Circular Motion The position of an object in circular motion...

Circular Motion 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. $$ x=\sin 2 t, \quad y=\cos 2 t $$

Key Concepts

Circular Motion with Parametric Equations

In mathematics, circular motion is often modeled using parametric equations that involve sine and cosine functions. These functions map an angular parameter, usually time, to x and y coordinates on a circle. The parametric form is useful because it directly links the angular position to a point on the circular path, allowing one to analyze the motion's properties such as speed, direction, and periodicity.

Radius of the Circle

The radius of a circle in parametric equations of the form x = r sin(t) and y = r cos(t) is determined by the constant multiplier 'r'. By squaring and adding these equations, one obtains x² + y² = r², signifying that the distance from the center to any point on the path is constant. Recognizing the radius is essential for understanding the size and scale of the circle along which the object moves.

Initial Position and Orientation

The initial position of an object in parametric circular motion is found by substituting the starting value of the parameter (often t = 0) into the equations, establishing where the motion begins. The orientation—whether the motion is clockwise or counterclockwise—can be determined by examining the derivatives of the parametric functions. This analysis relates to the instantaneous direction of movement and hence the overall direction of the object's traversal along the circular path.

Period of Circular Motion

The period in circular motion refers to the time required for the object to make one complete revolution around the circle. For standard sine and cosine functions, this period is typically 2?. Knowing the period is important for understanding the timing and repetition in periodic phenomena, which is a fundamental aspect of oscillatory and circular motion.

Transcript

00:01 All right, here on this problem, we want to talk about this parametric equations.

00:05 Now, we're told that this is the position of a circle.

00:08 Okay, so we know this is going to graph into a circle.

00:11 Now, to answer some details about it, though, let's actually go about graphing it.

00:17 So let me just start by plugging in zero for t, right, just to see where our starting point is on this circle.

00:28 Now, because we have ones out here, because it's a one sine 2t and a 1 cosine 2t, we do know our radius is going to be 1.

00:37 And because it's a plus zero on each of these, we do know our center is going to be zero, zero.

00:44 And so we do know those parts.

00:46 But let's plug in some points here so we can figure out which direction that it's moving in.

00:50 All right, and where we start at.

00:52 So if t is zero, the sign of zero is zero, the cosine of zero is one.

01:00 And so it means when t is zero, we're starting at the point zero one.

01:04 And so let's put that value up there.

01:08 Now let me plug in pi over 4 next.

01:15 So when we plug in pi over 4, we're going to end up with the sign of pi over 2, whenever you take 2 times that, and the cosine of pi over 2, again, when you take 2 times pi over 4, the sign of pi over 2 is 1, the cosine of pi over 2 is 0.

01:36 And so that means our next point there is going to be 1 .0...

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