Q3. When two objects of unequal mass are hung vertically over a frictionless pulley of negligible mass as in the figure below, the arrangement is called an Atwood machine. The device is sometimes used in the laboratory to determine the value of g. Determine the magnitude of the acceleration of the two objects and the tension in the lightweight string.

Question

Q3. When two objects of unequal mass are hung vertically over a frictionless pulley of negligible mass as in the figure below, the arrangement is called an Atwood machine. The device is sometimes used in the laboratory to determine the value of g. Determine the magnitude of the acceleration of the two objects and the tension in the lightweight string.

Answer 1

To determine the magnitude of the acceleration of the two objects and the tension in the lightweight string, we will first draw free body diagrams for the two masses and then use Newton's second law.

Let m1 and m2 be the two masses, with m1 > m2.

For mass m1:
The force pulling the mass downward is m1*g, and the force pulling the mass upward is the tension T in the string. So we can write:
m1*g - T = m1*a (1)

For mass m2:
The force pulling the mass downward is m2*g, and the force pulling the mass upward is the tension T in the string. We can write:
T - m2*g = m2*a (2)

Now let's solve equations (1) and (2) for the acceleration a and the tension T.

Add equation (1) and equation (2) to eliminate T:
m1*g - m1*a = m2*g + m2*a

Now solve for a:
a = (m1*g - m2*g) / (m1 + m2)
a = g*(m1 - m2) / (m1 + m2)

Now to find tension T, substitute the value of a we found in equation (1):
T = m1*g - m1*a
T = m1*g - m1*g*(m1 - m2) / (m1 + m2)
T = m1*m2*g / (m1 + m2)

Therefore, the magnitude of the acceleration of the two objects is a = g*(m1 - m2) / (m1 + m2), and the tension in the lightweight string is T = m1*m2*g / (m1 + m2).