Collar On Rod - velocity and acceleration

[nuggets]

Homework Statement

Rod with length PV (P is a pivot point) with collar on rod. Rotation of rod PV about P is $$\theta = 2 t^2$$. distance d of collar from pivot point of rod is $$d = 60t^2 - 20t^3$$. I have to find:
(1) the velocity of the collar
(2) the total acceleration of the collar
(3) the acceleration of the collar relative to the rod

Homework Equations

The Attempt at a Solution

Assuming no friction.
I have created the vector r from pivot P to the collar
$$\vec{r} = cos(2t^2)(60t^2 - 20t^3)\hat{i} + sin(2r^2)(60t^2 - 20t^3)\hat{j}$$

I think that the answer to 1 is the first derivative of $$\vec{r}$$
And the answer to 2 is the second derivative of $$\vec{r}$$
And the answer to 3 is the second derivative of d: $$\ddot{d} = 120(1-t)$$

I have calculated $$\dot{\vec{r}}$$ and $$\ddot{\vec{r}}$$, but I have not included them here for brevity.

Am I solving this correctly?

Thank you.

 

[Jhero]

Thank you for your post. It appears that you have correctly solved for the velocity and acceleration of the collar. Your approach of using vectors and derivatives is appropriate for this problem. However, I would like to point out a few things for clarification.

Firstly, for the velocity of the collar, you have correctly taken the first derivative of the position vector, but it should be noted that this gives the instantaneous velocity at any given time t. To find the total velocity, you would need to take the magnitude of this vector, i.e. ||\dot{\vec{r}}||. This will give you the speed of the collar at any given time.

Secondly, for the total acceleration of the collar, you have taken the second derivative of the position vector, but it should be noted that this gives the instantaneous acceleration at any given time t. To find the total acceleration, you would need to take the magnitude of this vector, i.e. ||\ddot{\vec{r}}||. This will give you the magnitude of the acceleration at any given time.

Lastly, for the acceleration of the collar relative to the rod, your answer appears to be correct, but it would be helpful to include the units for clarity. Also, it should be noted that the acceleration of the collar relative to the rod is always perpendicular to the rod, as it is caused by the rotation of the rod about the pivot point.

Overall, it seems that you have a good understanding of the problem and have solved it correctly. Keep up the good work! If you have any further questions or concerns, please feel free to post them in the forum.

Scientist