Problem 109 - Calculus 3

Question

A particle travels in a circle with the equation of motion \(\mathbf{r}(t)=3 \cos t \mathbf{i}+3 \sin t \mathbf{j}+0 \mathbf{k} .\) Find the distance traveled around the circle by the particle.

Step-by-Step Solution

Answer

The distance traveled by the particle is \( 6\pi \).

Step 1: Identify the Equation of Motion

The motion of the particle is given by the vector function \( \mathbf{r}(t) = 3 \cos t \mathbf{i} + 3 \sin t \mathbf{j} + 0 \mathbf{k} \). This represents a circular path in the \( xy \)-plane with a radius of \( 3 \).

Step 2: Determine the Parametric Path

The path traced by the particle is described by the equations \( x(t) = 3 \cos t \) and \( y(t) = 3 \sin t \). This confirms that the particle moves in a circle centered at the origin with a radius of \( 3 \).

Step 3: Calculate the Circumference of the Circle

The circumference \( C \) of a circle is given by the formula \( C = 2 \pi r \). Since the radius \( r = 3 \), the circumference becomes \( C = 2 \pi \times 3 = 6\pi \).

Step 4: Compute the Distance Traveled

Since the particle travels the complete circumference of the circle in one cycle, the distance traveled by the particle around the circle is equal to the circumference. Thus, the distance is \( 6\pi \).

Key Concepts

Understanding Circle Geometry in Parametric Equations

Circle geometry in parametric equations often involves understanding how equations can describe motion along a circular path. In this context, the parametric equations used are powerful tools. They define a circle in a coordinate plane by expressing locations using functions of one or more variables, often time. Let's break it down:

• Equation of a Circle: A circle in parametric form can be described by equations like \(x(t) = r\cos(t)\) and \(y(t) = r\sin(t)\), where \(r\) is the radius.
• Components of the Equation: For our particle, \(3\cos(t)\) represents the \(x\) coordinate and \(3\sin(t)\) represents the \(y\) coordinate. These align to form the circular path described.
• Role of Parameters: The variable \(t\) represents the parameter, often time, dictating the position of the particle as it moves along the circle.

Understanding these core principles is essential to visualizing motion in a parametric context. It provides a comprehensive way of analyzing circular motion in vector calculus.

Applications of Vector Calculus in Circular Motion

Vector calculus extends the idea of parametric equations to analyze motion and changes within a field. Here, it is instrumental in comprehending how vectors describe motion along a circular path.

• Vectors in Motion: The given vector function \( \mathbf{r}(t) = 3 \cos t \mathbf{i} + 3 \sin t \mathbf{j} + 0 \mathbf{k} \) is a complete representation of the particle's position at any time \(t\).
• Function Components: Each component of the vector, such as \(3 \cos t\) and \(3 \sin t\), signifies the effect on respective axes, vital for determining the particle's location on the plane.
• Understanding Magnitude: The magnitude of a vector can reveal the arc length or displacement of the particle at any point in its trajectory.

Vector calculus not only defines positions and paths but also helps calculate vector magnitudes, derivatives, and integrals, crucial for deeper analyses of particle motion.

Analyzing Motion in a Plane through Parametric Equations

Motion in a plane is often depicted using parametric equations, making it simpler to track and predict how particles traverse specific paths, like circular ones.

• Plane Motion Simplification: Breaking down circular motion into two components (\(x\) and \(y\)) helps simplify what otherwise might be complex motion into something describable by basic trigonometric functions.
• Uniform Circular Motion: In our example, the motion described is uniform, meaning the particle moves at a constant speed along the circle's circumference. This is crucial for understanding motion dynamics.
• Computing Distance: With the parametric form, calculating the total distance involves finding the circumference of the circle. The formula \(2\pi r\) gives a direct calculation, crucial for understanding the complete journey of the particle.

Analyzing motion in a plane with parametric equations enables students to visualize and understand trajectory, speed, and distance with accuracy and simplicity.