[Problem] A Pin In Constrained Motion

QUESTION

A pin is constrained to move in a circular slot of radius 64mm. At the same time a slotted bar also constrains the pin to move down with a constant velocity of 4mm/s as shown. What is the magnitude of the acceleration of the pin for the position shown?

SOLUTION

1. The time it takes for the pin to move to the position is:

2. The position of the pin can be expressed as follows:

3. the pin is moving downward in a costant acceleration, therefor we know that the magnitude of the acceleration of the pin comes from the sideways movement, namely x-direction movement. Therefore, we can re-write the equation as: 

and, taking the differential, we get: 

taking the differential again, we get:

4. Substituding the time t we have caculated in 1., we get the position x. From t and x we can obtain the vlocity (dx/dt). And finally, the acceleration (d2x/dt2).

NOTE

When I first saw the question my immediate thought was to use relative motion. I had a hard time finding the relationship between the pin and the slotted bar or the center of motion. Then I thought I could do it by using energy conservation. But then I'd have to first calculate the velocity of the pin and then calculate the kinematic energy and make life hard. The above solution, using differentials, is by far the most direct and foolproof way I can think of. It does, nonetheless, involve quite some calculating and reducing, which might more easily result in mindless mistakes. So I wonder if it could be solved without having to take the differentials and use the Newton's second law instead?