We will cover following topics

Orbits Under Central Forces

The orbit under the influence of a central force is an orbit and can be better understood by considering the polar coordinates. We observe, for orbital motion:

Orbit Trajectory

\[r=\dfrac{h^{2} / \mu}{1+e \cos \theta} \underbrace{\left(=\dfrac{a\left(1-e^{2}\right)}{1+e \cos \theta}\right)}_{\text { elliptical orbits }}\]

where \(\mu\) is the gravitational perimeter given by \(G(M+m)\).

Conservation of angular momentum

\[h=r^{2} \dot{\theta}=\vert r \times v \vert\]

Relationship between the major semi-axis and the period

\[\mu=\left(\dfrac{2 \pi}{\tau}\right)^{2} a^{3}\]

PYQs

Central Orbits

1) A particle moves with a central acceleration which varies inversely as the cube of the distance. If it is projected from an apse at a distance \(a\) from the origin with a velocity which is \(\sqrt{2}\) times the velocity for a circle of radius \(a\), then find the equation to the path.

[2016, 10M]

2) A particle moves in a plane under a force, towards a fixed centre, proportional to the distance. If the path of the particle has two apsidal distances \(a\), \(b\) (\(a>b\)), find the equation of the path.

[2015, 13M]

3) A particle moves with a central acceleration \(\mu\left(r^{5}-9 r\right)\), being projected from and apse at a distance \(\sqrt{3}\) with velocity 3\(\sqrt{(2 \mu)}\). Show that its path is the curve \(x^{4}+y^{4}=9\).

[2010, 20M]

4) A body is describing an ellipse of eccentricity \(e\) under the action of a central force directed towards a focus and when at the nearer apse, the centre of force is transferred to the other focus. Find the eccentricity of the new orbit in terms of the eccentricity of the original orbit.

[2009, 12M]

5) A particle of mass \(m\) moves under a force \(m \mu\left\{3 a u^{4}-2\left(a^{2}-b^{2}\right) u^{5}\right\}, u=1 / r, a>b, a, b\) and \(\mu (>0)\) being given constants. It is projected from an apse at a distance \(a+b\) with velocity \(\dfrac{\sqrt{\mu}}{a+b}\). Show that its orbit is given by the equation \(r=a+b \cos \theta\), where \((r, \theta)\) are the plane polar coordinates of a point.

[2008, 15M]

6) A particle \(\mathrm{P}\) moves in a plane such that it is acted on by two constant velocities \(\mathrm{u}\) and \(\mathrm{v}\) respectively along the direction \(\mathrm{O} \mathrm{X}\), and along the direction perpendicular to \(OP\), where \(\mathrm{O}\) is some fixed point, that is the origin. Show that the path traversed by \(\mathrm{P}\) is a conic section with focus at \(\mathrm{O}\) and eccentricity \(u/v\).

[2008, 15M]

7) A particle describes the curve

\[r=\dfrac{a(1+ \cos h \theta)}{(\cos h \theta-2)}\]

under a force \(F\) to the pole. Show that the law of force is \(F \propto \dfrac{1}{r^4}\).

[2003, 12M]

8) A particle describes a curve with constant velocity and its angular velocity about a given point \(O\) varies inversely as its distance from \(O\). Show that the curve is an equiangular spiral.

[2002, 15M]

9) Find the law of force to the pole when the path of a particles is the cardioids \(r=(1-\cos \theta)\) and prove that if \(F\) be the force at the apse and \(v\) the velocity there, then \(3 v^{2}=4 a F\).

[2001, 12M]

< Previous Next >