2) The rod AB rotates about point A with an angular...

2) The rod AB rotates about point A with an angular displacement of $\theta = 3t^2 - 2t$, where $\theta$ and $t$ are expressed in radians and seconds, respectively. The collar C slides along the rod so that its distance from point A is $r = 4t^2 - 3t^2$, where $r$ and $t$ are expressed in meters and seconds, respectively. Determine the velocity and acceleration of the collar when $t = 2$ s. Hint: $v = \dot{r} + r\dot{\theta}$ $a = (\ddot{r} - r\dot{\theta}^2)\hat{e}_r + (r\ddot{\theta} + 2\dot{r}\dot{\theta})\hat{e}_\theta$

Transcript

00:01 In this question, we have a double color c is pin connected along the elliptical track.

00:08 And then another side is pinned along the rod ab.

00:11 We are given the angular velocity of ab here, which is e to the 0 .5 t square.

00:20 We are also given the radio distance of the color and any data.

00:27 So in this question we are given the, we are asked to find the radio and transverse components of velocity and acceleration of c at t equals to one second.

00:37 So since we need to, so based on that, based on what we are asked to find, and we look at our radio function, there is a data.

00:48 We need to actually calculate the data at t equals to one second and we will be using simpson's rule to determine that.

00:55 Okay, so based on this, based on the requirements of the question, we'll first find data.

01:05 Okay, so solution.

01:10 Okay, so simpson's rule states that given a definite integral, a to b, fx, x, x is equal to delta x over 3, f a plus four times of f a plus four times of f a plus delta x plus two times f of a plus to delta x until you reach f of b and delta x is equal to b minus a over n and then the four and two just alternate the factor four and two just alternate okay so this is it allows us to find a approximate the value of the definite integral, okay, so in our case, the angle data adds t equals to one second is equal to 0 to 1 data dot.

02:36 Okay, so this is 0 to 1, e2 0 .5 t squared, okay, so our n is 50, so our delta t is 1 over 50 which is 0 .02.

02:54 Okay and what i've done over here is to show the table of the numbers from t equals to 0 to 1 in steps of 0 .02 and then including the factor and then multiply the function with the factor and then if you sum the last column on both tables, okay then by delta t over tree right so since we'll say that data t equals to one second is equal to delta t okay so this is delta t over tree and then you do the sum okay so f plus four times f of 0 .02 plus 2 times f of 0 .04 plus 2 .4 plus dot dot dot until f where f t is uh...

04:08 Zero uh...

04:10 Exponential zero point five t square and you calculate these you get uh...

04:14 One point one ninety five regents and this is uh...

04:18 Sixty eight point five degrees okay so this is one of the answer that we need to find using simpson true so just check the table you can just use excel to to calculate to generate all these numbers and then you just sum the last column and you get 1 .195 radiance, okay, sum and multiply by delta t over 3, okay? right, so now we can get back to do some calculations.

04:54 Okay, so before that, let us record a formula.

05:00 So we are, this is the radial component of the velocity, this is r dots, and then our transverse components is v -data is r -data dot, and then ar is the radio components of the radio of the acceleration is r double dots minus r -data -dott square, and then a -data is r -data double dot plus 2r -data -dott...

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