An Atwood machine is constructed using a disk of mass 2 kg and radius 24.8 cm. What is the acceleration of the system? The acceleration of gravity is 9.8 m/s 2. Answer in units of m/s 2.

a = (m₂-m₁) / (m₂ + m₁ + ½ m)

a = (m₂-m₁) / (m₂ + m₁ + 1)

Explanation:

An Atwood machine consists of two masses of different m1 and m2 value that pass through a pulley, in this case with mass. Let's use Newton's second law for this problem.

Assume that m₂> m₁, so the direction of descent of m₂ is positive, this implies that the direction of ascent of m₁ is positive

Equation of the side of m₁

T₁ - W₁ = m₁ a

Equation of the side of m₂

W₂ - T₁ = m₂ a

Mass pulley equation m; by convention the counterclockwise rotation is positive

τ = I α

T₁ R - T₂ R = I (-α)

The moment of inertia of a disk is

I = ½ m R²

Angular and linear acceleration are related

a = α R

α = a / R

The rotation is clockwise, so it is negative

We replace

(T₁ - T₂) R = ½ m R² (-a / R)

T₁ - T₂ = - ½ m a

Let's write our three equations together

T₁ - m₁ g = m₁ a

m₂ g - T₂ = m₂ a

T₁ - T₂ = - ½ m a

Let's multiply the last equation by (-1) and add

m₂ g - m₁ g = m₂ a + m₁ a + ½ m a

a = (m₂-m₁) / (m₂ + m₁ + ½ m)

calculate

a = (m₂ - m₁) / (m₁ + m₂ + 1)