Physics help from experts.
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.
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.
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, τf, from
τnet = τa + τf
0.313 = 0.320 -τf
τf = 0.007 N*m