Problem 33 - Master Resource Book in JEE Main Physics

Question

A planet of mass \(m\) is revolving round the sun (of mass \(m_{s}\) ) in an elliptical orbit. If \(\mathbf{v}\) is the velocity of the planet when its position vector from sun \(r\) then if the planet rotates in counter clockwise direction then areal velocity has direction (a) given by "Right Hand Thumb Rule" (b) given by "Left Hand Thumb Rule" (c) normal to the plane of orbit upwrads (d) normal to the plane of orbit downwards

Step-by-Step Solution

Answer

(c) normal to the plane of orbit upwards.

Step 1: Understanding Areal Velocity

Areal velocity is defined as the rate at which area is swept out by the position vector of an object in orbit. When dealing with orbits in a plane, this is often represented by the cross product of the position vector \( \mathbf{r} \) and the velocity vector \( \mathbf{v} \). The areal velocity is perpendicular to the plane of rotation, following the right-hand rule for cross products.

Step 2: Applying the Right-Hand Rule

To determine the direction of the areal velocity, use the right-hand rule for the cross product \( \mathbf{r} \times \mathbf{v} \). When you point your right-hand fingers in the direction of \( \mathbf{r} \) and curl them towards \( \mathbf{v} \) (the direction of rotation), your thumb will point in the direction of the areal velocity.

Step 3: Direction of Areal Velocity in Counterclockwise Rotation

For a counterclockwise rotation in the plane of orbit, using the right-hand rule where the fingers follow the position vector to the velocity direction (counterclockwise), the thumb points outwards, meaning it is normal to the plane of the orbit upwards (away from the plane). This is the direction of the areal velocity.

Key Concepts

Right-Hand Rule

When dealing with vectors and rotational directions, the Right-Hand Rule is a quick and straightforward method to determine the orientation of the resulting vector from a cross product. Imagine you are performing a cross product of two vectors, such as the position vector \( \mathbf{r} \) and the velocity vector \( \mathbf{v} \). To apply the rule, extend your right hand and point your fingers in the direction of the first vector, \( \mathbf{r} \). Then, curl your fingers towards the second vector, \( \mathbf{v} \). Here are the steps simplified:

• Point your fingers in the direction of the first vector (\( \mathbf{r} \)).
• Curl your fingers towards the second vector (\( \mathbf{v} \)).
• Your extended thumb indicates the direction of the resultant vector.

This method is particularly useful in determining the direction of the areal velocity in orbital mechanics. For an object moving in an elliptical orbit in a counterclockwise direction, the areal velocity will point upwards, normal to the plane of motion.

Elliptical Orbit

An elliptical orbit is a closed, oval-shaped path through which a celestial object travels around a focal point. Typically, one of these foci is occupied by the sun or another massive body, like a planet. Elliptical orbits are governed by Kepler's laws of planetary motion, which help explain the movement of planets and satellites. In such orbits:

• Planets sweep out equal areas in equal times, which relates directly to areal velocity.
• The speed of a planet varies, increasing as it approaches the focal point and decreasing as it moves away.
• An elliptical orbit means the total energy of a planet is conserved, with potential energy highest at furthest points and kinetic energy highest at the closest points to the focus.

Understanding an elliptical orbit helps clarify why areal velocity remains constant and why its direction is determined via the right-hand rule.

Cross Product

The cross product is a mathematical operation that takes two vectors and produces a third vector. This new vector is perpendicular to the plane formed by the original two. Here's how:Imagine two vectors, \( \mathbf{a} \) and \( \mathbf{b} \), with a cross product denoted by \( \mathbf{a} \times \mathbf{b} \). The steps to find this include:

• Calculate the magnitude of the cross product: \( |\mathbf{a}||\mathbf{b}|\sin\theta \), where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \).
• The direction is found using the right-hand rule—perpendicular to the plane of \( \mathbf{a} \) and \( \mathbf{b} \).

In the context of areal velocity,

• The cross product of the position vector \( \mathbf{r} \) and the velocity vector \( \mathbf{v} \) gives the areal velocity vector.
• This explains the direction of areal velocity being perpendicular to the motion plane, crucial when describing an orbit.

Hence, the cross product is integral in understanding vector directions in physics, particularly in rotational dynamics like those seen in elliptical orbits.