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Frictional Torque

Frictional torque is a complicating factor in the motion of pulleys. Here a mass is hanging from such a pulley.

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

τnet = τa + τf

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

Frictional torque is calculated: the free body diagram illustrates weight and tension acting on a mass hanging from a pulley.

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

Iα = 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 = Ia/R2 [eqn 2]

Combining equations 1 and 2:

mg - ma = Ia/R2

So acceleration can be calculated from

Tangential acceleration of a real pulley can be calculated from this formula, involving moment of inertia, radius, and the mass of a hanging object.

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*m2 will accelerate at 9.59 m/s2. Since α = a/R and τ = Iα , the pulley has an angular acceleration of 320 rad/s2 and is experiencing a torque of 0.0320 N*m.

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/s2. Since α = a/R and τ = Iα the angular acceleration is 9.4/0.03 = 313 rad/s2; the net torque is 0.313 N*m. We can find frictional torque, τf, from

τnet = τa + τf

Substituting,

0.313 = 0.320 -τf

Therefore

τf = 0.007 N*m

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