Circular Motion and Rotational Dynamics Study Notes

Introduction to Circular Motion and Rotational Dynamics

• Definition of Circular Motion: The motion of an object along a circular path. Circular motion is ubiquitous in daily life, appearing in both revolving and rotating rigid objects.

• Revolution vs. Rotation:

  • Revolution: Every particle in the object undergoes circular motion about a point outside the object or about a different object.

  • Rotation: Motion occurs about an axis of rotation passing through the object itself.

• Centre of Mass: The point at which the entire mass of a body is assumed to be concentrated.

• Kinematical Equations of Motion: These are equations used to describe the motion of an object. In rotational dynamics, these are analogous to the equations used for translational (linear) motion.

• Real vs. Pseudo Forces:

  • Real Forces: Arise from actual physical interactions (e.g., gravitational, friction).

  • Pseudo Forces: Arise due to the acceleration of the frame of reference (e.g., centrifugal force).

Characteristics of Circular Motion

1. Accelerated Motion: Even if the speed remains constant, the direction of the velocity vector changes at every instant. Since acceleration is the rate of change of velocity, circular motion is intrinsically an accelerated motion.

2. Periodic Motion: During circular motion, the particle repeats its path along the same trajectory at regular intervals of time.

Kinematics of Circular Motion

• Angular Quantities:

  • Angular Displacement ($\theta$): Analogous to linear displacement ($s$).

  • Angular Velocity ($\omega = \frac{d\theta}{dt}$): Analogous to linear velocity ($v = \frac{ds}{dt}$).

  • Angular Acceleration ($\alpha = \frac{d\omega}{dt}$): Analogous to linear acceleration ($a = \frac{dv}{dt}$).

• Relation between Linear and Angular Velocity: $\vec{v} = \vec{\omega} \times \vec{r}$. The magnitude is $v = \omega r$, where $r$ is the radius vector from the centre.

• Direction of Angular Velocity ($\omega$):

  • Determined by the Right-Hand Thumb Rule: Curl the fingers of the right hand along the sense of rotation; the outstretched thumb indicates the direction of $\vec{\omega}$ along the axis of rotation.

• Uniform Circular Motion (UCM): Motion where the speed of the particle remains constant.

  • Velocity direction changes constantly, being always tangential to the path.

  • Centripetal Acceleration ($a_r$): Also called radial acceleration. It is directed towards the centre (along $-\vec{r}$).

  • Vector form: $\vec{a} = -\omega^2 \vec{r}$.

  • Magnitude: $a = \omega^2 r = \frac{v^2}{r} = v\omega$.

• Non-Uniform Circular Motion: Motion where the speed of the particle changes over time (e.g., a fan switching ON/OFF).

  • Tangential Acceleration ($a_t$): Responsible for changing the magnitude of velocity. It is directed along or opposite to the velocity.

  • Resultant Acceleration: The combination of radial acceleration ($a_r$) and tangential acceleration ($a_t$).

  • Direction of $\vec{\alpha}$: For increasing speed (\alpha > 0), it is along $\vec{\omega}$. For decreasing speed (\alpha < 0), it is opposite to $\vec{\omega}$.

Dynamics of Circular Motion: Centripetal and Centrifugal Forces

• Centripetal Force (CPF):

  • The resultant of all real forces acting on a body toward the centre to maintain circular motion.

  • Formula: $\text{CPF} = -m\omega^2 \vec{r}$.

  • Magnitude: $F = mr\omega^2 = \frac{mv^2}{r} = mv\omega$.

  • Important Note: "Centripetal" is an adjective describing the direction (centre-seeking), not a new specific type of force like gravity.

• Centrifugal Force (CFF):

  • A pseudo force that arises in a non-inertial (rotating) frame of reference to explain the state of rest of an object within that frame.

  • Formula: $\text{CFF} = +m\omega^2 \vec{r}$. Directed away from the centre.

  • It is non-real but measurable by instruments in the rotating frame.

• Force Equations:

  • In an inertial frame: $\text{Resultant Force} = -m\omega^2 \vec{r}$.

  • In a non-inertial frame: $\sum \text{Real Forces} + \text{Pseudo Force} = 0$.

Applications of Uniform Circular Motion

1. Vehicle Along a Horizontal Circular Track

• Forces acting: Weight ($mg$) vertically down, Normal Reaction ($N$) vertically up, and Static Friction ($f_s$) between road and tyres.

• Equations: $N = mg$ and $f_s = \frac{mv^2}{r}$.

• The limit of static friction is $(f_s)_{max} = \mu_s N = \mu_s mg$.

• Maximum Safety Speed: $v_{max} = \sqrt{\mu_s rg}$.

2. Well (Wall) of Death

• A vehicle (considered a point mass) moves in horizontal circles inside a vertical cylinder of radius $r$.

• Forces: Normal reaction $N$ horizontally toward the centre, weight $mg$ vertically down, and static friction $f_s$ vertically up.

• Equations: $N = \frac{mv^2}{r}$ and $f_s = mg$.

• Condition: $f_s \le \mu_s N \implies mg \le \mu_s \frac{mv^2}{r}$.

• Minimum Speed: $v_{min} = \sqrt{\frac{rg}{\mu_s}}$.

3. Vehicle on a Banked Road

• Banking is the tilting of road surfaces with the horizontal at an angle $\theta$ to reduce reliance on friction.

• No Friction case (Most Safe Speed): Only components of Normal reaction and gravity are considered.

  • $N \cos(\theta) = mg$

  • $N \sin(\theta) = \frac{mv^2}{r}$

  • $\tan(\theta) = \frac{v^2}{rg} \implies v_s = \sqrt{rg \tan(\theta)}$.

• Banking Angle: $\theta = \tan^{-1}(\frac{v^2}{rg})$.

• Speed Limits with Friction:

  • Lower Limit ($v_{min}$): $v_1 = \sqrt{rg \frac{\tan(\theta) - \mu_s}{1 + \mu_s \tan(\theta)}}$. If $\mu_s \ge \tan(\theta)$, $v_{min} = 0$.

  • Upper Limit ($v_{max}$): $v_2 = \sqrt{rg \frac{\tan(\theta) + \mu_s}{1 - \mu_s \tan(\theta)}}$. If $\mu_s = \cot(\theta)$, $v_{max} = \infty$.

Conical Pendulum

• A bob of mass $m$ suspended by a string of length $L$ revolving in a horizontal circle.

• Forces: Tension ($T_0$) and weight ($mg$).

• Equations: $T_0 \cos(\theta) = mg$ and $T_0 \sin(\theta) = mr\omega^2$.

• Radius: $r = L \sin(\theta)$.

• Angular Speed: $\omega = \sqrt{\frac{g}{L \cos(\theta)}}$.

• Period ($T$): $T = 2\pi \sqrt{\frac{L \cos(\theta)}{g}}$.

• Frequency ($n$): $n = \frac{1}{2\pi} \sqrt{\frac{g}{L \cos(\theta)}}$.

• Note: The string can never be horizontal ($\theta = 90^\circ$) because tension would have no vertical component to balance $mg$, and it would require infinite kinetic energy.

Vertical Circular Motion (VCM)

Case 1: Mass Tied to a String

• Motion governed by gravity; kinetic energy matches potential energy changes.

• Uppermost Position (A): Minimum speed for string not to slack is $v_A = \sqrt{rg}$. Tension $T_A \ge 0$.

• Lowermost Position (B): Minimum speed is $v_B = \sqrt{5rg}$.

• Horizontal Position (C/D): Minimum speed is $v_C = \sqrt{3rg}$.

• Tension Difference: $T_B - T_A = 6mg$.

Case 2: Mass Tied to a Rod

• A rigid rod can support the mass even at zero speed at the top.

• Uppermost Position: $(v_A)_{min} = 0$.

• Lowermost Position: $(v_B)_{min} = \sqrt{4rg} = 2\sqrt{rg}$.

• Horizontal Position: $(v_C)_{min} = \sqrt{2rg}$.

Other VCM Examples

• Sphere of Death: Dynamics similar to mass on a string (Normal reaction $N$ replaces Tension $T$).

• Convex Over-Bridge: Weight ($mg$) and Normal reaction ($N$) act at the top.

  • $mg - N = \frac{mv^2}{r}$.

  • Maximum Speed to maintain contact ($N=0$): $v_{max} = \sqrt{rg}$.

Moment of Inertia (MOI)

• Definition: MOI ($I$) is the rotational analogue of mass. It depends on the mass distribution around the axis of rotation.

• Formula: $I = \sum_{i=1}^N m_i r_i^2$ (discrete) or $I = \int r^2 \,dm$ (continuous).

• Rotational Kinetic Energy: $K.E._{rot} = \frac{1}{2} I \omega^2$.

• MOI of Uniform Ring: $I = MR^2$ (about central axis).

• MOI of Uniform Disc: $I = \frac{1}{2} MR^2$ (about central axis).

• Radius of Gyration ($K$): The distance at which the entire mass can be considered to be concentrated for the same MOI. $I = MK^2 \implies K = \sqrt{\frac{I}{M}}$.

Theorems of Moment of Inertia

1. Theorem of Parallel Axes: The MOI of an object about any axis ($I_O$) is the sum of its MOI about a parallel axis passing through its centre of mass ($I_C$) and the product of its mass and the square of the distance ($h$) between the axes.

   • $I_O = I_C + Mh^2$

2. Theorem of Perpendicular Axes: Specifically for laminar (2D) objects. The MOI about an axis ($z$) perpendicular to its plane is the sum of MOIs about two mutually perpendicular axes ($x$ and $y$) in its plane, concurrent at the same point.

   • $I_Z = I_X + I_Y$

Angular Momentum and Torque

• Angular Momentum ($L$): Moment of linear momentum.

  • $\vec{L} = \vec{r} \times \vec{p}$. Magnitude: $L = I\omega$.

• Torque ($\tau$): Moment of force.

  • $\vec{\tau} = \vec{r} \times \vec{f}$. Magnitude: $\tau = I\alpha$.

• Conservation of Angular Momentum: If the external unbalanced torque is zero ($\tau = 0$), then $\vec{L}$ remains constant ($I_1 \omega_1 = I_2 \omega_2$).

  • Examples:

    • Ballet Dancers: Stretches limbs to increase $I$ (decrease $\omega$) and brings limbs close to decrease $I$ (increase $\omega$).

    • Divers: Fold body mid-air to increase frequency of rotation ($\omega$).

Rolling Motion

• Simultaneous translational and circular motion.

• Total Kinetic Energy: $E = K.E._{trans} + K.E._{rot} = \frac{1}{2} Mv^2 + \frac{1}{2} I\omega^2 = \frac{1}{2} Mv^2 (1 + \frac{K^2}{R^2})$.

• Rolling Down an Inclined Plane:

  • Linear Velocity: $v = \sqrt{\frac{2gh}{1 + K^2/R^2}}$.

  • Linear Acceleration: $a = \frac{g \sin(\theta)}{1 + K^2/R^2}$.

Questions & Discussion

• Can circular motion be described purely as translation?: No, it requires angular displacement and angular velocity which have specific inter-relations ($v = \omega r$).

• Example 1 (Fan Problem): A fan at $90\,rpm$ stops after $21$ revolutions.

  • $\omega_0 = 1.5\,rps = 3\pi\,rad/s$.

  • $\theta = 2\pi \times 21 = 42\pi\,rad$.

  • Using $\omega^2 = \omega_0^2 + 2\alpha\theta$, calculated time $t = 28\,s$.

• Example 1.8 (Balloon Problem): Spherica balloon revolves at $60\,rpm$. $48.8\%$ of water leaks out. Calculate new frequency.

  • $m_1 = 1$, $m_2 = 0.512$. Since $m \propto R^3$, $R_2/R_1 = 0.8$.

  • By conservation of angular momentum, $n_2 \approx 3.052\,rps$.