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5.6 Newton’s Second Law in Rotational Form

Syllabus
2024
Topic
5.6
Level
—

Learning objectives

5.6A—Describe the conditions under which a system’s angular velocity changesDescribe the conditions under which a system’s angular velocity changes.• Angular velocity changes when the net torque exerted on the object or system is not equal to zero.• The rate at which the angular velocity of a rigid system changes is directly proportional to the net torque exerted on the rigid system and is in the same direction. The angular acceleration of the rigid system is inversely proportional to the rotational inertia of the rigid system. Relevant equation:• T o fully describe a rotating rigid system, linear and rotational analyses may need to be performed independently.

Predict angular acceleration from net torque

Identify when angular velocity changes

A rigid system's angular velocity changes when the net torque about the chosen axis is nonzero. The resulting angular acceleration has the same signed rotational sense as the net torque.

Connect net torque and rotational inertia

αsys=∑iτiIsys=τnetIsys\alpha_{\mathrm{sys}}=\frac{\sum_i\tau_i}{I_{\mathrm{sys}}}=\frac{\tau_{\mathrm{net}}}{I_{\mathrm{sys}}}

Quantity Meaning SI unit
τnet\tau_{\mathrm{net}} Signed sum of torques about the axis N m\text{N}\,\text{m}
IsysI_{\mathrm{sys}} Rotational inertia about that axis kg m2\text{kg}\,\text{m}^2
αsys\alpha_{\mathrm{sys}} Rate of change of angular velocity rad s−2\text{rad}\,\text{s}^{-2}

Predict proportional changes

Change Angular acceleration response Why
Double τnet\tau_{\mathrm{net}} at fixed II Doubles α∝τnet\alpha\propto\tau_{\mathrm{net}}
Double II at fixed τnet\tau_{\mathrm{net}} Halves α∝1/I\alpha\propto1/I
Make τnet=0\tau_{\mathrm{net}}=0 Becomes zero Angular velocity stays constant

Calculate angular acceleration

Rigid system: take counterclockwise as positive. If τnet=+6.0 N m\tau_{\mathrm{net}}=+6.0\,\text{N}\,\text{m} and Isys=2.0 kg m2I_{\mathrm{sys}}=2.0\,\text{kg}\,\text{m}^2,

α=+6.02.0=+3.0 rad s−2\alpha=\dfrac{+6.0}{2.0}=+3.0\,\text{rad}\,\text{s}^{-2}.

The positive result means the angular velocity changes in the counterclockwise sense under this convention.

Separate rotational and linear dynamics

Rotational analysis alone does not fully describe every rigid system. Use ∑τ=Iα\sum\tau=I\alpha for rotation about the chosen axis and, when the center of mass also translates, perform a separate linear analysis with ∑F⃗=ma⃗cm\sum\vec F=m\vec a_{\mathrm{cm}}. One equation does not replace the other.