Problem 11 - Classical Mechanics

Question

The position of a moving particle is given as a function of time \(t\) to be $$\mathbf{r}(t)=\hat{\mathbf{x}} b \cos (\omega t)+\hat{\mathbf{y}} c \sin (\omega t)$$ where \(b, c,\) and \(\omega\) are constants. Describe the particle's orbit.

Step-by-Step Solution

Answer

The particle follows an elliptical path centered at the origin with axes lengths determined by \( b \) and \( c \).

Step 1: Identify the Position Function Components

The position function provided is \( \mathbf{r}(t)=\hat{\mathbf{x}} b \cos (\omega t)+\hat{\mathbf{y}} c \sin (\omega t) \). This indicates that the position of the particle in the \( x \)-direction at any time \( t \) is \( b \cos(\omega t) \) and in the \( y \)-direction is \( c \sin(\omega t) \).

Step 2: Eliminate Time Parameter

To find the shape of the orbit, we need to eliminate the time variable \( t \) by expressing both \( x \) and \( y \) in terms of each other. From the position function, set \( x = b \cos(\omega t) \) and \( y = c \sin(\omega t) \).

Step 3: Use Trigonometric Identity

Using the Pythagorean identity \( \cos^2(\theta) + \sin^2(\theta) = 1 \), solve for \( \cos(\omega t) \) and \( \sin(\omega t) \). Substitute these back into the identity: \( \left(\frac{x}{b}\right)^2 + \left(\frac{y}{c}\right)^2 = 1 \).

Step 4: Recognize the Standard Equation of an Ellipse

The equation \( \left(\frac{x}{b}\right)^2 + \left(\frac{y}{c}\right)^2 = 1 \) is the standard form of an ellipse centered at the origin with semi-major axis \( b \) and semi-minor axis \( c \). Thus, the particle's motion traces out an elliptical path.

Key Concepts

Position Function

In the world of physics, a position function provides essential information about the location of a particle at any given point in time. The position function in the exercise is \[\mathbf{r}(t)=\hat{\mathbf{x}} b \cos (\omega t)+\hat{\mathbf{y}} c \sin(\omega t)\]Here, the particle's position is captured in two parts:

• The term \(\hat{\mathbf{x}} b \cos (\omega t)\) describes the particle's movement in the \(x\)-direction
• The term \(\hat{\mathbf{y}} c \sin (\omega t)\) denotes its movement in the \(y\)-direction

The constants \(b\) and \(c\) represent the maximum reach or amplitude of the particle's movement in the respective directions. Meanwhile, \(\omega\) is the angular frequency, which tells us how quickly the particle completes a cycle of its motion.
This position function combines trigonometric components to define precisely where the particle is as time \(t\) progresses. It is an example of parametric equations, where two separate equations describe a singular movement of a particle in space.

Ellipse

An ellipse is a geometric shape that resembles an elongated circle. The equation \(\left(\frac{x}{b}\right)^2 + \left(\frac{y}{c}\right)^2 = 1\) as derived in the exercise is a standard representation of an ellipse.

• Centered at the origin.
• With a semi-major axis of length \(b\) in the x-direction
• And a semi-minor axis of length \(c\) in the y-direction

The significant property of an ellipse is that for any point on it, the sum of the distances from two fixed points (foci) is constant.
This property helps us understand the particle's path in the exercise. By eliminating the parameter \( t \) from the position function, we can visualize the particle's motion more broadly as a distinct, recognizable shape.
The particle's path is clearly illustrated as it travels, demonstrating how mathematics and geometry intersect to provide a clear picture of motion.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate trigonometric functions to one another. In this exercise, they are crucial for understanding the relationship between the components of the position function.A powerful identity used is the Pythagorean identity:\[\cos^2(\theta) + \sin^2(\theta) = 1\]This identity allows us to transform our position functions from involving the explicit parameter \( t \) to a statement about \( x \) and \( y \), converting the problem into a geometric interpretation.
For instance, when we have:

• \(x = b \cos(\omega t)\)
• \(y = c \sin(\omega t)\)

We can use the Pythagorean identity to say:\[\left(\frac{x}{b}\right)^2 + \left(\frac{y}{c}\right)^2 = \cos^2(\omega t) + \sin^2(\omega t) = 1\]Thus, the trigonometric identities serve as fundamental building blocks, helping us derive a comprehensive picture of the path a particle takes, such as in this motion along an ellipse.