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Physics Tutorial: Frictional Torque

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

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

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, τ_f, from

τ_net = τ_a + τ_f

Substituting,

0.313 = 0.320 -τ_f

Therefore

τ_f = 0.007 N*m

For examples of physics problems involving rotational motion or frictional torque, try

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