Atwood machine

The free body diagrams of the two hanging masses of the Atwood machine. Our sign convention, depicted by the acceleration vectors is that m₁ accelerates downward and that m₂ accelerates upward, as would be the case if m₁ > m₂

An equation for the acceleration can be derived by analyzing forces. Assuming a massless, inextensible string and an ideal massless pulley, the only forces to consider are: tension force (T), and the weight of the two masses (W₁ and W₂). To find an acceleration, consider the forces affecting each individual mass. Using Newton's second law (with a sign convention of ${\displaystyle m_{1}>m_{2}}$) derive a system of equations for the acceleration (a).

As a sign convention, assume that a is positive when downward for ${\displaystyle m_{1}}$ and upward for ${\displaystyle m_{2}}$. Weight of ${\displaystyle m_{1}}$ and ${\displaystyle m_{2}}$ is simply ${\displaystyle W_{1}=m_{1}g}$ and ${\displaystyle W_{2}=m_{2}g}$ respectively.

Forces affecting m₁:

$${\displaystyle m_{1}g-T=m_{1}a}$$

Forces affecting m₂:

$${\displaystyle T-m_{2}g=m_{2}a}$$

and adding the two previous equations yields

$${\displaystyle m_{1}g-m_{2}g=m_{1}a+m_{2}a,}$$

and the concluding formula for acceleration

$${\displaystyle a=g{\frac {m_{1}-m_{2}}{m_{1}+m_{2}}}}$$

The Atwood machine is sometimes used to illustrate the Lagrangian method of deriving equations of motion.[2]

It can be useful to know an equation for the tension in the string. To evaluate tension, substitute the equation for acceleration in either of the two force equations.

$${\displaystyle a=g{m_{1}-m_{2} \over m_{1}+m_{2}}}$$

For example, substituting into ${\displaystyle m_{1}a=m_{1}g-T}$, results in

$${\displaystyle T={2gm_{1}m_{2} \over m_{1}+m_{2}}={2g \over 1/m_{1}+1/m_{2}}=m_{h}\,g}$$

where ${\displaystyle m_{h}={\frac {2m_{1}m_{2}}{m_{1}+m_{2}}}}$ is the harmonic mean of the two masses. The numerical value of ${\displaystyle m_{h}}$ is closer to the smaller of the two masses.

For very small mass differences between m₁ and m₂, the rotational inertia I of the pulley of radius r cannot be neglected. The angular acceleration of the pulley is given by the no-slip condition:

$${\displaystyle \alpha ={\frac {a}{r}},}$$

where ${\displaystyle \alpha }$ is the angular acceleration. The net torque is then:

$${\displaystyle \tau _{\mathrm {net} }=\left(T_{1}-T_{2}\right)r-\tau _{\mathrm {friction} }=I\alpha }$$

Combining with Newton's second law for the hanging masses, and solving for T₁, T₂, and a, we get:

Acceleration:

$${\displaystyle a={{g(m_{1}-m_{2})-{\tau _{\mathrm {friction} } \over r}} \over {m_{1}+m_{2}+{{I} \over {r^{2}}}}}}$$

Tension in string segment nearest m₁:

$${\displaystyle T_{1}={{m_{1}g\left(2m_{2}+{\frac {I}{r^{2}}}+{\frac {\tau _{\mathrm {friction} }}{rg}}\right)} \over {m_{1}+m_{2}+{\frac {I}{r^{2}}}}}}$$

Tension in string segment nearest m₂:

$${\displaystyle T_{2}={{m_{2}g\left(2m_{1}+{\frac {I}{r^{2}}}+{\frac {\tau _{\mathrm {friction} }}{rg}}\right)} \over {m_{1}+m_{2}+{\frac {I}{r^{2}}}}}}$$

Should bearing friction be negligible (but not the inertia of the pulley nor the traction of the string on the pulley rim), these equations simplify as the following results:

Acceleration:

$${\displaystyle a={{g\left(m_{1}-m_{2}\right)} \over {m_{1}+m_{2}+{\frac {I}{r^{2}}}}}}$$

Tension in string segment nearest m₁:

$${\displaystyle T_{1}={{m_{1}g\left(2m_{2}+{\frac {I}{r^{2}}}\right)} \over {m_{1}+m_{2}+{\frac {I}{r^{2}}}}}}$$

Tension in string segment nearest m₂:

$${\displaystyle T_{2}={{m_{2}g\left(2m_{1}+{\frac {I}{r^{2}}}\right)} \over {m_{1}+m_{2}+{\frac {I}{r^{2}}}}}}$$

• Frictionless plane
• Kater's pendulum
• Spherical cow
• Swinging Atwood's machine

Wikimedia Commons has media related to Atwood's machine.

• A treatise on the rectilinear motion and rotation of bodies; with a description of original experiments relative to the subject by George Atwood, 1764. Drawings appear on page 450.
• Professor Greenslade's account on the Atwood Machine
• Atwood's Machine by Enrique Zeleny, The Wolfram Demonstrations Project