Topic 5.6 Newton's Second Law in Rotational Form: relate net torque, rotational inertia and angular acceleration through $\tau_{net} = I\alpha$, and apply it to pulleys and combined translational-rotational systems.

What this topic is asking

The College Board (Topic 5.6) wants you to relate net torque, rotational inertia and angular acceleration through $\tau_{net} = I\alpha$, and to apply it to massive pulleys and combined translational-rotational systems. This is the rotational counterpart of $F = ma$ and the dynamical heart of Unit 5: an unbalanced torque produces angular acceleration, and many problems couple a rotating body to a translating one through a string.

The rotational form of Newton's second law

This is the rotational mirror of $F_{net} = ma$. Torque plays the role of force (the cause of angular acceleration), rotational inertia plays the role of mass (the resistance to it), and angular acceleration replaces linear acceleration. An unbalanced net torque spins a body up or slows it down at a rate set by its rotational inertia: the same torque produces a smaller angular acceleration on a body with larger $I$. The analogy is exact, so the problem-solving habits from translational dynamics carry straight over.

Massive pulleys

A classic application is a pulley with mass, which the idealized massless pulley ignores. A real pulley resists being spun up, so the string tensions on its two sides are not equal; the difference provides the torque that angularly accelerates it. For a string wrapped around a pulley of radius $R$, the tension acts at the rim, giving a torque $\tau = TR$, so the rotational equation is $TR = I\alpha$. The string does not slip on the pulley, so the linear acceleration of the string (and the hanging block) equals the rim's tangential acceleration: $a = R\alpha$. This constraint links the rotational and translational equations.

Combined translational and rotational systems

Many problems couple a translating object to a rotating one: a block on a string over a massive pulley, a yo-yo, a cylinder rolling down a ramp. The method is to treat each part with its own second law:

and then close the system with a kinematic constraint relating $a$ and $\alpha$ (typically $a = R\alpha$ from a string that does not slip or a body that rolls without sliding). You get as many equations as unknowns and solve simultaneously. The constraint is the step students most often forget; without it the translational and rotational equations cannot be combined.

Try this

Q1. A net torque of $6.0$ N m acts on a wheel of rotational inertia $3.0$ kg m squared. Calculate its angular acceleration. [2 points]

• Cue. $\alpha = \tau_{net}/I = 6.0/3.0 = 2.0$ rad/s squared.

Q2. State the constraint that links the linear acceleration of a block to the angular acceleration of the non-slipping pulley it hangs from. [2 points]

• Cue. $a = R\alpha$, where $R$ is the pulley's radius.