CaCh Case uelerile [le grapn Ullal DCSt Tepresells valTaliOnl OL llle Lallgenllal acceleralon uC _ 1nC VCrical dcs represent the tangential acceleration, the horizo...

For each of the following problems, find the tangential and normal components of acceleration.
$$
\mathbf{r}(t)=3 \cos (2 \pi t) \mathbf{i}+3 \sin (2 \pi t) \mathbf{j}
$$

Given problem position, Doctor. From point A to a our way will be nine minus sign 30. I kept minus cost. 30 jacket. Okay, so from here are we equals to nine. Minus one by two. I kept minus route three by two. Jacob. Okay, Now velocity appoint a V equals two. We know less omega, but they are so substituting values we get. Or maybe we're not is equal to zero. And Omega is five. K cap. And are you A Is nine minus one by two. I came minus three by two. Jacob. Now, using cross product rules, we get the but the quit school 39 0.0. I get minus 22.5. Jacob, meet up for second. So this is the velocity at point A for the part of the problem. Okay. No moving forward to calculate acceleration of point A. We can write that acceleration at point A, but there will be equal to a note. Plus, alpha are away. Bless omega, but glad, by the way. Okay. So now, substituting the values we get, acceleration at point A will be equal to a note is equal to zero. Plus, alpha is minus 10-K cab cross and cross our way across nine minus one by two. I get minus Route three by two Jacob Bless Omega Cross. The Omega is five. K cab. Of course we is this value for From part A 39.0 I kept managed 22.5. Jacob. Okay, so now solving it, we get a vector. Acceleration of point. A factor comes out to 34.6 plus 2. 14 Jacob inch per second square. This is not in meter. Okay, this is indeed inch, but second. Okay, Inch per second squared. So this is the answer for the but a of the problem. Okay. Now, for the velocity and acceleration of part B weekend, calculate the position vector R o b. As are we, plus our baby. So now, substituting the values we get R o B question. Are you A s nine minus sign 30. I kept minus cost 30 Jacob and our baby is three. Because 30 I get minus sign 30. Jacob. Okay, So after solving we get r o B vector, it goes to three by two. Go three minus three. I get minus one plus 3 to 3 Jacob inch. Okay, so this is the uh huh. Hobie Vector. So the velocity point to be will be equals two. We not plus Omega Cross R o B factor. Okay, so substituting the values we notice equals to zero, So we'll be vector will be calls to zero. Plus Omega is five. K. Cap cross three by two. Bracket under three, minus three. Jacob bless. Minus mhm. Minus one plus 3 to 3. Jacob. Okay, so now solving it, we get we b equals two. 46.5. I kept minus 9.51 Jacob inch per second. So this is the velocity of point to be of the given problem. Okay. No moving to the acceleration we can write. Acceleration at point B will be close to you. Less Alpha Alpha cross R o B plus Omega Cross week B Victor. Okay, these are all in the vectors. Okay. So substituting values we get a B vector equals to notice again. Zero and alpha is minor. Stand K cap across our O. B. S three by two. Through three minus three. I kept minus one minus 33 G k. Okay. Plus omega is where you kick it. Cross we be is 46.5 cap minus 9.5 on Jacob. Okay, so after my plane, we get a director equals two minus 45.4 I kept, plus 20 2 51 j k. Inch per second squared. So this is the acceleration of point.

Okay, so we have. Archie is equal to to over 31 plus G three or two you have to do with you. 23 What? Minus G Marjorie. Over two and a duty. Okay, Now let's find our velocity. It's extra motive of above. He gets one. Puts t to the power of one over two negative one modesty to fire of one over two. And where would have Now let's look for a celebration That's just the derivative off our velocity. So we get 1/2 one plus t power of negative 1/2 1/2 one. Modesty thinking of 1/2 and don't Now it's find a t ethical to be dot a over bang to the beat, so this simplifies to just 1/2. One plus two went over to minus 1/2. When it's 1/2 one plus two, you have 1/2 money, too. Well, the suit cancel and then that's post. Also, we get there. No, let's look for a and ethical tubes growed over magnitude of a squared, minus a of teeth word that's equal to the square root of one. What it's for one plus teeth and part of negative one plus one over +41 wanted t to fire. Negative one No, that simplifies to 1/2 to root of two minus one minus C squared.

Good point A at a the acceleration of everyone will be equals two cellars and Of M one is equals to our Omega Square. So here Omega is equals two pi divided by T were multiplied with our this is directed along, oh, The acceleration of M two is also directed in the same direction. Therefore, relative acceleration of MT with respect to anyone he can read it, uh, relative excellence. And this will be calls to a B minus A. A. It is a close to a v minus A is equal to pi divided by t. Holy square When he played with our let us take for the option. So we see option option is correct. Relative excellence and will be close to a B -2 pi divided by the Holy square Man declared with our option.

Okay, so we have Marty with equal to co sign duty. Sign of two things and one, Well, it's final velocity ethic. Look too Needed to sign of to t to co sign of to tea and jump acceleration. It's equal. D'oh! You got a four coast side of sooty, heretofore sign of duty. And don't you know we're asked to find Katie. That's just equal to dot product of being me over the magnitude of B that gives me Seeley. It's makeup to sign to the T ducal sign to T and Daryl dotted with negative four co sign to t. I think the fourth sign of city and Joe all over the magnitude of being which is the square root of negative to sign duty squared too close Sign squared to t squared. Okay, so that gives me eight. Sign of to t co sign of to think minus takes sign of toothy coast attitude. But this is already just cancel. So whatever is on the I did not made it doesn't matter if we just get this. Is he good to go and then our normal going on a n. You could describe it of our celebration the magnitude of. Okay, so that's a relationship minus 80 squared. Hey, what we get to square it. 16 names, co sign squared, toothy, closed sign where duty, This cancer to one, We just get square roots of 16 which is equal to four.