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Physics

Mechanics

Easy

Question

The magnitude of the gravitational field at distances r₁ & r₂ from the centre of a uniform sphere of radius R & mass M are F₁ & F₂ respectively then

$\frac{F_{1}}{F_{2}} = \frac{r_{1}}{r_{2}} if r_{1} < R & r_{2} < R$
$\frac{F_{1}}{F_{2}} = \frac{r_{1}^{2}}{r_{2}^{2}} if r_{1} > R & {sr}_{2} > R$
$\frac{F_{1}}{F_{2}} = \frac{r_{1}}{r_{2}} if r_{1} > R \otimes r_{2} > R$
$\frac{F_{1}}{F_{2}} = \frac{r_{1}^{2}}{r_{2}^{2}} if r_{1} < R \otimes r_{2} < R$

The correct answer is: $F subscript 1 over F subscript 2 equals r subscript 1 over r subscript 2 text if end text r subscript 1 less than R straight & r subscript 2 less than R$

$W e space k n o w space f o r space u n i f o r m space s p h e r e space w h e n space r space less than space R table attributes columnalign left columnspacing 1em rowspacing 4 pt end attributes row cell straight F equals GM over straight R cubed straight r comma space when space straight r greater than straight R end cell row cell straight F equals GM over straight r squared end cell row cell stack text So end text with _ below text when end text r subscript 1 less than R straight & subscript 0 r subscript 2 less than R end cell row cell F subscript 1 over F subscript 2 equals r subscript 1 over r subscript 2 straight & text when end text r subscript 1 greater than R circled times r subscript 2 greater than R end cell row cell F subscript 1 over F subscript 2 equals fraction numerator r subscript 2 superscript 2 over denominator r subscript 1 superscript 2 end fraction end cell end table$

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A spherical hole is made in a solid sphere of radius R. The mass of the sphere before hollowing was M. The gravitational field at the centre of the hole due to the remaining mass is

A spherical hole is made in a solid sphere of radius R. The mass of the sphere before hollowing was M. The gravitational field at the centre of the hole due to the remaining mass is

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$text The gravitational field in a region is end text 10 straight N divided by kg left parenthesis i with not stretchy hat on top minus j with not stretchy hat on top right parenthesis. text The work done by end text g r a v i t a t i o n a l space f o r c e space t o space s h i f t space s l o w l y space a space p a r t i c l e space o f space m a s s space 1 space k g space f r o m space p o i n t space left parenthesis 1 m comma space 1 m right parenthesis space t o space a space p o i n t space left parenthesis 2 m comma space minus 2 m right parenthesis space i s space space space space space$

$text The gravitational field in a region is end text 10 straight N divided by kg left parenthesis i with not stretchy hat on top minus j with not stretchy hat on top right parenthesis. text The work done by end text g r a v i t a t i o n a l space f o r c e space t o space s h i f t space s l o w l y space a space p a r t i c l e space o f space m a s s space 1 space k g space f r o m space p o i n t space left parenthesis 1 m comma space 1 m right parenthesis space t o space a space p o i n t space left parenthesis 2 m comma space minus 2 m right parenthesis space i s space space space space space$

PhysicsMechanics

There are two bodies of masses 100 kg and 10000 kg separated by a distance 1 m. At what distance from the smaller body, the intensity of gravitational field will be zero?

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A mass m is placed in the cavity inside a uniform hollow sphere of mass M as shown in the figure. What is the gravitational force on the mass m?

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$T w o space l e a d space s p h e r e s space o f space s a m e space r a d i u s space a r e space i n space c o n t a c t space w i t h space e a c h space o t h e r. space T h e space g r a v i t a t i o n a l space f o r c e space o f space a t t r a c t i o n space b e t w e e n space t h e m space i s space F. space I f space t w o space l e a d space s p h e r e s space o f space d o u b l e space t h e space p r e v i o u s space r a d i u s space a r e space i n space c o n t a c t space w i t h space e a c h space o t h e r comma space t h e space g r a v i t a t i o n a l space f o r c e space o f space a t t r a c t i o n space b e t w e e n space t h e m space w i l l space b e$

$T w o space l e a d space s p h e r e s space o f space s a m e space r a d i u s space a r e space i n space c o n t a c t space w i t h space e a c h space o t h e r. space T h e space g r a v i t a t i o n a l space f o r c e space o f space a t t r a c t i o n space b e t w e e n space t h e m space i s space F. space I f space t w o space l e a d space s p h e r e s space o f space d o u b l e space t h e space p r e v i o u s space r a d i u s space a r e space i n space c o n t a c t space w i t h space e a c h space o t h e r comma space t h e space g r a v i t a t i o n a l space f o r c e space o f space a t t r a c t i o n space b e t w e e n space t h e m space w i l l space b e$

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$T w o space m e t a l space s p h e r e s space e a c h space o f space r a d i u s space ‘ r ’ space a r e space k e p t space i n space c o n t a c t space w i t h space e a c h space o t h e r. space I f space d space i s space t h e space d e n s i t y space o f space t h e space m a t e r i a l space o f space t h e space s p h e r e comma space t h e n space t h e space g r a v i t a t i o n a l space f o r c e space b e t w e e n space t h o s e space s p h e r e s space i s space p r o p o r t i o n a l space t o$

$T w o space m e t a l space s p h e r e s space e a c h space o f space r a d i u s space ‘ r ’ space a r e space k e p t space i n space c o n t a c t space w i t h space e a c h space o t h e r. space I f space d space i s space t h e space d e n s i t y space o f space t h e space m a t e r i a l space o f space t h e space s p h e r e comma space t h e n space t h e space g r a v i t a t i o n a l space f o r c e space b e t w e e n space t h o s e space s p h e r e s space i s space p r o p o r t i o n a l space t o$

PhysicsMechanics

$T w o space p a r t i c l e s space e a c h space o f space m a s s space ‘ m ’ space a r e space p l a c e d space a t space A space a n d space C space a r e space s u c h space A B equals B C equals L. space T h e space g r a v i t a t i o n a l space f o r c e space o n space t h e space t h i r d space p a r t i c l e space p l a c e d space a t space D space a t space a space d i s tan c e space L space o n space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space t h e space l i n e space A C space i s$

$T w o space p a r t i c l e s space e a c h space o f space m a s s space ‘ m ’ space a r e space p l a c e d space a t space A space a n d space C space a r e space s u c h space A B equals B C equals L. space T h e space g r a v i t a t i o n a l space f o r c e space o n space t h e space t h i r d space p a r t i c l e space p l a c e d space a t space D space a t space a space d i s tan c e space L space o n space t h e space p e r p e n d i c u l a r space b i s e c t o r space o f space t h e space l i n e space A C space i s$

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$T h r e e space i d e n t i c a l space p a r t i c l e s space e a c h space o f space m a s s space “ m ” space a r e space a r r a n g e d space a t space t h e space c o r n e r s space o f space a n space e q u i l a t e r a l space t r i a n g l e space o f space s i d e space “ L ”. space I f space t h e y space a r e space t o space b e space i n space e q u i l i b r i u m comma space t h e space s p e e d space w i t h space w h i c h space t h e y space m u s t space r e v o l v e space u n d e r space t h e space i n f l u e n c e space o f space o n e space a n o t h e r ’ s space g r a v i t y space i n space a space c i r c u l a r space o r b i t space c i r c u m s c r i b i n g space t h e space t r i a n g l e space i s$

$T h r e e space i d e n t i c a l space p a r t i c l e s space e a c h space o f space m a s s space “ m ” space a r e space a r r a n g e d space a t space t h e space c o r n e r s space o f space a n space e q u i l a t e r a l space t r i a n g l e space o f space s i d e space “ L ”. space I f space t h e y space a r e space t o space b e space i n space e q u i l i b r i u m comma space t h e space s p e e d space w i t h space w h i c h space t h e y space m u s t space r e v o l v e space u n d e r space t h e space i n f l u e n c e space o f space o n e space a n o t h e r ’ s space g r a v i t y space i n space a space c i r c u l a r space o r b i t space c i r c u m s c r i b i n g space t h e space t r i a n g l e space i s$

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$I f space t h e space d i s tan c e space b e t w e e n space t h e space s u n space a n d space t h e space e a r t h space i s space i n c r e a s e d space b y space t h r e e space t i m e s comma space t h e n space t h e space g r a v i t a t i o n a l space f o r c e space b e t w e e n space t h e space t w o space w i l l$

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PhysicsMechanics

$I f space t h e space m a s s space o f space o n e space p a r t i c l e space i s space i n c r e a s e d space b y space 50 percent sign space a n d space t h e space m a s s space o f space a n o t h e r space p a r t i c l e space i s space d e c r e a s e d space b y space 50 space percent sign comma space t h e space f o r c e space b e t w e e n space t h e m$

$I f space t h e space m a s s space o f space o n e space p a r t i c l e space i s space i n c r e a s e d space b y space 50 percent sign space a n d space t h e space m a s s space o f space a n o t h e r space p a r t i c l e space i s space d e c r e a s e d space b y space 50 space percent sign comma space t h e space f o r c e space b e t w e e n space t h e m$

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$T w o space m a s s e s space ‘ M ’ space a n d space ‘ 4 M ’ space a r e space a t space a space d i s tan c e space ‘ r ’ space a p a r t space o n space t h e space l i n e space j o i n i n g space t h e m comma space ‘ P ’ space i s p o i n t space w h e r e space t h e space r e s u l tan t space g r a v i t a t i o n a l space f o r c e space i s space z e r o space left parenthesis s u c h space a space p o i n t space i s space c a l l e d space a s space n u l l space p o i n t right parenthesis. space T h e space d i s tan c e space o f space ‘ P ’ space f r o m space t h e space m a s s space ‘ M ’ space i s$

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PhysicsMechanics

$I f space t h r e e space p a r t i c l e s comma space e a c h space o f space m a s s space M space a r e space p l a c e d space a t space t h e space t h r e e space c o r n e r s space o f space a n space e q u i l a t e r a l space t r i a n g l e space o f space s i d e space a comma space t h e space f o r c e space e x e r t e d space b y space t h i s space s y s t e m space o n space a n o t h e r space p a r t i c l e space o f space m a s s space M space p l a c e d space left parenthesis i right parenthesis space a t space t h e space m i d p o i n t space o f space a space s i d e space a n d space left parenthesis i i right parenthesis space a t space t h e space c e n t r e space o f space t h e space t r i a n g l e space a r e comma space r e s p e c t i v e l y.$

$I f space t h r e e space p a r t i c l e s comma space e a c h space o f space m a s s space M space a r e space p l a c e d space a t space t h e space t h r e e space c o r n e r s space o f space a n space e q u i l a t e r a l space t r i a n g l e space o f space s i d e space a comma space t h e space f o r c e space e x e r t e d space b y space t h i s space s y s t e m space o n space a n o t h e r space p a r t i c l e space o f space m a s s space M space p l a c e d space left parenthesis i right parenthesis space a t space t h e space m i d p o i n t space o f space a space s i d e space a n d space left parenthesis i i right parenthesis space a t space t h e space c e n t r e space o f space t h e space t r i a n g l e space a r e comma space r e s p e c t i v e l y.$

PhysicsMechanics

$G r a v i t a t i o n a l space f o r c e space b e t w e e n space t w o space p o i n t space m a s s e s space m space a n d space M space s e p a r a t e d space b y space a space d i s tan c e space r space i s space F. space N o w space i f space a space p o i n t space m a s s space 3 m space i s space p l a c e d space n e x t space t o space m comma space t h e space t o t a l space f o r c e space o n space M space w i l l space b e$

$G r a v i t a t i o n a l space f o r c e space b e t w e e n space t w o space p o i n t space m a s s e s space m space a n d space M space s e p a r a t e d space b y space a space d i s tan c e space r space i s space F. space N o w space i f space a space p o i n t space m a s s space 3 m space i s space p l a c e d space n e x t space t o space m comma space t h e space t o t a l space f o r c e space o n space M space w i l l space b e$

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