Rotational Motion: Concepts, Equations, and Applications

Rotational Motion

Introduction to Rotational Motion

Rotational motion concerns the movement of rigid bodies about a fixed axis. Many physical systems, from wheels to planets, exhibit rotational dynamics. Understanding the analogies between linear and rotational motion is essential for solving problems in mechanics.

• Angle of Rotation (\(\theta\)): Measured in radians (dimensionless), defined as the ratio of arc length \(s\) to radius \(r\): $\theta = \frac{s}{r}$

• Radians and Degrees: \(\pi\) radians = 180°, so 1 radian ≈ 57.3°.

• Rotational Position: Analogous to linear position \(x\); positive angles are counterclockwise (CCW).

Angular Velocity and Angular Acceleration

Angular velocity and acceleration describe how quickly an object rotates and how its rotational speed changes.

• Angular Velocity (\(\omega\)): Rate of change of angle with respect to time. $\omega = \frac{d\theta}{dt}$ Units: radians per second (rad/s)

• Angular Acceleration (\(\alpha\)): Rate of change of angular velocity. $\alpha = \frac{d\omega}{dt}$ Units: rad/s2

• Sign Convention: Positive \(\omega\) and \(\alpha\) for CCW, negative for CW.

• Right-Hand Rule: Direction of vector \(\omega\) is along the axis of rotation, determined by curling fingers in the direction of rotation; thumb points in the direction of \(\omega\).

Relationship Between Linear and Angular Quantities

Linear and angular variables are closely related for points on a rotating object.

• Tangential Velocity (\(v\)): $v = r\omega$

• Tangential Acceleration (\(a_{\text{tan}}\)): $a_{\text{tan}} = r\alpha$

• Radial (Centripetal) Acceleration (\(a_r\)): $a_r = \frac{v^2}{r} = r\omega^2$

• Total Acceleration: $|\vec{a}| = \sqrt{a_{\text{tan}}^2 + a_r^2}$

Frequency and Period

Frequency and period describe how often a rotation repeats.

• Frequency (\(f\)): Number of revolutions per second (Hz). $f = \frac{1}{T}$

• Period (\(T\)): Time for one complete revolution. $T = \frac{2\pi}{\omega}$

• Relationship: $\omega = 2\pi f$

Rotational Kinematics (Constant Angular Acceleration)

The equations for constant angular acceleration mirror those for linear motion.

• $\omega = \omega_0 + \alpha t$

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$

• $\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$

Example: A wheel with \(\omega_0 = 50\) rad/s slows at \(\alpha = -10\) rad/s2. Number of revolutions before stopping: $0 = (50)^2 + 2(-10)\Delta\theta \implies \Delta\theta = 125$ rad $\approx 19.9$ rev

Torque and Rotational Dynamics

Definition of Torque

Torque (\(\tau\)) is the rotational analogue of force; it causes angular acceleration.

• Magnitude: $\tau = r F_{\perp} = r F \sin\theta$

• Vector Definition: $\vec{\tau} = \vec{r} \times \vec{F}$

• Units: Newton-meter (N·m)

• Lever Arm (\(r\)): Distance from axis to point of force application.

• Sign Convention: Positive for CCW, negative for CW.

Example: Pulling on a door handle 0.8 m from the hinge with 20 N at 30°: $\tau = (0.8)(20)\sin 30^\circ = 8.0$ N·m

Net Torque and Angular Acceleration

• Net torque causes angular acceleration: $\sum \tau = I \alpha$

• Torque is a vector; direction given by right-hand rule.

Moment of Inertia (\(I\))

The moment of inertia quantifies an object's resistance to angular acceleration, analogous to mass in linear motion.

• Definition for Point Masses: $I = \sum m_i r_i^2$

• For Extended Objects: Integral over mass distribution.

• Examples:

  • Two masses at distance \(r\): $I = 2mr^2$

  • Hoop (mass \(M\), radius \(R\)): $I = MR^2$

  • Solid disk: $I = \frac{1}{2}MR^2$

• Parallel Axis Theorem: For axis a distance \(d\) from center of mass: $I = I_{\text{cm}} + Md^2$

Table: Moments of Inertia for Common Shapes

Shape	Axis	Moment of Inertia (I)
Hoop	Through center	$MR^2$
Disk	Through center	$\frac{1}{2}MR^2$
Solid sphere	Through center	$\frac{2}{5}MR^2$
Thin spherical shell	Through center	$\frac{2}{3}MR^2$
Thin rod	Center	$\frac{1}{12}ML^2$
Thin rod	End	$\frac{1}{3}ML^2$

Rotational Kinetic Energy and Rolling Motion

Rotational Kinetic Energy

• Rotational KE: $KE_{\text{rot}} = \frac{1}{2}I\omega^2$

• Translational KE: $KE_{\text{trans}} = \frac{1}{2}Mv^2$

• Total KE for Rolling Object: $KE_{\text{tot}} = \frac{1}{2}Mv_{\text{cm}}^2 + \frac{1}{2}I_{\text{cm}}\omega^2$

• For rolling without slipping: $v_{\text{cm}} = \omega R$

Conservation of Energy in Rolling Motion

When objects roll down an incline, both translational and rotational kinetic energy must be considered.

• Energy Conservation: $Mgh = \frac{1}{2}Mv^2 + \frac{1}{2}I\omega^2$

• Objects with smaller moment of inertia reach the bottom faster (e.g., sphere faster than disk, disk faster than hoop).

• Example: For a disk, $I = \frac{1}{2}MR^2$, so $v = \sqrt{\frac{4gh}{3}}$

• For a sliding (non-rotating) mass: $v = \sqrt{2gh}$ (faster than rolling objects)

Special Cases and Applications

• Falling Rod: For a rod of length \(L\), mass \(M\), pivoted at one end, the speed of the tip when it hits the ground is: $v = \sqrt{3gL\sin\theta}$

• Center of Mass in Gravitational Potential Energy: $PE_{\text{grav}} = Mgh_{\text{cm}}$

Summary Table: Linear vs. Rotational Quantities

Linear	Rotational
Position (x)	Angle (\(\theta\))
Velocity (v)	Angular velocity (\(\omega\))
Acceleration (a)	Angular acceleration (\(\alpha\))
Force (F)	Torque (\(\tau\))
Mass (M)	Moment of inertia (I)
$F = Ma$	$\tau = I\alpha$
$KE = \frac{1}{2}Mv^2$	$KE = \frac{1}{2}I\omega^2$

Additional info:

• Proofs for the parallel axis theorem and kinetic energy of rolling objects are provided in the appendices, using vector algebra and properties of the center of mass.

• For more complex shapes, moments of inertia require integration over the mass distribution.