Question

An Atwood machine is constructed using two
wheels (with the masses concentrated at the
rims). The left wheel has a mass of 1.8 kg and
radius 20.01 cm. The right wheel has a mass
of 2.8 kg and radius 28.05 cm. The hanging
mass on the left is 1.55 kg and on the right
1.27 kg. What is the acceleration of the system?
The acceleration of gravity is 9.8 m/s2. Answer in units of m/s2.
Don't round answer

Answer 1

For each pulley:

$$\tau=I\alpha=mra$$

For left pulley:

$$T_1r_1-T_2r_1=m_1r_1a\to m_1a=T_1-T_2$$

For right pulley:

$$T_2r_2-T_3r_2=m_2r_2a\to m_2a=T_2-T_3$$

Then,

$$m_3a=m_3g-T_1$$

$$m_4a=-m_4g+T_3$$

Adding these four equations:

$$(m_1+m_2+m_3+m_4)a=(m_3-m_4)g$$

$$(1.8+2.8+1.55+1.27)a=(1.55-1.27)9.8$$

$$a=0.370\frac{m}{s^2}$$