Finding the Correct Acceleration in a Spinning Circle: What Am I Doing Wrong?

[BryMan92]

Homework Statement

See image.

Homework Equations

ωx (ωx r) = anormal
αx r =atangent

x=rcos*theta
y=-rsin*theta

The Attempt at a Solution

I solved for w=2 k rad/s and α= -1.5 k rad/s

I also got the correct answer using the cross products. My problem is I am trying to do this problem in rectangular coordinates (I really like rectangular), but I am doing something wrong and I cannot see it.

So, I assume r is constant:
x''=-r*cos(45)[ω^2-α] = Correct answer= -15.566

BUT, y'' is giving me a headache:
I keep getting rsin(45)[ω^2-α] and I get = 15.66 and NOT the right answer of 7.07 in/s. There is a lurking negative sign and I cannot find it. That - should be a + and then I get a correct answer. Am I just deriving incorrectly?

Thanks!
And apologizes if this is pretty novice: its from my Junior-level engineering class...or a freshman Physics class. ;p

 

[jedishrfu]

from the problem you know that va = 8 in/sec and aa=6 in/sec/sec

and you know that va is tangent to the circle and that aa is along the radius

so the unit vector for aa=cos (theta) i + sin(theta) j

and the unit vector for va = - sin(theta) i + cos(theta) j

now figure out theta and factor in the vector magnitudes into the unit vector equations to get the vector equations of motion.

Lastly, if you do unit vector va dot aa = -sin(theta) cos(theta) + cos(theta) sin(theta) = 0 meaning they are perpendicular as a check

 

[BryMan92]

That make's sense. Shouldn't I be able to use y=-rsin*theta to kind of derive that? Everything you said makes perfect sense (I do like the dotting notion!), but I am still wondering why rectangular equation has issues.

 

[jedishrfu]

you should be able to start with s(x,y) and then diferentiate to get va
and differentiate again to aa.

s(x,y) = R*cos(w*t + offset)) i + R*sin(w*t + offset) j

make sure you're using radian measure for all angle values that could be where your problem lies.

 

[paranormal]

I would first like to commend you for your efforts in trying to solve this problem and for seeking help when you encounter difficulties. It demonstrates your determination and willingness to learn, which are key qualities in a scientist.

Now, let's get to the problem at hand. It seems like you have a good understanding of the equations and concepts involved in solving this problem. However, there are a few things that you may have overlooked.

Firstly, when solving problems in rectangular coordinates, it is important to remember that the x and y components of acceleration are independent of each other. This means that the equations for x and y acceleration should be solved separately.

Secondly, in the equation for y acceleration, you have used the wrong angle. The angle in the y direction is not 45 degrees, but rather 90 degrees. This is because the y direction is perpendicular to the x direction in a circle. So the correct equation for y acceleration would be y'' = -r*sin(90)[ω^2-α], which gives the correct answer of 7.07 in/s.

Lastly, it is always a good practice to double check your calculations and equations to make sure you have not made any mistakes. Sometimes, a simple error in calculation or a misplaced negative sign can lead to incorrect answers.

In conclusion, it seems like you have a good understanding of the concepts involved in solving this problem. Just make sure to pay attention to the details and double check your work to avoid any mistakes. Keep up the good work and don't be afraid to seek help when needed. Good luck!