A device known as an atwood's machine consist of two masses hanging from the ends of a vertical rope that passes over a pulley. Assume the rope and pulley are massless and dthere is no friction in the pulley. Mass mA is grater than mass Mb. Find the expressions for the magnitude of their acceleration, a, and the tension in the rope, T. Express your answers in terms of the masses and g, the acceleration due to gravity

Answer:

In this case we must draw a free body diagram in each mass. Using the traditional axis x and y. Once we do that we have an empty x axis, because there are no forces there. In the case of the y axis we have two forces opposing each other: the weight and the tension. If mass mA is greater that means that this one falls and the other one goes up, so:

Balance of forces (Mass A) mA*g - T = a*mA (1) ; (Mass B) T - mB*g = a*mB (2)

Here we have two equations and two unknown values (T and a) so both of then can be found. In fact, when we add both equations we have a solution for the acceleration.

(1) + (2) :::::: mA*g - mB*g = a*mA + a*mB :::::::::::: a = g*( mA - mB )/( mA + mB )

And with the acceleration we can use (1) or (2) to find the tension:

Using (2) :::::: T = mB*(a + g) ::::::: T = 2*g*(mB*mA)/(mA + mB)