Atwood machine.We are able to derive an equation for the acceleration by

using force analysis. If we consider a massless, inelastic string and an ideal massless pulley the only forces we

have to consider are: tension force (N), and the weight of the two masses (mg). To find an acceleration we need to consider the forces affecting each individual mass. Using

Newton's laws (if m1 > m2) we can derive a system of equations for the acceleration (a).

Forces affecting m1:

forces affecting m2:

and adding the two previous equations we obtain,

and at last

,

• 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:

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

• 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.

• For example substituting into m1a = N − m1g, we get

• The tension cannot accurately be found in using this method due to torque of the pulley.

.

Atwood Example:

• Using an Atwood machine m1 = 4kg

• m2 = 3.5kg• Find the acceleration of the

setup and the tension in the wire.

Answer:

• a = 9.81 m/s2 (4kg – 3.5 kg) (4kg + 3.5 kg)

• a = .654 m/s2

• N = 9.81 m/s2 2(4kg)(3.5 kg) (4kg + 3.5 kg)

• N = 36.624 N