Starting with "m1g - T1 = m1a ", "T2 - m2g = m2a" and "(T1 -...

Starting with

"m1g - T1 = m1a ", "T2 - m2g = m2a" and "(T1 - T2) R = I ?"

show the derivation steps of

"(m1 - m2)g = (m1 + m2 + I/R^2) a

Solutions

Expert Solution

Consider the Atwood machine shown at the right. It is an example of pure rotational motion; that is, the center of gravity of the pulley does not translate up/down or to the left/right.

The equations of motion for an Atwood machine that has a pulley with rotational inertia are:

For the smaller mass:

net F = ma
T₁ - mg = ma

For the larger mass:

net F = ma
Mg - T₂ = Ma

For the pulley:

net ? = I_CM?
(T₂ - T₁)r = I?

In this case net torque is calculated by finding the product of (T₂ - T₁)r where T₂ is the tension in the direction of the pulley's rotation (towards the larger mass) and T₁ is the tension in the cord on the other side of the pulley. The radius of the pulley is r. Once again, remember that T₁ and T₂ can only be equal if the pulley is "massless and frictionless."

The equation a = r? will allow you to relate the linear acceleration of the hanging masses with the angular acceleration of the pulley.

genius_generous answered 1 year ago

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