Physics Notes: Rotation of Rigid Bodies, Torque, and Angular Momentum

Fundamental Rotational Quantities

• Arc Length ($s$) and Angle ($\theta$):

  • The relationship between the linear distance traveled along a circular path (arc length) and the angular displacement is given by: $s = r\theta$.

  • Radius is denoted by $r$.

  • Angle ($\theta$) is typically measured in radians.

• Angular Velocity ($\omega$):

  • Represents the rate of change of the angle with respect to time: $\omega = \frac{d\theta}{dt}$.

  • The period ($T$) is the time required for one full revolution.

  • The frequency ($f$) is the number of revolutions per unit time ($f = \frac{1}{T}$).

  • The relationship between angular velocity and frequency is: $\omega = 2̀̀\pi f = \frac{2̀̀\pi}{T}$.

• Angular Velocity as a Vector:

  • Angular velocity is categorized as an axial vector or a pseudo-vector.

  • Definition: Vectors involving exactly one cross product in their definition (or relating to rotation about an axis) are axial vectors.

  • Other examples of axial vectors include:

    • Angular momentum ($L$).

    • Torque ($\tau$).

    • The magnetic field ($B$).

Angular Acceleration

• Definition:

  • Average angular acceleration is the change in angular velocity over time: $\alpha_{\text{av}} = \frac{\Delta\omega}{\Delta t}$.

  • Instantaneous angular acceleration: $\alpha = \frac{d\omega}{dt}$.

  • The SI unit for angular acceleration is $\text{rad/s}^2$.

• Acceleration and Speed Dynamics:

  • If \omega > 0 and \alpha > 0, angular speed increases.

  • If \omega > 0 and \alpha < 0, angular speed decreases.

  • If \omega < 0 and \alpha < 0, angular speed increases (in the negative direction).

  • If \omega < 0 and \alpha > 0, angular speed decreases.

Rotational Kinematics

• Constant Angular Acceleration:

  • If $\alpha$ is constant, the angular velocity varies linearly with time: $\omega = \omega_0 + \alpha t$.

  • The equations for rotational kinematics are direct analogues to linear kinematics:

    • Linear: $v = v_0 + at$ → Rotational: \omega = ̀̀\omega_0 + ̀̀\alpha t

    • Linear: $x = v_0 t + \frac{1}{2}at^2$ → Rotational: $\theta = \omega_0 t + \frac{1}{2}\alpha t^2$

    • Linear: $v^2 = v_0^2 + 2ax$ → Rotational: \omega^2 = \omega_0^2 + 2\alphà̀\theta

Connections Between Linear and Rotational Quantities

• Velocity:

  • The velocity of a point on a rotating rigid body is purely tangential: $v = r̀̀\omega$.

• Acceleration:

  • Points on a rotating object experience two types of linear acceleration:

    • Tangential Acceleration ($a_t$): Due to the changing magnitude of angular speed: $a_t = r̀̀\alpha$.

    • Centripetal (Radial) Acceleration ($a_c$): Due to the changing direction of the velocity vector: $a_c = \frac{v^2}{r} = r̀̀\omega^2$.

  • If an object rotates at a constant angular speed ($\alpha = 0$), it only has centripetal acceleration. If it is speeding up or slowing down, it has both tangential and centripetal components.

Rotational Kinetic Energy and Moment of Inertia

• Rotational Kinetic Energy ($K_{rot}$):

  • For a single point mass moving in a circle: $K = \frac{1}{2}mv^2 = \frac{1}{2}m(r̀̀\omega)^2$.

  • For an extended rigid body, which is a collection of point masses, the total rotational kinetic energy is the sum: $K_{rot} = \frac{1}{2}(\sum m_i r_i^2)\omega^2$.

  • This is rewritten as: $K_{rot} = \frac{1}{2}I\omega^2$.

• Moment of Inertia ($I$):

  • Definition for discrete masses: $I = \sum m_i r_i^2$.

  • Definition for continuous bodies: $I = \int r^2 dm$.

  • Factors determining $I$:

    1. The object's geometry and mass distribution.

    2. The chosen axis of rotation.

  • Physical Meaning: Mass provides resistance to translational acceleration; moment of inertia provides resistance to angular acceleration.

• Total Kinetic Energy:

  • The total kinetic energy of an object is the sum of its translational and rotational kinetic energies: $K_{total} = \frac{1}{2}Mv_{cm}^2 + \frac{1}{2}I\omega^2$.

  • For an object rolling without slipping: $v = r̀̀\omega$.

• Comparative Case Study: Race between Cylinder and Hoop:

  • Scenario: A cylinder and a hoop of the same mass $m$ race down an incline.

  • Result: The cylinder wins.

  • Explanation: The hoop has its mass distributed further from the axis, giving it a larger moment of inertia ($I_{hoop} = MR^2$ vs. $I_{cylinder} = \frac{1}{2}MR^2$). Since more energy must go into rotating the hoop, less is available for translational velocity ($v$).

Parallel-Axis Theorem and Potential Energy

• Parallel-Axis Theorem:

  • Relates the moment of inertia about an axis through the center of mass ($I_{cm}$) to the moment of inertia about a parallel axis located a distance $d$ away:

  • $I = I_{cm} + Md^2$

• Gravitational Potential Energy ($U_{grav}$):

  • The potential energy of an extended object is calculated as if all mass were concentrated at the center of mass: $U_{grav} = Mgy_{cm}$.

  • Application: In high jump (Fosbury Flop), an athlete arches their body so the center of mass ($y_{cm}$) passes under the bar. This minimizes the required increase in gravitational potential energy to clear the bar.

Torque ($\tau$)

• Definitions:

  • Vector definition: $\mathbf{\tau} = \mathbf{r} \times \mathbf{F}$.

  • Magnitude: $\tau = rF\sin(\theta)$.

  • Alternatively, $\tau = r F_{\perp}$ or $\tau = r_{\perp} F$, where $r_{\perp} = r\sin(\theta)$ is the moment arm (the perpendicular distance from the axis to the line of action of the force).

• Dependencies:

  • The chosen axis of rotation.

  • The point at which the force ($F$) is applied.

  • The perpendicular component of the force ($F_{\perp}$).

Newton's Second Law for Rotation

• For a mass constrained to move in a circle by a rod, a tangential force results in: $F = ma_t = m(r\alpha)$.

• Multiplying by $r$ gives the torque: $\tau = rF = (mr^2)\alpha = I\alpha$.

• The general form for any rigid body is: $\Sigma \tau = I\alpha$.

Angular Momentum ($L$)

• Definitions for a Point Mass:

  • Linear momentum is $\mathbf{p} = m\mathbf{v}$.

  • Angular momentum about an origin: $\mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{r} \times m\mathbf{v}$.

  • Magnitude: $L = rmv\sin(\theta) = rmv_{\perp} = r_{\perp}mv$.

• Definition for Rigid Bodies:

  • For an extended object undergoing circular motion or spinning: $L = I\omega$.

• Relationship to Torque:

  • Directly analogous to $F = \frac{dp}{dt}$, torque is the rate of change of angular momentum: $\tau = \frac{dL}{dt}$.

  • This is the general form of Newton's second law for rotation.

Conservation of Angular Momentum

• Principle: If the net external torque on a system is zero ($\Sigma \tau_{ext} = 0$), then the total angular momentum remains constant: $L_{initial} = L_{final}$.

• Example: The Falling Cat:

  • A cat can twist its body in mid-air to land on its feet even if it starts with zero angular momentum.

  • By rotating different parts of its body in opposite directions, the total angular momentum of the cat remains zero at all times throughout the process.

Rolling Motion

• Rolling Without Slipping:

  • The translational speed of the center of mass is linked to the angular speed: $v = R\omega$.

  • Rolling motion is mathematically equivalent to the sum of pure translation (all points move at $v$) and pure rotation (points move at $v = R\omega$ relative to the center).

Practice Problems and Applications

• Bicycle Tire Distance: A tire with radius $0.33\text{ m}$ travels $1.95\text{ m}$. The angle of rotation is $\theta = \frac{s}{r} = \frac{1.95}{0.33} = 5.91\text{ rad}$.

• Earth's Angular Speed:

  • (a) $\omega = \frac{2\pi}{24 \times 3600} \approx 7.27 \times 10^{-5}\text{ rad/s}$.

  • (b) $\frac{360^{\circ}}{24\text{ hours}} = 15^{\circ}\text{/hour}$.

• Laboratory Centrifuge:

  • Initial speed: $3850\text{ rpm}$.

  • Time to rest: $10.2\text{ s} = 0.17\text{ min}$.

  • Deceleration ($\alpha$): $\frac{0 - 3850\text{ rpm}}{0.17\text{ min}} \approx -22,647\text{ rev/min}^2$.

• High-Speed Pilot G-Forces:

  • Limit for consciousness: a_c < 7.00\text{ g} = 7.00 \times 9.8\text{ m/s}^2 = 68.6\text{ m/s}^2.

  • Plane speed: $v = 245\text{ m/s}$.

  • Minimum radius: $r_{min} = \frac{v^2}{a_c} = \frac{245^2}{68.6} \approx 875\text{ m}$.

• Merry-Go-Round (Conservation of $L$):

  • A child of mass $28.0\text{ kg}$ moves from the edge (radius $1.60\text{ m}$) to the center.

  • System total inertia decreases, so angular velocity $\omega$ must increase: $(I_{platform} + I_{children, initial})\omega_{initial} = (I_{platform} + I_{children, final})\omega_{final}$.