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. The tip of the 15 -inch second hand of a clock completes one revolution in 60 seconds.

Short answer

Answer: The parametric equations are \(x(t) = 15 \cos(\frac{\pi}{30}t)\) and \(y(t) = 15 \sin(\frac{\pi}{30}t)\).

Step-by-step solution

Identify the Radius and Total Time

In this problem, the radius of the circle is given as 15 inches, so \(R = 15\). The total time it takes for the tip of the second hand to complete one revolution is also given as 60 seconds.

Determine the Angular Speed

Since it completes one revolution in 60 seconds, we can determine the angular speed \(\omega\) by dividing the total angle of a circle, which is \(2\pi\) radians, by the total time it takes to complete one revolution, which is 60 seconds. So, \(\omega = \frac{2 \pi}{60}\).

Parametric Equations for x(t) and y(t)

We now have all the necessary information to write the parametric equations for the circular path of the tip of the second hand. Substitute the values of \(R\) and \(\omega\) into the parametric equations for a circle: \(x(t) = R\cos(\omega t) = 15 \cos(\frac{2 \pi}{60}t)\) \(y(t) = R\sin(\omega t) = 15 \sin(\frac{2 \pi}{60}t)\)

Final Parametric Equations

So the parametric equations that describe the circular path of the 15-inch second hand of the clock are: \(x(t) = 15 \cos(\frac{\pi}{30}t)\) \(y(t) = 15 \sin(\frac{\pi}{30}t)\)

Key concepts

Angular Speed

Angular speed is a measure of how fast an object rotates around a fixed point or axis. It is an essential concept in understanding rotational motion, particularly when dealing with circular paths. Angular speed is denoted by the Greek letter omega (\( \omega \)) and is typically measured in radians per second or degrees per second.

• To calculate angular speed, you need to know the total angle of rotation for one complete revolution, which is typically \(2\pi\) radians for a circle, and the time the rotation takes.
• In the provided problem, the second hand of the clock completes one full circle, or \(2\pi\) radians, in 60 seconds.
• This results in an angular speed of \(\omega = \frac{2\pi}{60}\) radians per second.

This concept is crucial for translating rotational motion into parametric equations that describe the path of an object moving in a circle.

Circular Motion

Circular motion occurs when an object moves along a circular path. The radius of the circle remains constant, and every point on the object follows a circular trajectory around a fixed center known as the origin.

• In the context of the exercise, the second hand of a clock is undergoing uniform circular motion.
• The radius of this motion is 15 inches, indicating the distance from the center of the clock to the tip of the second hand.
• The circular path can be described using parametric equations involving trigonometric functions.

These equations use the circle's radius and the angular speed to provide a precise description of the motion, which in this case, correspond to the repetitive, predictable circle traced by the clock's second hand.

Trigonometric Functions

Trigonometric functions, specifically sine and cosine, are integral in expressing circular relationships mathematically. They help translate the circular motion into a parametric form, enabling calculations and analyses of the object's path.

• The cosine function, \(\cos(\omega t)\), expresses the horizontal (x-axis) position of the object on the circle as a function of time.
• The sine function, \(\sin(\omega t)\), corresponds to the vertical (y-axis) position of the object.
• By using these functions, the circular motion is represented by two parametric equations: \(x(t) = R \cos(\omega t)\) and \(y(t) = R \sin(\omega t)\).

In the given exercise, this results in the equations \(x(t) = 15 \cos\left(\frac{\pi}{30}t\right)\) and \(y(t) = 15 \sin\left(\frac{\pi}{30}t\right)\), illustrating the cyclical journey of the clock’s second hand. These equations are effective not only because they simplify complex motion, but also because they make it predictable and analyzable.