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physics

mechanics

Easy

Question

A solid cylinder of mass M and radius R rotates about its axis with angular speed ω. Its rotational kinetic energy is

$M R^{2} ω^{2}$
$\frac{1}{8} M R^{2} ω^{2}$
$\frac{1}{4} M R^{2} ω^{2}$
$\frac{1}{2} M R^{2} ω^{2}$

The correct answer is: $\frac{1}{4} M R^{2} ω^{2}$

Moment of inertia of the solid cylinder about its axis is
$I equals fraction numerator M R squared over denominator 2 end fraction$
$therefore$ Rotational kinetic energy of the cylinder is

$K equals 1 half I omega squared equals 1 half fraction numerator M R squared over denominator 2 end fraction omega squared equals 1 fourth M R squared omega squared$

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A wheel of moment of inertia $2.0 cross times 10 cubed kgm squared$ is rotating at uniform angular speed of $4 rads to the power of negative 1 end exponent$ . What is the torque required to stop it in one second.

PhysicsMechanics

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PhysicsMechanics

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PhysicsMechanics

A beam of length $l$ lies on the +x axis with its left end on the origin. A cable pulls the beam in y-direction with a force $straight F equals straight F subscript 0 left parenthesis 1 minus straight x over ell right parenthesis$ .If the axis is fixed at x = 0 then find the torque

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PhysicsMechanics

$text A couple produces end text$

$text A couple produces end text$

PhysicsMechanics

parallel

$T w o space p a r t i c l e s space A space a n d space B space a r e space s i t u a t e d space a t space a space d i s tan c e space d equals 2 space m space a p a r t. space P a r t i c l e space A space h a s space a space v e l o c i t y space o f space 10 space m space divided by space s space a t space a n space a n g l e space o f space 60 to the power of 0 space a n d space p a r t i c l e space B space h a s space a space v e l o c i t y space v space a t space a n space a n g l e 30 to the power of 0 space a s space s h o w n space i n space f i g u r e. space T h e space d i s tan c e space d space b e t w e e n space A space a n d space B space i s space c o n s tan t. space T h e space a n g u l a r space v e l o c i t y space o f space B space w i t h space r e s p e c t space t o space A space i s colon$

$T w o space p a r t i c l e s space A space a n d space B space a r e space s i t u a t e d space a t space a space d i s tan c e space d equals 2 space m space a p a r t. space P a r t i c l e space A space h a s space a space v e l o c i t y space o f space 10 space m space divided by space s space a t space a n space a n g l e space o f space 60 to the power of 0 space a n d space p a r t i c l e space B space h a s space a space v e l o c i t y space v space a t space a n space a n g l e 30 to the power of 0 space a s space s h o w n space i n space f i g u r e. space T h e space d i s tan c e space d space b e t w e e n space A space a n d space B space i s space c o n s tan t. space T h e space a n g u l a r space v e l o c i t y space o f space B space w i t h space r e s p e c t space t o space A space i s colon$

PhysicsMechanics

$A space h o o p space r o l l s space o n space a space h o r i z o n t a l space g r o u n d space w i t h o u t space s l i p p i n g space w i t h space l i n e a r space s p e e d space v. space S p e e d space o f space a space p a r t i c l e space P space o n space t h e space c i r c u m f e r e n c e space o f space t h e space h o o p space a t space a n g l e space theta space i s colon$

$A space h o o p space r o l l s space o n space a space h o r i z o n t a l space g r o u n d space w i t h o u t space s l i p p i n g space w i t h space l i n e a r space s p e e d space v. space S p e e d space o f space a space p a r t i c l e space P space o n space t h e space c i r c u m f e r e n c e space o f space t h e space h o o p space a t space a n g l e space theta space i s colon$

PhysicsMechanics

$A space d i s c space o f space r a d i u s space r space r o l l s space o n space a space h o r i z o n t a l space g r o u n d space w i t h space l i n e a r space a c c e l e r a t i o n space a space a n d space a n g u l a r space a c c e l e r a t i o n space alpha space a s space s h o w n space i n space f i g u r e. space T h e space m a g n i t u d e space o f space a c c e l e r a t i o n space o f space p o i n t space P space s h o w n space i n space f i g u r e space a t space a n space i n s tan t space w h e n space i t s space l i n e a r space v e l o c i t y space i s space v space a n d space a n g u l a r space v e l o c i t y space i s omega space w i l l space b e colon$

$A space d i s c space o f space r a d i u s space r space r o l l s space o n space a space h o r i z o n t a l space g r o u n d space w i t h space l i n e a r space a c c e l e r a t i o n space a space a n d space a n g u l a r space a c c e l e r a t i o n space alpha space a s space s h o w n space i n space f i g u r e. space T h e space m a g n i t u d e space o f space a c c e l e r a t i o n space o f space p o i n t space P space s h o w n space i n space f i g u r e space a t space a n space i n s tan t space w h e n space i t s space l i n e a r space v e l o c i t y space i s space v space a n d space a n g u l a r space v e l o c i t y space i s omega space w i l l space b e colon$

PhysicsMechanics

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$I n space f i g u r e comma space w h e e l space A space o f space r a d i u s space r subscript A equals 10 space c m space i s space c o u p l e d space b y space b e l t space B space t o space w h e e l space C space o f space r a d i u s space r subscript C equals space 25 space c m. space T h e space a n g u l a r space s p e e d space o f space w h e e l space A space i s space i n c r e a s e d space f r o m space r e s t space a t a space c o n s tan t space r a t e space o f space 1.6 space r a d space divided by space s squared. space F i n d space t h e space t i m e space n e e d e d space f o r space w h e e l space C space t o space r e a c h space a n space a n g u l a r space s p e e d space o f space 12.8 space r a d space divided by space s comma space a s s u m i n g space t h e space b e l t space d o e s space n o t space s l i p.$

$I n space f i g u r e comma space w h e e l space A space o f space r a d i u s space r subscript A equals 10 space c m space i s space c o u p l e d space b y space b e l t space B space t o space w h e e l space C space o f space r a d i u s space r subscript C equals space 25 space c m. space T h e space a n g u l a r space s p e e d space o f space w h e e l space A space i s space i n c r e a s e d space f r o m space r e s t space a t a space c o n s tan t space r a t e space o f space 1.6 space r a d space divided by space s squared. space F i n d space t h e space t i m e space n e e d e d space f o r space w h e e l space C space t o space r e a c h space a n space a n g u l a r space s p e e d space o f space 12.8 space r a d space divided by space s comma space a s s u m i n g space t h e space b e l t space d o e s space n o t space s l i p.$

PhysicsMechanics

$A space d i v e r space m a k e s space 2.5 space r e v o l u t i o n s space o n space t h e space w a y space f r o m space a space 10 minus space m space minus h i g h space p l a t f o r m space t o space t h e space w a t e r. space A s s u m i n g space z e r o space i n i t i a l space v e r t i c a l space v e l o c i t y comma space t h e space a v e r a g e space a n g u l a r space v e l o c i t y space d u r i n g space t h e space d i v e space i s$

$A space d i v e r space m a k e s space 2.5 space r e v o l u t i o n s space o n space t h e space w a y space f r o m space a space 10 minus space m space minus h i g h space p l a t f o r m space t o space t h e space w a t e r. space A s s u m i n g space z e r o space i n i t i a l space v e r t i c a l space v e l o c i t y comma space t h e space a v e r a g e space a n g u l a r space v e l o c i t y space d u r i n g space t h e space d i v e space i s$

PhysicsMechanics

$I n space p r e v i o u s space p r o b l e m comma space t h e space c a r apostrophe s space f u e l space e c o n o m y space i n space m i l e s space p e r space g a l l o n space o r space k m space divided by space L a p p e a r space t o space b e space h e t t e r$

$I n space p r e v i o u s space p r o b l e m comma space t h e space c a r apostrophe s space f u e l space e c o n o m y space i n space m i l e s space p e r space g a l l o n space o r space k m space divided by space L a p p e a r space t o space b e space h e t t e r$

PhysicsMechanics

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