Constraint Relations Homework: Acceleration of Block

[ritwik06]

Homework Statement

http://img510.imageshack.us/img510/5505/systemet4.jpg

What I wish to do is to relate the accelerations of the loop an the massive block. I know the angle theta at any instant. I also know that the acceleration of the loop on the fixed support is a. I have been given no other information except the figure and I have to write out the acceleration of the block.
My teacher says its a cos theta

But I cannot express the expression of the block in just two variable of a and theta. I also need the perpendicular distance of the fixed support from the pulley.

please help me!

 

[ron_jay]

Referring to the diagram:

The Perpendicular distance from the pulley to the axis of the ring is say a constant L. The part
of the string that is being pulled (variable) is say y. The movement of the ring on the X-axis is also changing (variable) and we take it as x.

Now, using the hypotenuse theorem (!) :

$$x^2+ L^2 = y^2$$

what we want is now the instantaneous change of the variable x and y with respect to time and the way to find that is by differentiating:

$$2x\frac{dx}{dt} + 0 = 2y\frac{dy}{dt}$$

or $$\frac{dy}{dt} = \frac{x}{y} . \frac{dx}{dt}$$

From the diagram $$\frac{x}{y} = cos\theta$$

therefore . $$v_{ring} = v_{block} . cos\theta$$

We can say now that:

$$a_{ring} = a_{block} . cos\theta$$

Hope that solves it.