Atwood machine

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Illustration of Atwood machine, 1905.

The Atwood machine (or Atwood's machine) was invented in 1784 by the English mathematician George Atwood as a laboratory experiment to verify the mechanical laws of motion with constant acceleration. Atwood's machine is a common classroom demonstration used to illustrate principles of classical mechanics.

The ideal Atwood Machine consists of two objects of mass m1 and m2, connected by an inextensible massless string over an ideal massless pulley. [1]

When m1 = m2, the machine is in neutral equilibrium regardless of the position of the weights.

When m1 ≠ m2 both masses experience uniform acceleration.

Equation for constant acceleration[edit]

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

We are able to derive an equation for the acceleration by using force analysis. If we consider a massless, inextensible string and an ideal massless pulley, the only forces we have to consider are: tension force (T), and the weight of the two masses (W1 and W2). To find an acceleration we need to consider the forces affecting each individual mass. Using Newton's second law (with a sign convention of m_1>m_2) we can derive a system of equations for the acceleration (a).

As a sign convention, we assume that a is positive when downward for m_1, and that a is positive when upward for m_2. Weight of m_1 and m_2 is simply W_1 = m_1 g and W_2 = m_2 g respectively.

Forces affecting m1:

\; m_1g-T=m_1a

Forces affecting m2:

\; T-m_2g=m_2a

and adding the two previous equations we obtain

\; m_1g-m_2g=m_1a+m_2a,

and our concluding formula for acceleration

a = g{m_1-m_2 \over m_1+m_2}

Conversely, the acceleration due to gravity, g, can be found by timing the movement of the weights, and calculating a value for the uniform acceleration a:  d = {1 \over 2} at^2 .

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

Equation for tension[edit]

It can be useful to know an equation for the tension in the string. To evaluate tension we substitute the equation for acceleration in either of the 2 force equations.

a = g{m_1-m_2 \over m_1 + m_2}

For example substituting into m_1 a = m_1 g-T, we get

T={2 g m_1 m_2 \over m_1 + m_2}

Equations for a pulley with inertia and friction[edit]

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:

 \alpha = {a\over r},

where  \alpha is the angular acceleration. The net torque is then:

\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:

 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:

 T_1 = {{m_1 g (2 m_2 + {{I} \over {r^2}} + {{\tau_{\mathrm{friction}}} \over {r g}})} \over {m_1 + m_2 + {{I} \over {r^2}}}}

Tension in string segment nearest m2:

 T_2 = {{m_2 g (2 m_1 + {{I} \over {r^2}} + {{\tau_{\mathrm{friction}}} \over {r g}})} \over {m_1 + m_2 + {{I} \over {r^2}}}}

Should bearing friction be negligible (but not the inertia of the pulley and not the traction of the string on the pulley rim), these equations simplify as the following results:

Acceleration:

 a = {{g (m_1 - m_2)} \over {m_1 + m_2 + {{I} \over {r^2}}}}

Tension in string segment nearest m1:

 T_1 = {{m_1 g (2 m_2 + {{I} \over {r^2}})} \over {m_1 + m_2 + {{I} \over {r^2}}}}

Tension in string segment nearest m2:

 T_2 = {{m_2 g (2 m_1 + {{I} \over {r^2}})} \over {m_1 + m_2 + {{I} \over {r^2}}}}

Practical implementations[edit]

Atwood's original illustrations show the main pulley's axle resting on the rims of another four wheels, to minimize friction forces from the bearings. Many historical implementations of the machine follow this design.

An elevator with a counterbalance approximates an ideal Atwood machine and thereby relieves the driving motor from the load of holding the elevator cab — it has to overcome only weight difference and inertia of the two masses. The same principle is used for funicular railways with two connected railway cars on inclined tracks.

See also[edit]

Notes[edit]

  1. ^ Tipler, Paul A. (1991). Physics For Scientists and Engineers, Third Edition, Extended Version. New York: Worth Publishers. ISBN 0-87901-432-6.  Chapter 6, example 6-13, page 160.
  2. ^ Goldstein, Herbert (1980). Classical Mechanics, second Edition. New Delhi: Addison-Wesley/Narosa Indian Student Edition. ISBN 81-85015-53-8.  Section 1-6, example 2, pages 26-27.

External links[edit]