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 =[ ±3√2 ]32
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