DP IB Physics: SL

A. Space, time and motion

A.4 Rigid body mechanics

DP IB Physics: SL

A. Space, time and motion

A.4 Rigid body mechanics 

Understandings
Students should understand:
a) The torque τ of a force about an axis as given by [math]τ = Fr sin θ[/math]
b) That bodies in rotational equilibrium have a resultant torque of zero
c) That an unbalanced torque applied to an extended, rigid body will cause angular acceleration
d) That the rotation of a body can be described in terms of angular displacement, angular velocity and angular acceleration
e)

That equations of motion for uniform angular acceleration can be used to predict the body’s angular position θ, angular displacement [math]\Delta \theta[/math] angular speed ω and angular acceleration α, as given by

[math]\begin{gather}
\Delta \theta = \frac{\omega_f + \omega_i}{2} t \\
\omega_f = \omega_i + a t \\
\Delta \theta = \omega_i t + \frac{1}{2} a t^2 \\
\omega_f^2 = \omega_i^2 + 2 a \Delta \theta
\end{gather}[/math]
f) That the moment of inertia I depends on the distribution of mass of an extended body about an axis of rotation
g)

The moment of inertia for a system of point masses as given by

[math]I = Σmr^2[/math]
h) Newton’s second law for rotation as given by τ = Iα where τ is the average torque
i)

That an extended body rotating with an angular speed has an angular momentum L as given by

[math]L = Iω[/math]
j) That angular momentum remains constant unless the body is acted upon by a resultant torque
k)

That the action of a resultant torque constitutes an angular impulse ΔL as given by

[math]ΔL = τΔt = Δ(Iω)[/math]
l)

The kinetic energy of rotational motion as given by

[math]E_k = \frac{1}{2} I \omega^2 = \frac{L^2}{2I}[/math]

  • (a) Torque (τ) of a Force About an Axis

  • Torque is the rotational equivalent of force. It is the measure of the turning effect of a force about a pivot or axis of rotation. The magnitude of torque is given by:
  • [math]τ = Frsinθ[/math]
  • Where:
  • – τ = Torque (Newton-meters, Nm)
  • – F = Applied force (Newtons, N)
  • – r = Perpendicular distance from the axis of rotation to the line of action of the force (meters, m)
  • – θ = Angle between the force and the lever arm
  • Figure 1 Torque of a force about an axis
  • ⇒  Features of Torque:
  • If [math]θ = 90^0[/math], then
  • [math]τ = Frsinθ \\ τ = Frsin 90^0[/math]
  • [math]sin 90^0 = 1.[/math]
  • [math]τ = Fr ( \text{Maximum Torque})[/math]
  • If [math]θ = 0^0 [/math] or [math]180^0[/math], then
  • [math]τ = Frsinθ \\ τ = Frsin0^0[/math]
  • [math]sin0^0 = 0 \text{ or } sin180^0 = 0[/math]
  • [math]τ = Fr(0) \\τ = 0[/math]
  • So, no torque is produced.
  • The greater the force or distance, the larger the torque.
  • ⇒  Example 1: Torque Calculation
  • A wrench applies a 20 N force at a distance of 5 m from a bolt, perpendicular to the wrench.
  • [math]\begin{gather}
    \tau = F r \sin \theta \\
    \tau = (20)(0.5) \sin 90^\circ \\
    \tau = 10 \, \text{Nm}
    \end{gather}[/math]
  • If the force was applied at 30° instead:
  • [math]\begin{gather}
    \tau = F r \sin \theta \\
    \tau = (20)(0.5) \sin 30^\circ \\
    \tau = 5 \, \text{Nm}
    \end{gather}[/math]
  • Torque is reduced when force is applied at an angle.
  • Figure 2 Torque applies on a body with distance effect

  • (b) Rotational Equilibrium: Resultant Torque is Zero

  • A body is in rotational equilibrium if the sum of all torques acting on it is zero:
  • [math]\sum \tau_{\text{clockwise}} = \sum \tau_{\text{anticlockwise}}[/math]
  • ⇒  Example 2: Beam in Equilibrium
  • A uniform 2 m long beam is pivoted at the center. A 10 N force acts 1 m to the left of the pivot, and a 5 N force acts 2 m to the right.
  • Clockwise Torque:
  • [math]\begin{gather}
    \sum \tau_{\text{clockwise}} = 5 \times 2 \\
    \sum \tau_{\text{clockwise}} = 10 \, \text{Nm}
    \end{gather}[/math]
  • Anticlockwise Torque:
  • [math]\begin{gather}
    \sum \tau_{\text{anticlockwise}} = 10 \times 1 \\
    \sum \tau_{\text{anticlockwise}} = 10 \, \text{Nm}
    \end{gather}[/math]
  • Since both torques are equal, the beam is in equilibrium.
  • – If torques are balanced, the object remains stationary or rotates at constant speed.
  • – If unbalanced, angular acceleration
  • Figure 3 Torque

  • (c) Unbalanced Torque and Angular Acceleration

  • If an unbalanced torque (τ) is applied, the rigid body undergoes angular acceleration (α) according to:
  • [math]τ = Iα[/math]
  • Where:
  • I = Moment of inertia (kg·m²) – the rotational equivalent of mass
  • α = Angular acceleration (rad/s²)
  • – A large torque on a small moment of inertia causes rapid angular acceleration.
  • – A large moment of inertia (e.g., a spinning flywheel) requires more torque to accelerate.
  • Example 3: Rotating Disc
  • A disc has I = 0.2 kg·m² and a torque of 5 Nm is applied. Find its angular acceleration.
  • [math]\begin{gather}
    \tau = I \alpha \\
    \alpha = \frac{\tau}{I} \\
    \alpha = \frac{5}{0.2} \\
    \alpha = 25 \, \text{rad/s}^2
    \end{gather}[/math]
  • Higher torque leads to faster rotation.

  • (d) Angular Quantities in Rotational Motion

  • (I) Angular Displacement (θ)
  • Definition:
  • The angle through which a point or line has been rotated in a given direction around a fixed axis.
  • Unit:
  • Radians (rad)
  • Formula:
  • [math]\theta = \frac{r}{s}[/math]
  • Where:
  • – s = arc length (m)
  • – r = radius (m)
  • (II) Angular Velocity (ω)
  • Definition:
  • The rate of change of angular displacement with respect to time.
  • Unit:
  • Radians per second (rad/s)
  • Formula:
  • [math]ω = \frac{dθ}{dt}[/math]
  • Where:
  • – ω = angular velocity
  • – dθ = change in angular displacement
  • – dt = time interval
  • (III) Angular Acceleration (α)
  • Definition:
  • The rate of change of angular velocity with respect to time.
  • Unit:
  • Radians per second squared (rad/s2)
  • Formula:
  • [math]α = \frac{dω}{dt}[/math]
  • Where:
  • – α = angular acceleration
  • – dω = change in angular velocity
  • – dt = time interval

  • (e) Equations of Motion for Uniform Angular Acceleration

  • Just like linear motion, we have four equations of motion for rotational motion, assuming constant angular acceleration:
  • (i) Angular Displacement Formula
  • [math]\Delta \theta = \frac{\omega_f + \omega_i}{2} t[/math]
  • This equation is useful when we know the initial and final angular velocity and the time taken.
  • (ii) Angular Velocity Formula
  • [math]ω_f = ω_i + αt[/math]
  • This equation calculates final angular velocity after time t, given initial angular velocity and angular acceleration.
  • (iii) Angular Displacement with Initial Angular Velocity
  • [math]\Delta \theta = \omega_i t + \frac{1}{2} \alpha t^2[/math]
  • This equation helps find angular displacement when the initial angular velocity, acceleration, and time are known.
  • (iv) Angular Velocity–Angular Displacement Relation
  • [math]\omega_f^2 = \omega_i^2 + 2 \alpha \Delta \theta[/math]
  • This equation is useful when time is unknown but we know initial velocity, final velocity, and angular displacement.
  • ⇒ Solving Problems Using Rotational Equations of Motion
  • Example 1: Rotating Wheel
  • A wheel starts from rest and accelerates at [math]α = 2 \text{ rad/s}^2[/math] for 5 seconds. Find:
  • (a) Final angular velocity
  • (b) Angular displacement
  • Solution:
  • (a) Using
  • [math]\begin{gather} ω_f = ω_i + αt \\ ω_f = 0 + (2)(5) \\ ω_f = 10 rad/s \end{gather}[/math]
  • (b) Using
  • [math]\begin{gather}
    \Delta \theta = \omega_i t + \frac{1}{2} \alpha t^2 \\
    \Delta \theta = (0)(5) + \frac{1}{2}(2)(5)^2 \\
    \Delta \theta = 25 \, \text{rad}
    \end{gather}[/math]
  • ⇒  Relationship Between Linear and Angular Motion
  • For a rotating object with radius r, the following relationships hold:
Angular Motion Linear Motion
[math]θ = \frac{s}{r}[/math] [math]s = rθ[/math]
[math]ω = \frac{v}{r}[/math] [math]v = rω[/math]
[math]α = \frac{a}{r}[/math] [math]a = rα[/math]
  • These equations help convert between linear and angular motion.

  • (f) Moment of Inertia (I)

  • ⇒  Definition
  • The moment of inertia (I) is the rotational equivalent of mass in linear motion. It quantifies an object’s resistance to changes in its rotational motion about a given axis.
  • Just as mass (m) resists changes in linear motion (Newton’s First Law), moment of inertia resists changes in rotational motion.
  • The larger the moment of inertia, the harder it is to start or stop the rotation of an object.
  • Figure 4 Moment of Inertia
  • ⇒  Formula for Moment of Inertia
  • For a single point mass rotating at a distance r from the axis:
  • [math]I = mr^2[/math]
  • For a system of multiple point masses:
  • [math]I = ∑mr^2[/math]
  • The total moment of inertia is found by summing up mr2m r^2mr2 for each mass in the system.
  • ⇒   Factors Affecting the Moment of Inertia
  • The moment of inertia depends on:
    1. Mass (m) – A greater mass leads to a larger moment of inertia.
    2. Distribution of Mass (r) – The further the mass is from the axis of rotation, the larger the moment of inertia.
  • Example:
  • A thin ring has a moment of inertia of [math]I = MR^2[/math] because all mass is concentrated at the edge.
  • A solid disk has a lower [math]I = \frac{1}{2} MR^2[/math], since some mass is closer to the axis.

  • (g) Moment of Inertia for Common Shapes

Object Moment of Inertia (I)
Point mass at radius r [math]I = mr^2[/math]
Thin ring about central axis [math]I = MR^2[/math]
Solid disk about central axis [math]I = \frac{1}{2} MR^2[/math]
Rod about center [math]I = \frac{1}{12} MR^2[/math]
Rod about one end [math]I = \frac{1}{3} MR^2[/math]
Solid sphere about center [math]I = \frac{2}{5} MR^2[/math]
  • The moment of inertia varies depending on the shape and the axis of rotation.
  • Objects with mass further from the axis (like a hoop) have a larger moment of inertia than objects with mass closer (like a disk).

  • (h) Newton’s Second Law for Rotation

  • ⇒  Torque (τ)
  • Torque is the rotational equivalent of force. It determines how effectively a force causes an object to rotate.
  • [math]τ = rFsinθ[/math]
  • Where:
  • τ = Torque (N·m)
  • r = Perpendicular distance from the axis to the force (m)
  • F = Applied force (N)
  • θ = Angle between the force and the lever arm
  • Newton’s Second Law for rotation states that the angular acceleration (α) of an object is proportional to the net torque (τ) applied to it and inversely proportional to its moment of inertia (I).
  • [math]τ = Iα[/math]
  • Where:
  • τ = Net torque (N·m)
  • I = Moment of inertia (kg·m²)
  • α = Angular acceleration (rad/s2)
  • Figure 5 Newton’s Law of Universal of gravitational
  • This is similar to Newton’s Second Law for linear motion: [math]F = ma[/math] , but for rotation.
  • Larger moment of inertia → smaller angular acceleration for the same torque.
  • ⇒  Solving Problems Using Rotational Dynamics
  • Example 1: Rotating Disc
  • A solid disk of mass 2 kg and radius 5 m has a force of 10 N applied tangentially to its edge. Find the angular acceleration.
  • Step 1: Calculate Moment of Inertia
  • For a solid disk, the formula is:
  • [math]\begin{gather}
    I = \frac{1}{2} M R^2 \\
    I = \frac{1}{2}(2)(0.5)^2 \\
    I = \frac{1}{2}(2)(0.25) \\
    I = 0.25 \, \text{kg} \cdot \text{m}^2
    \end{gather}[/math]
  • Step 2: Calculate Torque
  • [math] \begin{gather} τ = rF \\ τ = (0.5)(10) \\ τ = 5 N.m\end{gather}[/math]
  • Step 3: Use Newton’s Second Law for Rotation
  • [math]\begin{gather}
    \tau = I \alpha \\
    5 = (0.25) \alpha \\
    \alpha = \frac{5}{0.25} \\
    \alpha = 20 \, \text{rad/s}^2
    \end{gather}[/math]
  • The angular acceleration of the disk is 20 rad/s².
  • (i) Angular momentum and its conservation
  • ⇒  Definition of Angular Momentum (L)
  • Angular momentum is the rotational equivalent of linear momentum. It quantifies the amount of rotational motion an object possesses.
  • For a rotating body:
  • [math]L = Iω[/math]
  • Where:
  • L = Angular momentum (kg·m²/s)
  • I = Moment of inertia (kg·m²)
  • ω = Angular velocity (rad/s)
  • Figure 6 Angular momentum
  • Just as linear momentum (p = mv) depends on mass and velocity, angular momentum (L = Iω) depends on moment of inertia and angular velocity.

  • (j) Conservation of Angular Momentum

  • ⇒   Principle of Conservation
  • Angular momentum remains constant unless acted upon by an external torque.
  • [math]\begin{gather}
    L_{\text{initial}} = L_{\text{final}} \\
    I_i \omega_i = I_f \omega_f
    \end{gather}[/math]
  • If the moment of inertia decreases, angular velocity increases, and vice versa.
  • ⇒  Examples of Conservation of Angular Momentum
  • Example 1: Ice Skater’s Spin
  • An ice skater starts spinning with arms extended.
  • When she pulls her arms close to her body, her moment of inertia (I) decreases.
  • To conserve angular momentum, her angular velocity (ω) increases, making her spin faster.
  • Example 2: Neutron Stars
  • A massive star collapses into a small, dense neutron star.
  • Since its radius decreases dramatically, its moment of inertia decreases.
  • To conserve angular momentum, the rotation speed increases, leading to rapid spinning neutron stars (pulsars).
  • ⇒  Effect of Torque on Angular Momentum
  • Torque and Change in Angular Momentum
  • A torque (τ) applied to a rotating object changes its angular momentum:
  • [math]τ = \frac{dL}{dt}[/math]
  • Where:
  • τ = Torque (N·m)
  • [math]\frac{dL}{dt}[/math] = Rate of change of angular momentum
  • If no external torque is applied, L remains constant (Conservation of Angular Momentum).
  • If an external torque is applied, L changes over time, causing angular acceleration.
  • ⇒  Solving Problems Using Angular Momentum
  • Example 1: Spinning Disk
  • A solid disk with moment of inertia I=2 kg·m² rotates at ω=5 rad/s. If the moment of inertia is reduced to 5 kg·m², what is the new angular velocity?
  • Step 1: Apply Conservation of Angular Momentum
  • [math]\begin{gather}
    L_{\text{initial}} = L_{\text{final}} \\
    I_i \omega_i = I_f \omega_f \\
    (2)(5) = (1.5) \omega_f \\
    \omega_f = \frac{10}{1.5} \\
    \omega_f = 6.67 \, \text{rad/s}
    \end{gather}[/math]

  • (k) Angular impulse:

  • ⇒  Angular Impulse (ΔL)
  • Definition:
  • Angular impulse is the rotational equivalent of linear impulse. It represents the change in angular momentum when a torque is applied over a time interval.
  • [math]ΔL = τΔt[/math]
  • Where:
  • – ΔL = Change in angular momentum (kg·m²/s)
  • – τ = Torque (N·m)
  • – Δt = Time for which the torque acts (s)
  • Figure 7 Angular impulse
  • Similar to linear impulse:
  • [math]\begin{gather}
    \text{Linear impulse} = \text{Force} \times \text{Time} = \Delta p \\
    \text{Angular impulse} = \text{Torque} \times \text{Time} = \Delta L
    \end{gather}[/math]
  • ⇒  Angular Impulse and Change in Angular Momentum
  • Since angular momentum is given by:
  • [math]L = Iω[/math]
  • A change in angular momentum can be written as:
  • [math]ΔL = Δ(Iω)[/math]
  • If the moment of inertia remains constant, the equation simplifies to:
  • [math]ΔL = IΔω[/math]
  • If a torque acts for a longer time, the change in angular momentum is greater.
  • If no external torque acts (τ=0), then L remains constant (Conservation of Angular Momentum).

  • (l) Rotational Kinetic Energy ([math]E_k[/math]​)

  • ⇒  Definition:
  • A rotating body has kinetic energy due to its rotation, similar to how a moving object has translational kinetic energy.
  • [math]E_k = \frac{1}{2} I \omega^2[/math]
  • Where:
  • [math]E_k[/math]​ = Rotational kinetic energy (J)
  • I = Moment of inertia (kg·m²)
  • ω = Angular velocity (rad/s)
  • Figure 8 Rotational and translational kinetic energy
  • ⇒ Comparison with translational kinetic energy:
  • [math]\begin{gather}
    E_k = \frac{1}{2} m v^2 \\
    E_k = \frac{1}{2} I \omega^2
    \end{gather}[/math]
  • If I is large, more energy is needed to achieve a given ω.
  • ⇒  Alternative Formula Using Angular Momentum
  • Since angular momentum is given by:
  • [math]L = Iω[/math]
  • We can express kinetic energy in terms of L:
  • [math]E_k = \frac{1}{2} I \omega^2[/math]
  • Rewriting ω as L/I, we get:
  • [math]E_k = \frac{L^2}{2I}[/math]
  • This equation shows that an object with greater angular momentum (L) has more rotational kinetic energy.
  • For the same angular momentum, a smaller moment of inertia means higher rotational kinetic energy.
  • ⇒  Real-World Applications
  • Spinning Figure Skater
  • A skater extends their arms → Moment of inertia increases, slowing rotation.
  • The skater pulls their arms in → Moment of inertia decreases, increasing rotation speed.
  • Since L is constant, their kinetic energy also changes.
  • Rotating Flywheels
  • Used in mechanical energy storage.
  • A large flywheel with high moment of inertia stores energy efficiently.
  • Rotational Motion of Planets
  • Planets conserve angular momentum as they orbit the Sun.
  • If a planet moves closer to the Sun, its rotation speeds up.
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