Problem 26 - Algebra and Trigonometry

Question

\(25-28\) . The position of an object in circular motion is modeled by the given parametric equations. Describe the path of the object by stating the radius of the circle, the position at time \(t=0\) , the orientation of the motion (clockwise or counterclockwise), and the time \(t\) that it takes to complete one revolution around the circle. $$ x=2 \sin t, \quad y=2 \cos t $$

Step-by-Step Solution

Answer

Path: Circle with radius 2; Initial position: (0, 2); Motion: Counterclockwise; Time for one revolution: \(2\pi\) units.

Step 1: Identify the parametric equations

The parametric equations given are \(x = 2 \sin t\) and \(y = 2 \cos t\). These are expressions for the position of an object in terms of a parameter, \(t\), which usually represents time.

Step 2: Determine the path

The equations \(x = 2 \sin t\) and \(y = 2 \cos t\) represent a circle in the \(xy\)-plane. This is because the expressions are forms of the parametric equations for a circle, where \(x = a \sin t\) and \(y = a \cos t\) with \(a\) being the radius of the circle.

Step 3: Calculate the radius

The radius, \(a\), is given directly by the coefficients of \(\sin t\) and \(\cos t\) in the parametric equations. Here, it's 2, so the radius of the circle is 2.

Step 4: Find the initial position

At \(t = 0\), substitute into the equations: \(x = 2 \sin 0 = 0\) and \(y = 2 \cos 0 = 2\). Thus, the initial position of the object at \(t = 0\) is at the point (0, 2).

Step 5: Determine the orientation

To determine the motion's orientation, observe the parametric form: \(x = 2 \sin t\), \(y = 2 \cos t\). As \(t\) increases from 0 to \(\pi/2\), \(x\) increases from 0 to 2, and \(y\) decreases from 2 to 0, indicating counterclockwise motion.

Step 6: Calculate the time for one revolution

A full revolution on the circle means \(t\) goes from 0 to \(2\pi\). Therefore, the time it takes to complete one revolution is \(2\pi\).

Key Concepts

Circular Motion

Circular motion is a type of movement where an object travels along the circumference of a circle. It's a fascinating aspect of physics because, despite its simplicity, it can describe everything from the orbit of planets to the motion of electrons.
In our example, the object moves in a circular path in the plane. The key feature of circular motion is that the object stays at a constant distance from the center of the circle. This constant distance is known as the radius, and it is crucial in defining the path.

• The radius remains constant throughout the motion.
• The object completes a continuous loop around the center point.

The motion is described using parametric equations, which provide the positions in terms of another variable, often time, allowing us to track the object's current location as it moves.

Radians

Radians are a unit of angular measurement used extensively in trigonometry and in describing circular motion. Unlike degrees, radians provide a natural way to relate the arc length of a circle to its radius.
In circular motion, using radians simplifies the math significantly. For the circle described in the problem, time, denoted by \( t \), is measured in radians.
One full revolution, or 360 degrees, on a circle equates to \(2\pi\) radians.

• A quarter-turn corresponds to \(\pi/2\) radians.
• A half-turn is \(\pi\) radians.

This keeps equations neat and allows seamless integration with trigonometric functions, which are naturally defined using radians.

Trigonometric Functions

Trigonometric functions describe relationships between the angles and sides of triangles, and they play a crucial role in parametric equations, particularly in modeling circular motion.
In our example, the motion is defined by \(x = 2 \sin t\) and \(y = 2 \cos t\). These functions describe the oscillating motion in the x and y directions, akin to tracing out a circle.

• \(\sin t\) and \(\cos t\) vary between -1 and 1: multiplied by the radius, they scale to fit a circle.
• At \(t = 0\), \(\sin t\) is 0 and \(\cos t\) is 1, so the position is at (0, 2).

The trigonometric form is powerful because it seamlessly describes circular paths and can easily adjust to changes in radius or orientation.

Orientation of Motion

The orientation of motion refers to the direction in which an object travels along its path. In circular motion, this could be clockwise or counterclockwise.
In our example, the motion is determined by examining the changes in the parametric equations as time \(t\) increases.
As \(t\) goes from 0 to \(\pi/2\), the x-coordinate moves from 0 to 2, while the y-coordinate drops from 2 to 0, indicating counterclockwise motion.

• For clockwise motion, the pattern would be reversed.
• Determining the orientation helps predict future positions.

Understanding the orientation is key for predicting where the object will be at any given time.