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Find the tangential and normal components of the acceleration of a point


Tangential and Normal components of the acceleration

Find the tangential and normal components of the acceleration of a point 

describing the curve y2 – 2x = 1 -2y with the uniform speed v when the 

particle is at (0, -1 ± 2).







Solution:


    The particle moving along a curve

    Tangential component = aT ­= dvdt= 0

    And

    Normal component = aN = v2ρ

    We know that roh equation

            ρ =[1+(dydx)2d2ydx2]32- - - - - - - - -(A)

    Given curve

            Y2 – 2x = 1 – 2y

    Taking derivative w.r.t     x

            2ydydx - 2 =  - 2dydx

            2ydydx + 2dydx = 2

            2dydx(y+1) = 2

            dydx(y+1) = 1

            dydx=1y+1- - - - - -(1)

    At point ( 0 , - 1  ± 2)

            dydx=1-1±2+1

            dydx=1±2

    From equation (1)

             dydx = {y+1}-1

    Taking derivative w.r.t x

            d2ydx2=  -(y+1)-21(y+1) 

            d2ydx2=  -1(y+1)-2.1(y+1) 

            d2ydx2=  -1(y+1)3 

    At point (0, -1 ± 2)

            d2ydx2=  -1(-1 ±2 +1)3       

            d2ydx2=  -1( ±2 )3 

    Now equation (A) becomes

            ρ =[1+(1±2)2-1( ±2 )3]32

            ρ =[1+(1±(2)12)2-1( ±\2 )3]32

            ρ =[1+12-1( ±\2 )3]32

            ρ =[32-1( ±2 )3]32

            ρ =[3( ±\2 )]32

            ρ =[ ±32 ]32

    Normal component = aN = v2[ ±32]32





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