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Physics

Mechanics

Easy

Question

A wire of uniform cross-section is bent in the shape shown in figure. The co-ordinate of the centre of mass of each side are shown in (figure). The coordinates of the centre of mass of the system are

$(\frac{15 ℓ}{14}, \frac{6 ℓ}{7})$
$(\frac{15 ℓ}{14}, ℓ)$
$(ℓ, \frac{ℓ}{2})$
$(ℓ, ℓ)$

The correct answer is: $open parentheses fraction numerator 15 ell over denominator 14 end fraction comma fraction numerator 6 ell over denominator 7 end fraction close parentheses$

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A weightless rod of length 2l carries two equal masses ‘m’, one tied at lower end A and the other at the middle of the rod at B. The rod can rotate in vertical plane about a fixed horizontal axis passing through C. The rod is released from rest in horizontal position. The speed of the mass B at the instant rod, become vertical is:

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$A space m a n space left parenthesis m a s s space equals 50 space k g right parenthesis space a n d space h i s space s o n space left parenthesis m a s s space equals 20 space k g right parenthesis space a r e space s tan d i n g space o n space a space f r i c t i o n l e s s space s u r f a c e space f a c i n g space e a c h space o t h e r. space T h e space m a n space p u s h e s space h i s space s o n space s o space t h a t space h e space s t a r t s space m o v i n g space a t space a space s p e e d space o f space 0.70 space m s to the power of negative 1 end exponent space w i t h space r e s p e c t space t o space t h e space m a n. space T h e space s p e e d space o f space t h e space m a n space w i t h space r e s p e c t space t o space t h e space s u r f a c e space i s space colon$

$A space m a n space left parenthesis m a s s space equals 50 space k g right parenthesis space a n d space h i s space s o n space left parenthesis m a s s space equals 20 space k g right parenthesis space a r e space s tan d i n g space o n space a space f r i c t i o n l e s s space s u r f a c e space f a c i n g space e a c h space o t h e r. space T h e space m a n space p u s h e s space h i s space s o n space s o space t h a t space h e space s t a r t s space m o v i n g space a t space a space s p e e d space o f space 0.70 space m s to the power of negative 1 end exponent space w i t h space r e s p e c t space t o space t h e space m a n. space T h e space s p e e d space o f space t h e space m a n space w i t h space r e s p e c t space t o space t h e space s u r f a c e space i s space colon$

PhysicsMechanics

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$T h e space t r a j e c t o r y space o f space a space p r o j e c t i l e space n e a r space t h e space s u r f a c e space o f space t h e space e a r t h space i s space g i v e n space a s space y equals 2 space x minus 9 space x squared. space I f space i t space w e r e space l a u n c h e d space a t space a n space a n g l e space theta subscript 0 space w i t h space s p e e d space v subscript 0 space t h e n space left parenthesis g equals 10 space m space s to the power of negative 2 end exponent right parenthesis colon$

$T h e space t r a j e c t o r y space o f space a space p r o j e c t i l e space n e a r space t h e space s u r f a c e space o f space t h e space e a r t h space i s space g i v e n space a s space y equals 2 space x minus 9 space x squared. space I f space i t space w e r e space l a u n c h e d space a t space a n space a n g l e space theta subscript 0 space w i t h space s p e e d space v subscript 0 space t h e n space left parenthesis g equals 10 space m space s to the power of negative 2 end exponent right parenthesis colon$

PhysicsMechanics

$A space u n i f o r m space r o d space o f space l e n g t h space ell space space i s space b e i n g space r o t a t e d space i n space a space h o r i z o n t a l space p l a n e space w i t h space a space c o n s tan t space a n g u l a r space s p e e d space a b o u t space a n space a x i s space p a s sin g space t h r o u g h space o n e space o f space i t s space e n d s. space I f space t h e space t e n s i o n space g e n e r a t e d space i n space t h e space r o d space d u e space t o space r o t a t i o n space i s space T left parenthesis x right parenthesis space a t space a space d i s tan c e space x space f r o m space t h e space a x i s comma space t h e n space w h i c h space o f space t h e space f o l l o w i n g space g r a p h s space d e p i c t s space i t space m o s t space c l o s e l y ?$

$A space u n i f o r m space r o d space o f space l e n g t h space ell space space i s space b e i n g space r o t a t e d space i n space a space h o r i z o n t a l space p l a n e space w i t h space a space c o n s tan t space a n g u l a r space s p e e d space a b o u t space a n space a x i s space p a s sin g space t h r o u g h space o n e space o f space i t s space e n d s. space I f space t h e space t e n s i o n space g e n e r a t e d space i n space t h e space r o d space d u e space t o space r o t a t i o n space i s space T left parenthesis x right parenthesis space a t space a space d i s tan c e space x space f r o m space t h e space a x i s comma space t h e n space w h i c h space o f space t h e space f o l l o w i n g space g r a p h s space d e p i c t s space i t space m o s t space c l o s e l y ?$

PhysicsMechanics

$A space c i r c u l a r space d i s c space o f space r a d i u s space b space h a s space a space h o l e space o f space r a d i u s space a space a t space i t s space c e n t r e space left parenthesis s e e space f i g u r e right parenthesis. space I f space t h e space m a s s space p e r space u n i t space a r e a space o f space t h e space d i s c space v a r i e s space a s space open parentheses sigma subscript 0 over r close parentheses comma space t h e n space t h e space r a d i u s space o f space g y r a t i o n space o f space t h e space d i s c space a b o u t space i t s space a x i s space p a s sin g space t h r o u g h space t h e space c e n t r e space i s colon$

$A space c i r c u l a r space d i s c space o f space r a d i u s space b space h a s space a space h o l e space o f space r a d i u s space a space a t space i t s space c e n t r e space left parenthesis s e e space f i g u r e right parenthesis. space I f space t h e space m a s s space p e r space u n i t space a r e a space o f space t h e space d i s c space v a r i e s space a s space open parentheses sigma subscript 0 over r close parentheses comma space t h e n space t h e space r a d i u s space o f space g y r a t i o n space o f space t h e space d i s c space a b o u t space i t s space a x i s space p a s sin g space t h r o u g h space t h e space c e n t r e space i s colon$

PhysicsMechanics

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$A space s h e l l space i s space f i r e d space f r o m space a space f i x e d space a r t i l l e r y space g u n space w i t h space a n space i n i t i a l space s p e e d space u space s u c h space t h a t space i t space h i t s space t h e space t a r g e t space o n space t h e space g r o u n d space a t space a space d i s tan c e space R space f r o m space i t. space I f space t subscript 1 space a n d space t subscript 2 space a r e space t h e space v a l u e s space o f space t h e space t i m e space t a k e n space b y space i t space t o space h i t space t h e space t a r g e t space i n space t w o space p o s s i b l e space w a y s comma space t h e space p r o d u c t space t subscript 1 space t subscript 2 space i s space equals$

$A space s h e l l space i s space f i r e d space f r o m space a space f i x e d space a r t i l l e r y space g u n space w i t h space a n space i n i t i a l space s p e e d space u space s u c h space t h a t space i t space h i t s space t h e space t a r g e t space o n space t h e space g r o u n d space a t space a space d i s tan c e space R space f r o m space i t. space I f space t subscript 1 space a n d space t subscript 2 space a r e space t h e space v a l u e s space o f space t h e space t i m e space t a k e n space b y space i t space t o space h i t space t h e space t a r g e t space i n space t w o space p o s s i b l e space w a y s comma space t h e space p r o d u c t space t subscript 1 space t subscript 2 space i s space equals$

PhysicsMechanics

$A space p l a n e space i s space i n c l i n e d space a t space a n space a n g l e space alpha equals 30 degree space w i t h space r e s p e c t space t o space t h e space h o r i z o n t a l. space A space p a r t i c l e space i s space p r o j e c t e d space w i t h space a space s p e e d space u equals space 2 space m s space to the power of negative 1 end exponent comma space f r o m space t h e space b a s e space o f space t h e space p l a n e comma space m a k i n g space a n space a n g l e space theta equals 15 degree space w i t h space r e s p e c t space t o space t h e space p l a n e space a s space s h o w n space i n space t h e space f i g u r e. space t h e space d i s tan c e space f r o m space t h e space b a s e comma space a t space w h i c h space t h e space p a r t i c l e space h i t s space t h e space p l a n e space i s space c l o s e space t o space colon space left parenthesis T a k e space g equals 10 m s to the power of negative 2 end exponent right parenthesis$

$A space p l a n e space i s space i n c l i n e d space a t space a n space a n g l e space alpha equals 30 degree space w i t h space r e s p e c t space t o space t h e space h o r i z o n t a l. space A space p a r t i c l e space i s space p r o j e c t e d space w i t h space a space s p e e d space u equals space 2 space m s space to the power of negative 1 end exponent comma space f r o m space t h e space b a s e space o f space t h e space p l a n e comma space m a k i n g space a n space a n g l e space theta equals 15 degree space w i t h space r e s p e c t space t o space t h e space p l a n e space a s space s h o w n space i n space t h e space f i g u r e. space t h e space d i s tan c e space f r o m space t h e space b a s e comma space a t space w h i c h space t h e space p a r t i c l e space h i t s space t h e space p l a n e space i s space c l o s e space t o space colon space left parenthesis T a k e space g equals 10 m s to the power of negative 2 end exponent right parenthesis$

PhysicsMechanics

$T h e space t i m e space d e p e n d e n c e space o f space t h e space p o s i t i o n space o f space a space p a r t i c l e space o f space m a s s space m equals 2 space i s space g i v e n space b y space r with rightwards arrow on top left parenthesis t right parenthesis equals 2 t i with hat on top minus 3 t squared j with hat on top. space I t s space a n g u l a r space m o m e n t u m comma space w i t h space r e s p e c t space t o space t h e space o r i g i n comma space a t space t i m e space t equals 2 space i s$

$T h e space t i m e space d e p e n d e n c e space o f space t h e space p o s i t i o n space o f space a space p a r t i c l e space o f space m a s s space m equals 2 space i s space g i v e n space b y space r with rightwards arrow on top left parenthesis t right parenthesis equals 2 t i with hat on top minus 3 t squared j with hat on top. space I t s space a n g u l a r space m o m e n t u m comma space w i t h space r e s p e c t space t o space t h e space o r i g i n comma space a t space t i m e space t equals 2 space i s$

PhysicsMechanics

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