0 degree comma space r e s p e c t i v e l y. space I t space i s space o b s e r v e d space t h a t space m e r c u r y space g e t s space d e p r e s s e d space b y space a n space a m o u n t space h space i n space a space c a p i l l a r y space t u b e space o f space r a d i u s space r subscript 1 comma space w h i l e space w a t e r space r i s e s space b y space t h e space s a m e space a m o u n t space h space i n space a space c a p i l l a r y space t u b e space o f space r a d i u s space r subscript 2. space T h e space r a t i o space left parenthesis r subscript 1 divided by r subscript 2 right parenthesis comma space i s space t h e n space c l o s e space t o space colon$

PhysicsFluid-Mechanics

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Fluid-Mechanics

$Y o u n g apostrophe s space m o d u l i space o f space t w o space w i r e s space A space a n d space B space a r e space i n space t h e space r a t i o space 7 space colon space 4. space W i r e space A space i s space 2 space m space l o n g space a n d space h a s space r a d i u s space R. space W i r e space B space i s space 1.5 space m space l o n g space a n d space h a s space r a d i u s space 2 space m m. space I f space t h e space t w o space w i r e s space s t r e t c h space b y space t h e space s a m e space l e n g t h space f o r space a space g i v e n space l o a d comma space t h e n space t h e space v a l u e space o f space R space i s space c l o s e space t o$

$Y o u n g apostrophe s space m o d u l i space o f space t w o space w i r e s space A space a n d space B space a r e space i n space t h e space r a t i o space 7 space colon space 4. space W i r e space A space i s space 2 space m space l o n g space a n d space h a s space r a d i u s space R. space W i r e space B space i s space 1.5 space m space l o n g space a n d space h a s space r a d i u s space 2 space m m. space I f space t h e space t w o space w i r e s space s t r e t c h space b y space t h e space s a m e space l e n g t h space f o r space a space g i v e n space l o a d comma space t h e n space t h e space v a l u e space o f space R space i s space c l o s e space t o$

PhysicsFluid-Mechanics

Fluid-Mechanics

$A space s t e e l space w i r e space h a v i n g space a space r a d i u s space o f space 2.0 space m m comma space c a r r y i n g space a space l o a d space o f space 4 space k g comma space i s space h a n g i n g space f r o m space a space c e i l i n g. space G i v e n space t h a t space straight g equals 3.1 pi ms to the power of negative 2 end exponent comma space w h a t space w i l l space b e space t h e space t e n s i l e space s t r e s s space t h a t space w o u l d space b e space d e v e l o p e d space i n space t h e space w i r e ?$

$A space s t e e l space w i r e space h a v i n g space a space r a d i u s space o f space 2.0 space m m comma space c a r r y i n g space a space l o a d space o f space 4 space k g comma space i s space h a n g i n g space f r o m space a space c e i l i n g. space G i v e n space t h a t space straight g equals 3.1 pi ms to the power of negative 2 end exponent comma space w h a t space w i l l space b e space t h e space t e n s i l e space s t r e s s space t h a t space w o u l d space b e space d e v e l o p e d space i n space t h e space w i r e ?$

PhysicsFluid-Mechanics

Fluid-Mechanics

$A space b o y ’ s space c a t a p u l t space i s space m a d e space o f space r u b b e r space c o r d space w h i c h space i s space 42 space c m space l o n g comma space w i t h space 6 space m m space d i a m e t e r space o f space c r o s s minus s e c t i o n space a n d space o f space n e g l i g i b l e space space m a s s. space T h e space b o y space k e e p s space a space s t o n e space w e i g h i n g space 0.02 space k g space o n space i t space a n d space s t r e t c h e s space t h e space c o r d space b y space 20 space c m space b y space a p p l y i n g space a space c o n s tan t space f o r c e. space W h e n space r e l e a s e d comma space t h e space s t o n e space f l i e s space o f f space w i t h space a space v e l o c i t y space o f space 20 space m s to the power of negative 1 end exponent. space N e g l e c t space t h e space c h a n g e space i n space t h e space a r e a space o f space c r o s s minus s e c t i o n space o f space t h e space c o r d space w h i l e space s t r e t c h e d. space T h e space Y o u n g ’ s space m o d u l u s space o f space r u b b e r space i s space c l o s e s t space t o space minus space$

$A space b o y ’ s space c a t a p u l t space i s space m a d e space o f space r u b b e r space c o r d space w h i c h space i s space 42 space c m space l o n g comma space w i t h space 6 space m m space d i a m e t e r space o f space c r o s s minus s e c t i o n space a n d space o f space n e g l i g i b l e space space m a s s. space T h e space b o y space k e e p s space a space s t o n e space w e i g h i n g space 0.02 space k g space o n space i t space a n d space s t r e t c h e s space t h e space c o r d space b y space 20 space c m space b y space a p p l y i n g space a space c o n s tan t space f o r c e. space W h e n space r e l e a s e d comma space t h e space s t o n e space f l i e s space o f f space w i t h space a space v e l o c i t y space o f space 20 space m s to the power of negative 1 end exponent. space N e g l e c t space t h e space c h a n g e space i n space t h e space a r e a space o f space c r o s s minus s e c t i o n space o f space t h e space c o r d space w h i l e space s t r e t c h e d. space T h e space Y o u n g ’ s space m o d u l u s space o f space r u b b e r space i s space c l o s e s t space t o space minus space$

PhysicsFluid-Mechanics

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Fluid-Mechanics

$W a t e r space f r o m space a space p i p e space i s space c o m i n g space a t space a space r a t e space o f space 100 space l i t e r s space p e r space m i n u t e. space I f space t h e space r a d i u s space o f space t h e space p i p e space i s space 5 space c m comma space t h e space R e y n o l d s space n u m b e r space f o r space t h e space f l o w space i s space o f space t h e space o r d e r space o f space minus space left parenthesis d e n s i t y space o f space w a t e r space equals space 1000 space k g divided by m cubed comma space c o e f f i c i e n t space o f space v i s cos i t y space o f space w a t e r space equals space 1 space m P a space s right parenthesis space$

$W a t e r space f r o m space a space p i p e space i s space c o m i n g space a t space a space r a t e space o f space 100 space l i t e r s space p e r space m i n u t e. space I f space t h e space r a d i u s space o f space t h e space p i p e space i s space 5 space c m comma space t h e space R e y n o l d s space n u m b e r space f o r space t h e space f l o w space i s space o f space t h e space o r d e r space o f space minus space left parenthesis d e n s i t y space o f space w a t e r space equals space 1000 space k g divided by m cubed comma space c o e f f i c i e n t space o f space v i s cos i t y space o f space w a t e r space equals space 1 space m P a space s right parenthesis space$

PhysicsFluid-Mechanics

Fluid-Mechanics

$A space c y l i n d r i c a l space c a p i l l a r y space t u b e space o f space 0.2 space m m space r a d i u s space i s space m a d e space b y space j o i n i n g space t w o space c a p i l l a r i e s space T subscript 1 space a n d space T subscript 2 space o f space d i f f e r e n t space m a t e r i a l s space h a v i n g space w a t e r space c o n t a c t space a n g l e s space o f space 0 to the power of 0 space a n d space 60 to the power of 0 comma space r e s p e c t i v e l y. space T h e space c a p i l l a r y space t u b e space i s space d i p p e d space v e r t i c a l l y space i n space w a t e r space i n space t w o space d i f f e r e n t space c o n f i g u r a t i o n s comma space c a s e space I space a n d space I I space a s space s h o w n space i n space f i g u r e. space W h i c h space o f space t h e space f o l l o w i n g space o p t i o n left parenthesis s right parenthesis space i s left parenthesis a r e right parenthesis space c o r r e c t ? text (Surface tension of water end text equals 0.075 straight N divided by straight m comma text density of water end text equals 1000 kg divided by straight m cubed comma text take end text open straight g equals 10 straight m divided by straight s squared close parentheses$

$A space c y l i n d r i c a l space c a p i l l a r y space t u b e space o f space 0.2 space m m space r a d i u s space i s space m a d e space b y space j o i n i n g space t w o space c a p i l l a r i e s space T subscript 1 space a n d space T subscript 2 space o f space d i f f e r e n t space m a t e r i a l s space h a v i n g space w a t e r space c o n t a c t space a n g l e s space o f space 0 to the power of 0 space a n d space 60 to the power of 0 comma space r e s p e c t i v e l y. space T h e space c a p i l l a r y space t u b e space i s space d i p p e d space v e r t i c a l l y space i n space w a t e r space i n space t w o space d i f f e r e n t space c o n f i g u r a t i o n s comma space c a s e space I space a n d space I I space a s space s h o w n space i n space f i g u r e. space W h i c h space o f space t h e space f o l l o w i n g space o p t i o n left parenthesis s right parenthesis space i s left parenthesis a r e right parenthesis space c o r r e c t ? text (Surface tension of water end text equals 0.075 straight N divided by straight m comma text density of water end text equals 1000 kg divided by straight m cubed comma text take end text open straight g equals 10 straight m divided by straight s squared close parentheses$

PhysicsFluid-Mechanics

Fluid-Mechanics

$A space t e s t space p a r t i c l e space i s space m o v i n g space i n space a space c i r c u l a r space o r b i t space i n space t h e space g r a v i t a t i o n a l space f i e l d space p r o d u c e d space b y space a space m a s s space d e n s i t y space rho left parenthesis straight r right parenthesis equals straight K over straight r squared. space space I d e n t i f y space t h e space c o r r e c t space r e l a t i o n space b e t w e e n space t h e space r a d i u s space R space o f space t h e space p a r t i c l e apostrophe s space o r b i t space a n d space i t s space p e r i o d space T colon$

$A space t e s t space p a r t i c l e space i s space m o v i n g space i n space a space c i r c u l a r space o r b i t space i n space t h e space g r a v i t a t i o n a l space f i e l d space p r o d u c e d space b y space a space m a s s space d e n s i t y space rho left parenthesis straight r right parenthesis equals straight K over straight r squared. space space I d e n t i f y space t h e space c o r r e c t space r e l a t i o n space b e t w e e n space t h e space r a d i u s space R space o f space t h e space p a r t i c l e apostrophe s space o r b i t space a n d space i t s space p e r i o d space T colon$

PhysicsFluid-Mechanics

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