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1.17 Application to motion in an ellipse.

Suppose a particle moves in an ellipse whose semi-axes are a and b in such a manner that it obeys the law of areas with respect to the center of the ellipse as origin ; it is required to find the components of acceleration along and perpendicular to the radius vector. The equation of the ellipse may be written in the parametric form

x = ac osφ , y = b sin φ
(1.30)

for, if φ is eliminated, the ordinary equation

2 2 x--+ y-- a2 b2

is found. It follows from (1.30) that

dx- d-φ dy- dφ- dt = -a sin φ dt , dt = b cos φdt
(1.31)

On substituting (1.30) and (1.31) in the expression for the law of areas,

dy dx x ---- y ---= c dt dt

it is found that

dφ- -c- dt = ab = c1

The integral of this, equation is

φ = c1t + c2

and if φ = 0 when t = 0, then φ = c1t.

On substituting the final expression for φ in (1.30), it is found that

( | d 2x |{ ---2 = -c 21a cosφ = -c 21x dt2 ||( d-y- 2 2 dt2 = -c1b sinφ = -c 1y

If these values of the derivatives are substituted in (1.20) the components of acceleration along and perpendicular to the radius vector are found to be

( { αr = -c 21r ( αθ = 0

1.17.1 Problems on velocity and acceleration.

1. A particle moves with uniform speed along a helix traced on a circular cylinder whose radius is R ; find the components of velocity and acceleration parallel to the x, y and z axes. Answer :

x = R cosω , y = R sinω , z = hω

( { vx = -Rc sin(ct), vy = +Rc co s(ct) vz = hc 2 2 ( αz = -Rc co s(ct), αy = -Rc sin (ct) αz = 0

2. A particle moves in the ellipse whose parameter and eccentricity are p and e with uniform angular speed with respect to one of the foci as origin ; find the components of velocity and acceleration along and perpendicular to the radius vector and parallel to the x and y axes in terms of the radius vector and the time. Answer :

( ec ||| vr = --r2sin(ct), vθ = rc ||| p |||| -ec 2 ||| vx = -cr sin(ct) + 2p r sin(2ct) ||| ec |||| vy = cr cos(ct)+ --r2sin2(2ct) ||| p ||{ ec2 2 2e2c2 3 2 2 αr = ---r cos(ct)+ ---2-r sin (ct) - c r || p p ||| 2ec2-2 |||| αθ = p r sin(ct) ||| 2 2 2 2 ||| 2 ec-- 2 3ec--2 2 2e-c- 2 2 |||| αx = -c rco s(ct)+ p r - p r sin (ct)+ p2 r sin (ct)co s(ct) ||| 2 2 2 ||| α = -c 2rsin(ct) + 3ec-r2 sin (2ct)+ 2e-c-r3sin3(ct) ( y 2p p2

3. A particle moves in an ellipse in such a manner that it obeys the law of areas with respect to one of the foci as an origin ; it is required to find the components of velocity and acceleratio i along and perpendicular to the radius vector and parallel to the axes in terms of the coordinates. Answer :

( ||| vr = eA-sin θ vθ = 2A- ||| p r |||| eA A sin θ eA A cos θ ||{ vx = ---sin 2θ - ------- vy = ---sin2 θ + ------- 2p r p r || A 2 1 ||| αr = - -p-r2 αθ = 0 ||| |||| A-2cos-θ A-2sinθ- ( αx == - p r2 αy = - p r2

4. The accelerations along the x and y axes are the derivatives of the velocities along these axes ; why are not the accelerations along and perpendicular to the radius vector given by the derivatives of the velocities in these respective directions ? Find the accelerations along axes rotating with the angular velocity unity in terms of the accelerations with respect to fixed axes.

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