The coordinates of an object moving in the x y plane vary...

Key Concepts

Vector Kinematics in Two Dimensions

This concept involves describing motion using vectors in a two-dimensional plane. In problems like this one, the position, velocity, and acceleration of an object are represented as vectors with components along the x and y axes. The approach requires breaking down the motion into independent components, making it easier to analyze the behavior of the object using calculus and algebra.

Parametric Equations and Their Differentiation

Parametric equations express the coordinates of a point as functions of a variable, typically time. Differentiating these equations with respect to time yields the velocity components, and differentiating a second time provides the acceleration components. This method is fundamental in kinematics to transition from position functions to velocity and acceleration vectors.

Circular Motion Described by Parametric Equations

The given position equations suggest that the motion follows a circular path. Recognizing the relationship between sine and cosine functions in the component equations allows one to deduce that the path is circular (with a shifted center if necessary). This understanding helps in visualizing the trajectory on an xy graph and in verifying that the derived velocity and acceleration maintain the characteristics of circular motion.

Angular Frequency

Angular frequency, denoted by ?, represents the rate at which an object moves through an angle in circular motion and has units of radians per second. It is a key parameter in the equations of circular motion, linking the time-dependent trigonometric functions to the physical rotation experienced by the object. Its role is crucial in establishing the periodicity and dynamic properties of the motion.

Transcript

00:01 Hello.

00:02 In this question, we have a particle that is moving in the x, y, plane, and the equation for the x of this particle is basically negative 5 sine omega -t, where omega here is in, is a frequency, or it's in units of second inverse, and y is just 4.

00:33 4 minus 5 cosine omega t we are asked in part a to give expressions of the velocity and the acceleration of the particle at time zero and in part b we're asked to give a general expressions for the position velocity acceleration at any later point in time i will start by doing b first because it's much easier to do a if we already have b answered.

01:12 So now we're getting the general representation of the position vector, velocity vector, and the acceleration vector.

01:20 Let's start with the position vector r.

01:23 R is basically x in the direction of i and y in the direction of j.

01:28 So just plugging those here.

01:30 So that's negative 5 sine omega -t, sine sine omega t in the i direction plus four minus five cosine omega t in the j direction.

01:51 That's the general formula for the position in vector representation.

01:57 So let's get the velocity afterwards and as we know, that's just the derivative of the position vector with respect to time.

02:05 So let's let's do this.

02:08 We know that the derivative of sign is basically cosine and we have to differentiate what's inside the sign itself.

02:17 So here we will have that omega coming out from here and this will change to cosine in a step.

02:23 So that's basically negative 5 omega, cosine omega t.

02:33 That's in the direction of i.

02:36 And differentiating this, the four gives nothing for sure and the five the differentiation of the cosine function is just the negative of the sign so doing this and of course we will have omega coming out of the bracket as well so this will just give us a negative this will turn this negative here into a positive so that's five multiplied by omega sine omega t and that's in the direction of j now getting the representation of acceleration, just the same as the previous point, but instead we're differentiating the velocity vector itself this time.

03:24 So another differentiation.

03:26 And doing this, the cosine, the derivative is just negative signs.

03:31 So this will cancel the negative here.

03:33 So that's five.

03:34 And we have another omega coming out.

03:36 So that's omega squared.

03:37 And this will turn into sine omega -t.

03:41 That's in the i direction.

03:45 And here, the same design, the differentiation is just cosine, and we have another omega coming out, so that's 5 omega squared cosine omega -t in the j direction.

04:00 So here, let's box this here.

04:03 That's the answer for part b, and this is the representation of the acceleration velocity and the position vector.

04:11 Okay, now getting back to, if we know, if we need to know that.

04:16 Position or the velocity or the acceleration at any point in time, we will just need to replace each t here with the specific value we're looking for.

04:25 So now let's come back and do part e.

04:28 And it's asking us to give a representation for the velocity and the acceleration and zero...

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