Physics Tutorial – Atwood’s Machine

Atwood’s machine is commonly used in the classroom to demonstrate the mechanical laws of motion with constant acceleration. This device consists of two masses connected by string over a frictionless pulley. This device is commonly used in a variety of grade 11 physics problems dealing with classical mechanics.

Example:

Two unequal masses are hung vertically over a frictionless pulley. If $m_1$ is 1.00 Kg and $m_2$ is 2.00 Kg, calculate the acceleration of the system and the tension in the string $($Gravity, $g = 9.81m/s^2)$.

Solution:

Since $m_1 > m_2$ the system will accelerate in favour of $m_2$ and therefore assigned a positive value. For mass 1 we find that:

$(1)$ $ \Sigma F_y = T – m_1g = m_1a$

For mass 2 we find that:

$(2)$ $ \Sigma F_y = T – m_1g = m_1a$

When equation (2) is subtracted from equation (1), the result is:

$(3)$ $ -m_1g + m_2g = m_1a + m_2a$

Substitute all known variables and solve:

$(4)$ $ (-1.00 x 9.81) + (2 x 9.81) = 1a + 2a$

$(5)$ $ 9.81 = 3a$

$(6)$ $ a = 3.27m/s^2$

Now that the acceleration of the system is known we can substitute the value of “a” into equation (1) to solve for T:

$(7)$ $ T – m_1g = m_1a$

$(8)$ $ T – 1.00(9.81) = 1.00(3.27)$

$(9)$ $ T – 9.81 = 3.27$

$(10)$ $ T = 13.08$ Newtons