Why is acceleration of a cart on an incline divided by sin θ equal to g?

[choyphin]

Homework Statement

In a lab in physics, we were asked to roll a cart down an inclined plane. How would you prove that the acceleration of cart over SINE of theta (the angle of the incline) is equal to gravity?
(A hint equation was given of the Net Force parallel= mass times acceleration parallel)

3. I didn't know where to start, I'm not too savvy with formulas...

 

[CompuChip]

Draw a picture with the incline and the gravitational force. The component of the gravitational force along the incline and the arrow of the gravitational force are two sides of a triangle and the angle between them is also theta (your picture should look something like this; http://upload.wikimedia.org/wikipedia/commons/8/85/Free_body.svg -- try to convince yourself that the angle between the arrows labelled "mg" and "mg sin(theta)" is also theta). Then finding that the component along the plane is indeed sin(theta) * <magnitude of the gravitational force> is basic trigonometry.

 

[choyphin]

Why is it acceptable that you add another vector (g*sin of theta)?

 

[CompuChip]

Actually, you add two and remove one. That is: you replace the vector F that indicates the gravitational force by two vectors which add up (in a vectorial sum sense) to F. In other words, you forget about gravity, but introduce two new forces who together have the same effect.

Do you understand that (I don't have the idea that I'm very clear about this).