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**Frictional torque ** is the difference between applied torque and observed or net torque and is attributed to resistance to relative motion between surfaces.

**τ**_{net} = **τ**_{a} + **τ**_{f}

Real pulleys have mass and frictional torque. Both result in a lower acceleration for a hanging mass.

First, consider a pulley with mass but no friction. Applying Newton's Second Law to the hanging mass:

ma = mg - T

where m is the mass and a is the acceleration of the hanging mass, a is the acceleration of the hanging mass, and T is the tension in the rope. Rearranging,

T = mg - ma [eqn 1]

The torque on the pulley is given by:

* τ* = TR

This can be written as

** I**α = TR

were ** I** is the moment of inertia and α is the angular acceleration of the pulley. This can be rewritten as:

** I**(a/R) = TR

since the tangential acceleration of the pulley is the same as the acceleration of the hanging mass.

Rearranging,

T = ** I**a/R

Combining equations 1 and 2:

mg - ma = ** I**a/R

So acceleration can be calculated from

As an example, a mass of 5 kg hanging from a frictionless pulley with a 3 cm radius and a moment of inertia of 1x10^{-4} kg*m^{2} will accelerate at 9.59 m/s^{2}. Since α = a/R and * τ* =

If the same pulley also has friction, acceleration will be less. For example, suppose the mass in the above configuration is observed to accelerate at 9.4 m/s^{2}. Since α = a/R and * τ* =

**τ**_{net} = **τ**_{a} + **τ**_{f}

Substituting,

0.313 = 0.320 -**τ**_{f}

Therefore

**τ**_{f }= 0.007 N*m

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