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DISCUSSION THREAD
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DISCUSSION and HINTS
The solution for this problem is a straight-forward application of our two rigid body kinematics equations relating the motion of point A back to point O:
vA = vO + ω x rA/O
aA = aO + α x rA/O - ω2rA/O
where point O is fixed (zero velocity and zero acceleration).
In order to find the angular velocity we would take the derivative of theta with respect to time correct? And then the angular acceleration is the derivative of the angular velocity with respect to time?
Yeah, it will be just a straight derivative with respect to time, of theta(t) for angular velocity and then again for angular acceleration.
Is the picture of the problem when it time is equal to zero? I do not know where to start from when sketching when A is at t = 0 and t = 1/3 sec.
I don't believe it is specifically at t=0. This is location at t=0 should just be directly to the right of point O. I think the reason point A is there is to show what direction theta goes in. To find the locations at t=0 and t=1/3, you can use the theta function, then convert from radians to degrees to find how large the theta value is then find the point along the circle where that makes sense.
For the w^2 value in the a_A calculation, do we need to plug in the corresponding time to the w equation to obtain our w^2 value?
2
Yes, you need to put in the appropriate times t for Parts a) and b).
I am a little concerned that I keep getting a positive i component for the acceleration at t=0.3s. Shouldn't the acceleration vector always be pointing towards O? At the theta value corresponding with t=0.3, it seems like the acceleration would have a negative i component.
I am having the same problem. I checked my signs multiple times and I am getting a positive i component even though logic would make you think it should be negative.
Brittain and Alexander,
It is good that you are doing this reality check. I assume that you are using the kinematics equation of:
a_A = a_O + alpha x r_A/O -omega^2*r_A/O
with a_O = 0. If we now look at each of the other two terms individually for the two values of theta (both angles less than 90°):
* alpha x r_A/O is a vector with a non-positive x-component
* -omega^2*r_A/O is a vector with a negative x-component
Therefore, the right hand side of this acceleration equation has a negative x-component, which it should (for reasons that you say).
Check your vector algebra on each of these terms. Also, be sure that you are using the correct r_A/O vector. It is easy to switch the sign on this term by accident.
As a slight follow up--I also got a positive i component for t= 1/3. I was told by a TA that, this is okay because at the position/theta that "A" is at during t = 1/3 seconds means that the acceleration vector points down and to the right but remains within the path. Thus, the object is slowing down. Whether or not this is true, I am not 100% sure, but I thought it may be of use to share my thoughts based on discussion with a TA.
You are correct. Since the disk is slowing down (θ_ddot is negative), the tangential component of acceleration points downward and to the right while the normal component of acceleration points inward towards point O. This makes it possible to have a positive i component for the acceleration at time t = 1/3s. However, if the disk were speeding up (θ_ddot > 0), the tangential component of acceleration would point upwards and to the left, indicating a negative i component for the acceleration at time t = -1/3s.
In general, are we expected to put our final answers in Cartesian? I did for this problem but it was extra work and could've been expressed much simpler in polar, except for crossing with k. Just curious!
We will see throughout the course that, as a rule, expressing velocity and acceleration for rigid bodies in Cartesian components is more convenient than in polar coordinates. This is particularly true when we deal with multiple bodies in the same problem. It is best to start out the topic in this way on the first day.
When solving for velocity and acceleration, we need to break r_A/O into its i and j components correct?
Yes, that is correct.
Would it be ok for my final answer for part B be in the etheta, er form? i feel like it would be much easier to get the answer using the V = r(theta_dot)e_theta equation
Please use Cartesian coordinates, we are probing your understanding of RB kinematics.