Paper 4, Section II, B

Dynamics and Relativity | Part IA, 2012

Let (r,θ)(r, \theta) be polar coordinates in the plane. A particle of mass mm moves in the plane under an attractive force of mf(r)m f(r) towards the origin OO. You may assume that the acceleration a is given by

a=(r¨rθ˙2)r^+1rddt(r2θ˙)θ^\mathbf{a}=\left(\ddot{r}-r \dot{\theta}^{2}\right) \hat{\mathbf{r}}+\frac{1}{r} \frac{d}{d t}\left(r^{2} \dot{\theta}\right) \hat{\theta}

where r^\hat{\mathbf{r}} and θ^\hat{\theta} are the unit vectors in the directions of increasing rr and θ\theta respectively, and the dot denotes d/dtd / d t.

(a) Show that l=r2θ˙l=r^{2} \dot{\theta} is a constant of the motion. Introducing u=1/ru=1 / r show that r˙=ldudθ\dot{r}=-l \frac{d u}{d \theta} and derive the geometric orbit equation

l2u2(d2udθ2+u)=f(1u)l^{2} u^{2}\left(\frac{d^{2} u}{d \theta^{2}}+u\right)=f\left(\frac{1}{u}\right)

(b) Suppose now that

f(r)=3r+9r3f(r)=\frac{3 r+9}{r^{3}}

and that initially the particle is at distance r0=1r_{0}=1 from OO, moving with speed v0=4v_{0}=4 in a direction making angle π/3\pi / 3 with the radial vector pointing towards OO.

Show that l=23l=2 \sqrt{3} and find uu as a function of θ\theta. Hence or otherwise show that the particle returns to its original position after one revolution about OO and then flies off to infinity.

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