motor.
In a rotational system, the angular acceleration, due to the net torque acting on it, is determined by its moment‐of‐inertia J. The example below shows how to calculate the moment‐of‐inertia J of a rotating solid cylindrical mass.
1 Calculate the moment‐of‐inertia J of a solid cylinder that is free to rotate about its axis, as shown in Fig. 2-6a, in terms of its mass M and the radius r1.
2 Given that a solid steel cylinder has a radius r1 = 6 cm, length ℓ = 18 cm, and material density ρ = 7.85 × 103 kg/m3, calculate its moment‐of‐inertia J.
Solution
1 From Newton’s Law of Motion, in Fig. 2-6a, to accelerate a differential mass dM at a radius r, the net differential force df required in a perpendicular (tangential) direction, from Eq. (2-1), is(2-11)
where the linear speed u in terms of the angular speed ωm (in rad/s) is
Multiplying both sides of Eq. (2-11) by the radius r, recognizing that (r df) equals the net differential torque dT and using Eq. (2-12),
The same angular acceleration
where ρ is the material density in kg/m3. Substituting dM from Eq. (2-14) into Eq. (2-13),
(2-15)
The net torque acting on the cylinder can be obtained by integrating over all differential elements in terms of r, θ, and ℓ as
(2-16)
Carrying out the triple integration yields
where the quantity within the brackets in Eq. (2-17) is called the moment‐of‐inertia J, which for a solid cylinder is
Since the mass of the cylinder in Fig. 2-6a is
(2-20)
1 Substituting r1 = 6 cm, length ℓ = 18 cm, and ρ = 7.85 × 103 kg/m3 in Eq. (2-19), the moment‐of‐inertia Jcyl of the cylinder in Fig. 2-5a is
Fig. 2-6 Calculation of the inertia, Jcyl, of a solid cylinder.
The net torque TJ acting on the rotating body of inertia J causes it to accelerate. Similar to systems with linear motion, where fM = M a Newton’s Law in rotational systems becomes
where the angular acceleration α(=dω/dt) in rad/s2 is
which is similar to Eq. (2-18) in the previous example. In MKS units, a torque of 1 Nm acting on the inertia of 1 kg ⋅ m2 results in an angular acceleration of 1 rad/s2.
In systems such as the one shown in Fig. 2-7a, the motor produces an electromagnetic torque Tem. The bearing friction and wind resistance (drag) can be combined with the load torque TL opposing the rotation. In most systems, we can assume that the rotating part of the motor with inertia JM is rigidly coupled (without flexing) to the load inertia JL. The net torque, which is the difference between the electromagnetic torque developed by the motor and the load torque opposing it, causes the combined inertias of the motor and the load to accelerate in accordance with Eq. (2-22):
where
