Newton's second law for rotation Knowledge-Based Study

Study notes

1. Translational vs. Rotational Motion

   In linear mechanics, Newton's second law states that a net external force acting on an object causes an acceleration inversely proportional to its mass: $\Sigma F = ma$.

   When analyzing rotational motion about a fixed axis, analogous physical quantities describe the state of motion:

   • Force ($F$) corresponds to Torque ($\tau$), the rotational effectiveness of a force.
   • Mass ($m$) corresponds to Moment of Inertia ($I$), the measure of an object's resistance to changes in its rotational state.
   • Linear Acceleration ($a$) corresponds to Angular Acceleration ($\alpha$), the rate of change of angular velocity.

   Formulation of the Rotational Law

   For a rigid body rotating about a fixed principal axis, Newton's second law for rotation is formulated as:

   $\Sigma \tau = I\alpha$

   Where:

   • $\Sigma \tau$ is the net external torque acting on the system, measured in newton-meters ($\text{N}\cdot\text{m}$).
   • $I$ is the moment of inertia about the axis of rotation, measured in kilogram-meters squared ($\text{kg}\cdot\text{m}^2$).
   • $\alpha$ is the resulting angular acceleration, measured in radians per second squared ($\text{rad/s}^2$).

   Just as in linear motion, internal torques between particles cancel in pairs due to Newton's third law. Therefore, only net external torques contribute to angular acceleration.

   Key Relationships

   To determine the net torque exerted by an applied force $F$ at a distance $r$ from the rotation axis:

   $\tau = r F \sin\theta$

   Here, $\theta$ is the angle between the position vector $\vec{r}$ and the force vector $\vec{F}$. Alternatively, torque equals force multiplied by the lever arm: $\tau = F r_\perp$.

   The moment of inertia depends on both the total mass of the body and how that mass is distributed relative to the axis of rotation:

   $I = \Sigma m_i r_i^2$ or $I = \int r^2 dm$

   Because $I$ depends on the square of the distance from the axis, shifting mass farther from the pivot drastically increases resistance to angular acceleration.

   Step-by-Step Problem Solving

   1. Identify the axis of rotation: Ensure all torques and the moment of inertia are evaluated about this exact same axis.
   2. Construct a free-body diagram: Draw all external forces acting at their specific points of application.
   3. Establish a sign convention: Choose one direction (typically counterclockwise) as positive for both torque and angular acceleration.
   4. Compute net torque: Calculate $\Sigma \tau = \Sigma (r_i F_i \sin\theta_i)$.
   5. Apply the rotational law: Substitute the net torque and rotational inertia into $\Sigma \tau = I\alpha$ to solve for the unknown angular acceleration or torque.