If ṙ= radial velocity of the particle then the total energy...

If $\dot{r}=$ radial velocity of the particle then the total energy of the particle at any instant is $$ \frac{1}{2} m \dot{r}^{2}+\frac{M^{2}}{2 m r^{2}}+k r^{2}=E $$ where the second term is the kinetic energy of angular motion about the centre $O$. Then the extreme values of $r$ are determined by $\dot{r}=0$ and solving the resulting quadratic equation $$ k\left(r^{2}\right)^{2}-E r^{2}+\frac{M^{2}}{2 m}=0 $$ we get $$ r^{2}=\frac{E \pm \sqrt{E^{2}-\frac{2 M^{2} k}{m}}}{2 k} $$ From this we see that $$ E=k\left(r_{1}^{2}+r_{2}^{2}\right) $$ where $r_{1}$ is the minimum distance from $\mathrm{O}$ and $r_{2}$ is the maximum distance. Then $$ \frac{1}{2} m v_{2}^{2}+2 k r_{2}^{2}=k\left(r_{1}^{2}+r_{2}^{2}\right) $$ Hence, $$ m=\frac{2 k r^{2}}{v_{2}^{2}} $$ -Note : Eq. (1) can be derived from the standard expression for kinetic energy and angular momentum in plane poler coordinates : $$ T=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2} $$ $M=$ angular momentum $=m r^{2} \dot{\theta}$

If $\dot{r}=$ radial velocity of the particle then the total energy of the particle at any instant is
$$
\frac{1}{2} m \dot{r}^{2}+\frac{M^{2}}{2 m r^{2}}+k r^{2}=E
$$
where the second term is the kinetic energy of angular motion about the centre $O$. Then
the extreme values of $r$ are determined by $\dot{r}=0$ and solving the resulting quadratic equation
$$
k\left(r^{2}\right)^{2}-E r^{2}+\frac{M^{2}}{2 m}=0
$$
we get
$$
r^{2}=\frac{E \pm \sqrt{E^{2}-\frac{2 M^{2} k}{m}}}{2 k}
$$
From this we see that
$$
E=k\left(r_{1}^{2}+r_{2}^{2}\right)
$$
where $r_{1}$ is the minimum distance from $\mathrm{O}$ and $r_{2}$ is the maximum distance. Then
$$
\frac{1}{2} m v_{2}^{2}+2 k r_{2}^{2}=k\left(r_{1}^{2}+r_{2}^{2}\right)
$$
Hence,
$$
m=\frac{2 k r^{2}}{v_{2}^{2}}
$$
-Note : Eq. (1) can be derived from the standard expression for kinetic energy and angular momentum in plane poler coordinates :
$$
T=\frac{1}{2} m \dot{r}^{2}+\frac{1}{2} m r^{2} \dot{\theta}^{2}
$$
$M=$ angular momentum $=m r^{2} \dot{\theta}$

Key Concepts

Angular Momentum Conservation

Angular momentum conservation is a key principle in systems with central forces, as the absence of external torques about the center ensures that the angular momentum remains constant. This conservation law plays a critical role in reducing the complexity of the equations of motion and in deriving the effective potential.

Conservation of Energy

In mechanics, conservation of energy is a fundamental principle stating that the total energy of an isolated system remains constant over time. This includes both kinetic and potential energy contributions, and it is a powerful tool for analyzing the motion of particles under various forces.

Central Force Motion

Central force motion refers to problems where the force acting on a particle is always directed towards or away from a fixed point (the center) and its magnitude depends only on the distance from that point. This setup simplifies the dynamics due to the symmetry and leads to conserved quantities such as angular momentum.

Effective Potential Energy

The concept of effective potential energy is used to combine the actual potential energy with the contribution from the particle's angular motion, typically represented as a centrifugal term. This effective potential helps in analyzing the radial motion separately by incorporating the effects of angular momentum.

Polar Coordinates in Mechanics

Polar coordinates are particularly useful in analyzing problems with central forces since they naturally separate the dynamics into radial and angular components. The kinetic energy can be expressed as the sum of a radial part and an angular part, with the latter being directly related to the conservation of angular momentum.

Turning Points in Radial Motion

Turning points are the positions where the radial component of velocity becomes zero, indicating the extrema of the radial motion (i.e., the closest and farthest distances from the center). Analyzing these points through the energy conservation equation helps determine the range of motion of the particle.

Transcript

Please give Ace some feedback