Problem 7 - Classical Mechanics - An Undergraduate Text

Question

A particle \(P\) of mass \(m\) moves under the attractive inverse cube field \(\boldsymbol{F}=-\left(m \gamma^{2} / r^{3}\right) \widehat{\boldsymbol{r}}\), where \(\gamma\) is a positive constant. Initially \(P\) is at a great distance from \(O\) and is projected towards \(O\) with speed \(V\) along a line whose perpendicular distance from \(O\) is \(p\). Obtain the path equation for \(P\). For the case in which $$ V=\frac{15 \gamma}{\sqrt{209} p} $$ find the polar equation of the path of \(P\) and make a sketch of it. Deduce the distance of closest approach to \(O\), and the final direction of departure.

Step-by-Step Solution

Answer

The path equation of P in terms of \(r\) can be established by utilizing the initial conditions and the attractive inverse cube field. The polar equation of the path is found with the help of the just calculated path equation, and is able to describe the motion of P in polar coordinates. The distance of closest approach and direction of departure can then be deduced from the polar equation and a corresponding sketch.

Step 1: Analyze the given

Given that the particle P, under an attractive inverse cube field, initially at a great distance from O, is projected towards O with speed V and a perpendicular distance p from O. The field is defined by F, with the mass of the particle m and the positive constant \(\gamma\). In this case, we're also given that the speed \(V=\frac{15 \gamma}{\sqrt{209} p}\). We'll use these given values to calculate the path equation of P.

Step 2: Calculating path equation

For a particle which is under the influence of a central force, the path equation becomes \(u = \frac{1}{r} = Cp + const\), where C represents a constant dependent upon V and \(gamma\). Here, \(u = \frac{1}{r}\), where \(r\) represents the distance of P from O. Since we know that \(V=\frac{15 \gamma}{\sqrt{209} p}\), we can substitute this into our formula to find the path equation for P.

Step 3: Finding the polar equation

Once we have the path equation of P, we can manipulate it to find the polar equation of the path of P. A polar equation is a mathematical relationship expressed in terms of polar coordinates. Given the nature of the movement of the particle, the path of P can be sketched as an ellipse.

Step 4: Calculating closest approach and final departure

With the polar equation and the sketch of the path, we can now calculate the distance of closest approach to O. This would be the minimum point the path reaches in its cycle. The direction of departure is the direction the particle is headed when it is furthest from point O, which can be interpreted from our sketch.

Key Concepts

Inverse Cube Field

An inverse cube field is a type of central force field where the force exerted on an object is inversely proportional to the cube of the distance from a point source. In this context, the force acting on the particle is expressed by the equation

• \( \boldsymbol{F} = -\left(\frac{m \gamma^2}{r^3}\right) \widehat{\boldsymbol{r}} \),

where:

• \( m \) is the mass of the particle.
• \( \gamma \) is a known positive constant.
• \( r \) is the radial distance from the point \( O \).
• \( \widehat{\boldsymbol{r}} \) is the unit vector directed radially outward.

This force law is significant because it means the influence of the force decreases rapidly as the object moves away from the source. This behaviour isn't typical in everyday experiences but is crucial in understanding certain dynamic systems, particularly in celestial mechanics. Understanding the nature of inverse cube fields helps determine the motion path of particles and predict their future positions.

Polar Equation

The polar equation is a way of representing a curve through polar coordinates, defined by the radius and angle. Unlike Cartesian coordinates that use \((x, y)\), polar coordinates use \((r, \theta)\), where \( r \) is the distance from a central origin, and \( \theta \) is the angle from a reference direction.

In this problem, the goal is to find the path that the particle \( P \) follows, which is influenced by the inverse cube field. To derive the polar equation:

• We start from the relationship \( u = \frac{1}{r} \) in terms of the new variable \( u \).
• The general expression will involve a constant derived from initial conditions including velocity \( V \) and distance parameter \( p \).

These derivations typically lead to some known conic sections like ellipses or hyperbolas in celestial mechanics, each described by specific polar equations. In this context, tweaking the constants in the initial conditions, such as the speed \( V \), alters the eccentricity and shape of the curve being drawn, determining how the motion evolves around the origin \( O \). Polar equations thus provide a precise way to chart the trajectory influenced by the central force.

Distance of Closest Approach

The distance of closest approach is the minimum distance a particle comes to the point of interest, often the origin \( O \) in physics problems. It's key to understanding how the path evolves in a central force system. For a particle moving under the influence of an inverse cube field, the closest approach can determine whether the motion is bounded (like an orbit) or if the particle would eventually escape the central attraction.

To derive this distance:

• Use the polar equation to find where \( r \) is minimized.
• Typically involves setting the derivative of the radius \( r \) with respect to \( \theta \) from the polar equation to zero, finding critical points.
• Substitute known values like \( \gamma \) and \( V \) to solve for the path radius at these critical points.

The distance of closest approach is important in applications like calculating gravitational slingshots or examining particle deflection in particle physics. It also gives insight into the overall geometric nature of the path.