A particle of mass m describes the parabola y=x^2 / 4 r0...

A particle of mass $m$ describes the parabola $y=x^{2} / 4 r_{0}$ under a central force $\mathbf{F}$ directed toward the center of force $C .$ Using Eq. ( 12.35) and Eq. $\left(12.37^{\prime}\right)$ with $\varepsilon=1,$ show that $\mathbf{F}$ is inversely proportional to the square of the distance $r$ from the particle to the center of force and that the angular momentum per unit mass $h=\sqrt{2 G M r_{0}}$

Key Concepts

Conservation of Angular Momentum

In central force problems, angular momentum is conserved because the force provides no torque about the center. This conservation law results in the angular momentum per unit mass remaining constant throughout the motion, which is critical in relating the shape of the trajectory to the initial conditions and determining orbital parameters.

Central Force

Central force refers to a force that is always directed along the line connecting the particle to a fixed center. In such systems, the force only depends on the distance from the center, leading to simplifications in the equations of motion and ensuring that the path of motion is confined to a plane.

Inverse-Square Law

The inverse-square law is a functional form where the magnitude of the force decreases as the square of the distance from the source. This type of force is characteristic of gravitational and electrostatic interactions and is the only central force law that leads to closed orbits for bound particles according to Bertrand's theorem.

Conic Section Orbits

Conic section orbits arise as solutions to the equations of motion for a particle under an inverse-square central force. Depending on the energy and angular momentum, these orbits can be ellipses, parabolas, or hyperbolas. A parabolic orbit, in particular, corresponds to the case where the eccentricity is equal to one, marking the transition between bound and unbound motion.

Transcript

00:01 Hello friends as shown in the figure a particle of mass describing the parabola y is equal to x square upon 4 or not under the central force f directed towards the center of the force c by using the equation 12 .35 and 12 .37 dies with a epsilon to be 1 we have to prove that f is inversely opposed to square of the distance and angular momentum per unit mass will be root or of 2 gm r not so from the figure you can find the value of x to be r sine theta and value of y to be r not minus r cost theta using equation 1 we can write r minus r r not cos of theta to be r square sine square theta upon 4 r0 and r minus r not cost theta will be equal to r square 1 minus cos square theta so we can write 1 minus cos square theta upon 4 r0 into r square plus cos of theta into r minus r not is equal to 0 solving this quality equation for r r r r can be written as 2 1 minus cos square theta into 1 minus cost theta plus minus root of on solving this equation it can be written as if you simplify this equation it can be written as or you may write 1 upon r is equal to assuming to be u say equation number 3 differentiating this equation with respect to theta you will get minus sign of theta upon 2 are not second derivative of if having the value cost theta upon 2 are not can be written as second derivative of u upon d theta 2 plus u is equal to 1 upon 2 2 2 are not and it can be written as f upon m h square u square so f you will get m h square u square upon 2 r not from equation 2 we can write u equal to 1 upon r so f is equal to m s square upon 2 r0 into r square so in equation 4 m r not and h are constant hence f is inversely proportional to r square.

07:23 This is the answer of one of the part by equation 1 .37 dash 1 upon r is equal to gm upon h square one plus epsilon cost theta that is equal to u.

08:09 This is given 1 so gm upon h square is equal to 1 upon and comparing with equation 3...

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