Question

Find parametric equations that describe the circular path of the following objects. Assume \((x, y)\) denotes the position of the object relative to the origin at the center of the circle. Use the units of time specified in the problem. There is more than one way to describe any circle. A bicyclist rides counterclockwise with constant speed around a circular velodrome track with a radius of \(50 \mathrm{m},\) completing one lap in \(24 \mathrm{s}\)

Short answer

Answer: The parametric equations for the circular path are \(x(t) = 50 \cos(\frac{\pi t}{12})\) and \(y(t) = 50 \sin(\frac{\pi t}{12})\).

Step-by-step solution

Understand given information and symbols

First, let's make sure we understand the given information and any symbols involved. We are given that the radius of the circular track is \(50 \mathrm{m}\), and the time it takes to complete one lap is \(24 \mathrm{s}\). We are also told to use units of time.

Determine the angle as a function of time

We need to find a function that describes the angle as a function of time (\(\theta(t)\)). Given that the bicyclist's motion is counterclockwise and constant speed, we can determine the rate of change of the angle with respect to time. Since it takes \(24 \mathrm{s}\) to complete a lap, the rate of change of the angle is: \(\frac{d\theta}{dt} = \frac{2\pi}{24 \mathrm{s}}\). Now we integrate the rate of change with respect to time to find the angle function: \(\theta(t) = \int \frac{d\theta}{dt} dt = \int \frac{2\pi}{24 \mathrm{s}} dt = \frac{2\pi}{24}t + C\) Since \(\theta(0)=0\), we have that \(C=0\) and \(\theta(t) = \frac{\pi t}{12}\)

Write parametric equations for the circular path

Now we can write down the parametric equations for the circular path using the given radius and the angle function we calculated in the previous step: \(x(t) = r \cos(\theta(t)) = 50 \cos(\frac{\pi t}{12})\) \(y(t) = r \sin(\theta(t)) = 50 \sin(\frac{\pi t}{12})\) The parametric equations for the circular path are: \(x(t) = 50 \cos(\frac{\pi t}{12})\) \(y(t) = 50 \sin(\frac{\pi t}{12})\) These equations describe the counterclockwise motion of the bicyclist in terms of their position \((x,y)\) relative to the origin at the center of the circular track.

Key concepts

Circular Motion

Circular motion refers to the motion of an object along the circumference of a circle or a circular path. A key characteristic of circular motion is that the object maintains a constant distance from a central point, known as the radius. This is a common motion type prevalent in many natural and engineered systems, from planets orbiting stars to rides at amusement parks.
For the bicyclist moving around the velodrome, the path they traverse forms a circle with a constant radius of 50 meters. Since they move counterclockwise with constant speed, this implies uniform circular motion. In uniform circular motion, the speed of the object remains the same; however, its velocity constantly changes direction due to the circular path. This distinction between speed (scalar quantity) and velocity (vector quantity) is crucial. Even though the bicyclist rides at constant speed, their velocity changes because the direction of motion changes as they circle around the track.

• Constant radius: 50 meters
• Path: Counterclockwise direction
• Uniform speed: Constant velocity magnitude

These characteristics help model circular motion mathematically using parametric equations which elegantly describe the motion along each coordinate axis.

Trigonometric Functions

Trigonometric functions are integral in representing circular motion, resulting in parametric equations that capture the circular path. For circular motion, the most commonly used trigonometric functions are sine and cosine. These functions relate to the circle in a manner that they can represent any angle as a function of time, showing how a point moves along the circle.
For the bicyclist's circular path, we utilized the angle function \[\theta(t) = \frac{\pi t}{12}\]which describes how the angle changes with time. Here, the use of sine and cosine functions connects the circular path to the Cartesian coordinates \((x, y)\).

• Cosine function: Determines horizontal position with \(x(t) = 50 \cos(\frac{\pi t}{12})\)
• Sine function: Determines vertical position with \(y(t) = 50 \sin(\frac{\pi t}{12})\)

These trigonometric functions actually depict the circular relationship between the angle and the positions across the axes. They offer a periodic and continuous representation of how the object's position evolves along its circular track. This bridge between angles and positions showcases the profound usefulness of trigonometric functions in modeling movements like circular orbits.

Bicyclist Path

The path a bicyclist takes around a track can be precisely described using parametric equations. These equations account for both the uniform circular motion and the trigonometric relationships defining the bicycle's path. In this scenario, the track has a known radius and the cyclist completes a lap in a specific time, allowing us to calculate their positional equations.
Parametric equations describe the whole path by associating a position at any given time. For the bicyclist with a radius of 50 meters, the equations are:

• \(x(t) = 50 \cos(\frac{\pi t}{12})\)
• \(y(t) = 50 \sin(\frac{\pi t}{12})\)

These define how the position changes over time as the cyclist travels the circular path. The beauty of parametric equations lies in their ability to handle complex motion where both the direction and the magnitude of movement are essential. They enable us to understand not just where the bicyclist is but also how fast they're moving and which direction they're headed at any point in time.