Complete Rotational Kinematics and Circular Motion Study Guide
4.1.1 Angular Position (θ)
Where S is the arc length and r is the radius.
4.1.2 Angular Displacement (Δθ)
Sign Convention
- Positive (+): Anti-clockwise motion.
- Negative (-): Clockwise motion.
Units
- SI Unit: Radian (rad).
- Other Units: Degrees (°) and revolutions (rev).
Key Conversions
- One complete revolution = 360° = 2Ï€ rad.
- 1 rad ≈ 57.3°.
Direction (Right Hand Rule)
- Grasp the axis of rotation with the right hand.
- Curl fingers in the direction of rotation.
- The thumb points in the direction of the angular displacement.
4.1.3 Angular Velocity (ω)
Direction
Matches the direction of angular displacement (determined by the Right Hand Rule).
Units
- SI Unit: Radian per second (rad s⁻¹).
- Other Units: deg/s, rev/s, or rev/min (rpm).
Types of Angular Velocity
Average Angular Velocity ($\omega_{av}$)
The total angular displacement divided by total time.
Instantaneous Angular Velocity ($\omega_{inst}$)
The limiting value of the ratio between small angular displacement and small time interval as time approaches zero.
Angular Acceleration ($\alpha$)
Units
- SI Unit: rad/s²
- Other units: deg/s² or rev/s²
Direction
Determined by the right-hand rule.
- Positive: When angular velocity increases (acceleration and velocity are in the same direction).
- Negative: When angular velocity decreases (acceleration and velocity are anti-parallel).
Types of Angular Acceleration
Average Angular Acceleration ($\alpha_{avg}$)
The total angular velocity divided by total time.
Instantaneous Angular Acceleration ($\alpha_{inst}$)
The limiting value of the ratio as the time interval approaches zero.
Relationship Between Linear and Angular Quantities
A. Linear and Angular Displacement
Relates arc length ($S$) to radius ($r$) and angular displacement in radians ($\theta$).
B. Linear and Angular Velocity
By taking the limit of displacement over time ($\lim_{\Delta t \to 0} \frac{\Delta S}{\Delta t}$), we derive the relationship between linear (tangential) velocity ($v$) and angular velocity ($\omega$).
C. Linear and Angular Acceleration
Linear acceleration in angular motion is composed of two distinct components:
1. Tangential Component ($a_t$)
- Parallel to the linear instantaneous velocity.
- Result of a change in the magnitude of linear velocity.
- Formula: $$a_t = r\alpha$$
2. Radial Component ($a_R$)
- Directed along the radius toward the center (centripetal acceleration).
- Result of a change in the direction of linear velocity.
Total Linear Acceleration
- Vector form: $\mathbf{a} = \mathbf{a}_t + \mathbf{a}_R$
- Magnitude: $$a = \sqrt{a_t^2 + a_R^2}$$
Table 4.1: Kinematic Equations Comparison
| Equations for Linear Motion | Equations for Angular Motion |
|---|---|
| $S = vt$ | $\theta = \omega t$ |
| $v_f = v_i + at$ | $\omega_f = \omega_i + \alpha t$ |
| $2aS = v_f^2 - v_i^2$ | $2\alpha\theta = \omega_f^2 - \omega_i^2$ |
| $S = v_it + \frac{1}{2}at^2$ | $\theta = \omega_it + \frac{1}{2}\alpha t^2$ |
Rotational Kinematics (Example Analysis)
When an object (like a tyre) rolls without slipping, linear motion and rotational motion are linked by the following relationships:
Linear to Angular Velocity:
The relationship is defined by $v = r\omega$ (or $\omega = \frac{v}{r}$).
Distance and Revolutions ($N$):
- One full revolution equals the circumference of the tyre ($2\pi r$).
- Total revolutions $N = \frac{S}{2\pi r}$, where $S$ is the total distance covered.
Angular Displacement ($\theta$):
Total displacement in radians is calculated as $\theta = N \times 2\pi$.
Angular Acceleration ($\alpha$):
Can be found using the equation independent of time: $$2\alpha\theta = \omega_{f}^{2} - \omega_{i}^{2}$$
Time ($t$):
Calculated using the formula $t = \frac{\omega_{f} - \omega_{i}}{\alpha}$.
Centripetal Acceleration ($a_c$)
Centripetal acceleration occurs when a particle moves in a circular path at a constant speed because its direction of velocity is constantly changing.
Direction
Always directed towards the center of the circle.
Vector Relationship
As a particle moves from point A to B, the change in velocity ($\Delta v$) and change in position ($\Delta r$) form similar triangles.
Mathematical Derivation
- The ratio of change is $\frac{\Delta v}{v} = \frac{\Delta r}{r}$.
- Instantaneous acceleration is defined as $a = \frac{v}{r} \lim_{\Delta t \to 0} \frac{\Delta r}{\Delta t}$.
- Since $v = \lim_{\Delta t \to 0} \frac{\Delta r}{\Delta t}$, the formula becomes $a_c = \frac{v^2}{r}$.
Dynamics of Circular Motion
Following Newton's Second Law, acceleration requires a net force:
Tangential Acceleration
Produced by an applied force (e.g., opening a door) which creates torque.
Radial (Centripetal) Acceleration
To keep an object moving in a circle, a force must be applied to provide the necessary radial acceleration.
Quick Calculations (Assignment 4.1)
To find angular velocity ($\omega$) when given revolutions ($N$) and time ($t$):
$\text{Angular Velocity} = \frac{\text{Total Displacement in radians}}{\text{Time}}$
Centripetal Force ($F_c$)
Key Formulas
Examples of Centripetal Force
Tension
A ball on a string; the tension ($T$) acts as the centripetal force to keep the ball from moving in a straight line.
Gravity
Keeps planets in orbit around the Sun and satellites around the Earth.
Friction
Provides the force for a car rounding a flat, un-banked curve.
Normal Force
Provides the force in specialized situations like banked tracks or hardware disks.
Banked Curves (Friction-Free)
When a road is tilted at an angle $\theta$, the Normal Force ($F_N$) is responsible for the turn:
- The horizontal component $F_N \sin \theta$ points toward the center, providing the centripetal force.
- The vertical component $F_N \cos \theta$ balances the weight of the car ($mg$).
- The Calculation: Dividing these components results in the formula:
$$\tan \theta = \frac{v^2}{rg}$$
Applications of Centrifugation
Centrifuge
A device that spins mixtures rapidly. Denser particles follow Newton's first law (moving in a straight line) to the bottom of the tube for collection.
Cream Separator
A centrifugal device that separates milk into cream and skimmed milk.
Washing Machine Dryer
A spinning cylinder with small holes that rotates rapidly to remove water from clothes.
Problem Solving: Example 4.2
Scenario: Find the radius of a curve for a car with mass 856 kg moving at 12.0 m/s with a centripetal force of 7250 N.
Formula:
$r = \frac{mv^2}{F_c}$
Calculation:
$r = \frac{856 \times (12)^2}{7250}$
Result:
$r = 17\text{ m}$
4.3 Moment of Inertia
Definition
- Moment of Inertia (Rotational Inertia): The property of a body by which it maintains its state of rest or uniform rotational motion about a fixed axis.
- It is the rotational equivalent of mass. Objects with a larger moment of inertia are harder to accelerate angularly.
Mathematical Representation
For a single mass ($m$) at distance ($r$):
For a rigid body:
The body is divided into many small portions with masses ($m_1, m_2, \dots$) and radii ($r_1, r_2, \dots$). The total is the sum of these parts:
Moments of Inertia for Uniform Objects (Mass 'M')
| Object | Axis of Rotation | Formula ($I$) |
|---|---|---|
| Cylinder rod | Center | $\frac{1}{12}ML^2$ |
| Cylinder rod | One end | $\frac{1}{3}ML^2$ |
| Rectangular plate | Center | $\frac{1}{12}M(a^2 + b^2)$ |
| Rectangular plate | Edge | $\frac{1}{3}M(a^2 + b^2)$ |
| Solid cylinder or disc | Center | $\frac{1}{2}MR^2$ |
| Ring or hoop | Center | $MR^2$ |
| Hollow cylinder | Center | $\frac{1}{2}M(R_1^2 + R_2^2)$ |
| Solid sphere | Center | $\frac{2}{5}MR^2$ |
| Hollow sphere | Center | $\frac{2}{3}MR^2$ |
4.4 Angular Momentum ($L$)
Definition
- The cross product of the position vector ($r$) with respect to the axis of rotation and the linear momentum ($p$) of an object.
- Formula: $L = r \times p$
- SI Unit: $\text{kg m}^2\text{s}^{-1}$
- Dimensions: $[ML^2T^{-1}]$
4.4.1 For a Point Mass
- Magnitude: If the angle $\theta = 90^\circ$, then $L = rp$.
- Linear Momentum: Since $p = mv$, then $L = r(mv)$.
- Angular Velocity: Using $v = r\omega$, the equation becomes $L = r(m \cdot r\omega)$.
- Final Form: $L = mr^2\omega = I\omega$.
4.4.2 For a Rigid Body
A rigid body is composed of many small masses ($m_1, m_2, \dots, m_n$) at distances ($r_1, r_2, \dots, r_n$).
- Total Angular Momentum ($L_{net}$): The sum of individual angular momenta: $$L_{net} = L_1 + L_2 + \dots + L_n$$
- Uniform Rotation: For a rigid body, all points rotate with the same angular velocity ($\omega$).
- Substitution: $L_{net} = (m_1r_1^2\omega + m_2r_2^2\omega + \dots + m_nr_n^2\omega)$.
- Factoring: $L_{net} = (\sum m_ir_i^2)\omega$.
- Conclusion: Since the sum in the parentheses is the moment of inertia ($I$), then: $$L_{net} = I\omega$$
4.4.3. Relation Between Torque and Angular Momentum
Angular momentum ($L$) is defined as the cross product of an object's position vector ($r$) and its linear momentum ($p$).
Mathematical Derivation
- Starting with $L = r \times p$, we multiply both sides by the rate of change over time ($\frac{\Delta}{\Delta t}$).
- This gives the relationship: $\frac{\Delta L}{\Delta t} = r \times \frac{\Delta p}{\Delta t}$.
- According to Newton's Second Law, $F = \frac{\Delta p}{\Delta t}$.
- Substituting force into the equation: $\frac{\Delta L}{\Delta t} = r \times F$.
- Since torque ($\tau$) is defined as $r \times F$, we conclude that: $\frac{\Delta L}{\Delta t} = \tau$.
4.4.4. Conservation of Angular Momentum
Definition: In the absence of any external torque, the angular momentum of a system remains constant.
- Mathematical Expression: If $\tau = 0$, then $\Delta L = 0$, meaning $L_f = L_i$ (Final angular momentum = Initial angular momentum).
- Formula: $I_f \omega_f = I_i \omega_i$ (where $I$ is rotational inertia and $\omega$ is angular velocity).
Practical Applications
Spinning Ice Skater
- When arms are extended, rotational inertia ($I$) is large and angular velocity ($\omega$) is small.
- When arms are pulled in, $I$ decreases, causing a much faster spin (increased $\omega$).
Diving and Gymnastics
Divers generate spins from a diving board and then vary their rotational inertia to perform somersaults and twists while angular momentum remains unchanged.
Gyroscopes
- Used to maintain orientation or resist changes in direction.
- A spinning gyroscope can stay balanced on a surface, but will fall when it stops spinning.
- Precession: When tilted, a gyroscope starts a "gravity-defying" motion called precession about the gravitational force axis. This happens because the change in angular momentum ($\Delta L$) follows the direction of the torque provided by gravity.
Flywheels
A flywheel is a mechanical component (a heavy disc or wheel) used to store energy in the form of rotational motion.
Function
- Applying torque increases its rotational speed, storing kinetic energy.
- This energy can be released when needed, such as during power outages or to provide additional power for machinery.
Common Uses
- Smoothing out power delivery in engines.
- Providing backup power and regulating machine speed in industrial settings.
- Storing excess energy from renewable sources (wind/solar) to balance the electricity grid.
Design Factors
Efficiency depends on the material, size, shape, bearings, and axle to minimize friction and maximize storage.
Student Activity: Conservation of Angular Momentum
- Experiment: Hold dumbbells while rotating on a turntable.
- Observation: Extending your arms decreases rotation speed. Drawing hands toward your chest increases rotation speed.
- Reasoning: This occurs because rotational inertia is changed, and angular velocity must adjust to keep the total angular momentum constant.
Angular Momentum ($L$) and Torque ($\tau$)
Angular Momentum Calculation
For a uniform cylindrical grinding wheel (disk), the moment of inertia is $I = \frac{1}{2} mR^2$. Angular momentum is then calculated as:
Relationship between Torque and Momentum
Torque is the rate of change of angular momentum over time:
Rotational Earth Example
To find the angular momentum of Earth, the average angular speed around its axis is given as $7.29 \times 10^{-5} \text{ rad s}^{-1}$.
Torque and Angular Acceleration
Just as force relates to acceleration in linear motion ($F = ma$), a similar relationship exists for rotational motion.
Torque for a Point Mass
For a mass '$m$' at a distance '$r$' from the axis of rotation:
When the force is tangential ($\theta = 90^{\circ}$), $\sin 90^{\circ} = 1$, so $\tau = rF$.
Deriving $\tau = I\alpha$
- From Newton's Second Law: $F = ma$.
- Since tangential acceleration $a = r\alpha$, then $F = m(r\alpha)$.
- Substitute $F$ into the torque equation: $\tau = r(mr\alpha) = mr^2\alpha$.
- Because $mr^2$ represents the moment of inertia ($I$), the final relationship is:
Weightlessness in Satellites
- Definition: Weightlessness occurs when there is zero apparent weight (the feeling of weight is absent).
Causes
Free-fall
Occurs when the force of gravity is balanced by inertial forces (like centrifugal force) from orbital flight.
Deep Space
Traveling millions of miles away from large objects where gravity reduces to nearly zero.
The "Zero Gravity" Misconception
Space stations are not in a gravity-free environment. Even 250 miles above Earth, the gravitational field is still roughly 95% as strong as it is on the surface. Astronauts feel weightless only because they are in constant free-fall.
Challenges of Weightlessness
Health Risks
Prolonged microgravity causes weakened bones/muscles, affects the cardiovascular system, and compromises the immune system.
Daily Activities
Standard tasks like eating, sleeping, showering, and digestion are significantly modified or difficult.
Proposed Solution
Rotational simulated gravity has been proposed to counteract these adverse effects during long-duration spaceflight.
Inertia Note
Even in a weightless setting, objects still possess mass and inertia, meaning they maintain straight-line motion unless acted upon by an outside force.
Artificial Gravity
Core Concepts & Definitions
- Artificial Gravity: Gravity produced artificially in satellites to counteract the effects of weightlessness.
- Mechanism: It is generated by rotating a space station around its own axis.
- The Physics of Weight: The inner surface of the rotating station exerts a force on objects in contact with it. This provides the centripetal force that keeps objects moving in a circular path, allowing astronauts to stand and work as if on a planet.
Mathematical Derivations
To simulate Earth-like gravity, the centripetal acceleration ($a_c$) must be set equal to the acceleration due to gravity ($g$).
1. Centripetal Acceleration:
2. Linear Velocity ($v$):
Setting $a_c = g$ and solving for $v$:
3. Angular Velocity ($\omega$):
Using the relationship $v = \omega R$:
4. Time Period ($T$):
The time for one full rotation:
5. Frequency ($f$):
The number of rotations per second ($f = 1/T$):
Example Problem Summary
Example 4.4: A station with a radius of $1.5\text{ km}$ ($1500\text{ m}$) aiming for $g = 9.8\text{ m/s}^2$.
Angular Velocity:
$\omega \approx 0.08\text{ rad/s}$
Time Period:
$T \approx 77.73\text{ s}$
Frequency:
$f \approx 0.013\text{ Hz}$
Variable Reference Table
| Symbol | Description | Unit |
|---|---|---|
| $g$ | Acceleration due to gravity | $m/s^2$ |
| $R$ | Radius of rotation | $m$ |
| $v$ | Linear velocity | $m/s$ |
| $\omega$ | Angular velocity | $rad/s$ |
| $T$ | Time period | $s$ |
Key Concepts & Definitions
Centrifugation
A process used for the separation of components through the mechanism of spinning.
Angular Acceleration
This is produced in a body by the application of a net torque.
Circumference in Radians
The total measure for the circumference of a circle is $2\pi$ rad.
Dynamics of Rotational Motion
Moment of Inertia
The moment of inertia of a spinning body is determined by the mass of the body, the distribution of mass around the axis, and the orientation of the axis. It does not depend on the angular velocity of the body.
Angular Momentum Change
The change in angular momentum is calculated based on the torque applied and the duration of time.
Example: A torque of $2.5 \text{ N m}$ acting for $2 \text{ s}$ results in a change of $5 \text{ J s}$.
Rotational Speed Factors
If the size (length) of fan wings is increased while keeping voltage and current constant, the rotational speed will decrease.
Motion and Forces
Overturning Tendency
The tendency for a car to overturn while turning a curve is proportional to the square of its speed. If the speed is doubled, the tendency to overturn is quadrupled.
Weightlessness in Space
Astronauts inside the International Space Station feel weightless because the station is in a state of free fall.
Physics Numerical Problem Study Guide
Angular Motion and Velocity
- Definition: Angular velocity ($\omega$) is the rate of change of angular displacement.
- Key Formula: $\omega = 2\pi f$, where $f$ is the frequency in revolutions per second.
- Example: A flywheel completing 3000 revolutions in a minute results in an angular velocity of 314 rad s⁻¹.
Circular Motion and Stability
- Maximum Speed Limit: The speed at which a vehicle can navigate a turn without losing contact with the road.
- Influencing Factors:
- Radius of the circular path ($r$).
- Height of the center of gravity from the ground.
Moment of Inertia ($I$)
- Core Concept: A measure of an object's resistance to changes in its rotation.
- Calculating for a Rod: Depends on the mass ($m$) and length ($L$) when rotating about an axis through its center.
- Calculated Value Example: A stick of 200 g and 0.8 m length has a moment of inertia of 0.01 kg m².
Angular Momentum ($L$)
- Definition: The rotational equivalent of linear momentum.
- Formula: $L = I\omega$.
- Conservation of Angular Momentum: If the moment of inertia is reduced (e.g., pulling arms closer while spinning), the angular velocity increases proportionally.
- Application: If $I$ is reduced to 1/3, $\omega$ triples.
Angular Acceleration ($\alpha$)
- Trigger: Produced when a force is applied at the edge of a rotating body (like a merry-go-round).
- Variable Factors:
- The magnitude of the applied force ($F$).
- The total mass and distribution of mass (e.g., an empty merry-go-round vs. one with a passenger).
Artificial Gravity
- Mechanism: Created in wheel-shaped space stations through rotation.
- Relation: Angular speed (rpm) is adjusted based on the station's diameter to achieve a specific acceleration (e.g., 5.00 m s⁻²).
Rotational Kinematics
- Arc Length and Radians: The relationship between the length of travel ($s$) and the radius ($r$) determines the radians traveled ($\theta = s/r$).
- Deceleration: Objects (like a fidget spinner) eventually come to rest over a specific time due to the loss of angular velocity.
- Total Revolutions: Calculated based on initial angular velocity and the time taken to stop.
Quick Concepts Review
4.1 Minute Hand Angular Acceleration
- The minute hand moves at a constant angular velocity.
- Because the rate of rotation does not change, the angular acceleration is zero.
4.2 Water Removal in Washing Machines
- There is no real force pushing water out; it is caused by a lack of centripetal force.
- The Process:
- The drum rotates at high speeds.
- The perforated walls provide centripetal force to the clothes.
- The water escapes through the holes due to inertia, moving in a straight tangential line.
4.3 Linear and Angular Relationships
- Displacement: $s = r\theta$
- Velocity: $v = r\omega$
- Acceleration: $a = r\alpha$
4.4 Nature of Centripetal Force
- It is not a fundamental force; it is a requirement provided by other forces (like tension or gravity).
- A combination of fundamental forces can provide the necessary centripetal force.
4.5 Moment of Inertia in Heavy Vehicles
- Double tyres increase the mass and its distribution relative to the axle.
- The moment of inertia will be different (specifically higher) compared to a single tyre.
4.6 Dual-Blade Helicopter Rotation
- Blades rotate in opposite directions to conserve angular momentum.
- This cancels out the torque that would otherwise cause the helicopter body to spin in the opposite direction of the blades.
4.7 Changes in Earth's Diameter
- If the diameter is halved with no change in mass, the moment of inertia decreases.
- To conserve angular momentum, the rotational speed must increase.
4.8 Tangential Acceleration
- Tangential acceleration acts in the direction of motion.
- It changes the magnitude of velocity (speed) but cannot change its direction.
4.9 Artificial Gravity Values
- Artificial gravity is usually set lower than $9.8\text{ m s}^{-2}$.
- This is typically done for structural efficiency and the physiological comfort of the crew.
4.10 Gyroscopes in Aeroplanes
- Used to maintain directional stability and navigation.
- A gyroscope stays fixed in its orientation, providing a steady reference point for the pilot.
4.11 Flywheels in Engines
- A flywheel stores rotational energy during power surges.
- It evens out power delivery by releasing energy when the engine is not in a power stroke, ensuring smooth rotation.
4.12 Clock Arm Angles at 09:15
- At 09:15, the angle is slightly more than $90^\circ$ because the hour hand has moved toward the 10.
- The position must be converted from degrees into radians for the final expression.
Complete Summary
Core Principles
- Rotational motion parallels linear motion with analogous quantities.
- Angular displacement, velocity, and acceleration follow similar kinematic equations.
- Centripetal force is not a separate force but a role played by net inward forces.
Key Relationships
- $s = r\theta$, $v = r\omega$, $a_t = r\alpha$
- $a_c = v^2/r = r\omega^2$
- $F_c = mv^2/r = mr\omega^2$
- $\tau = I\alpha$, $L = I\omega$
Conservation Laws
- Angular momentum conserved when $\tau_{net} = 0$
- $I_1\omega_1 = I_2\omega_2$ explains many real-world phenomena
Practical Applications
- Banked curves for safe turning
- Artificial gravity in space stations
- Centrifugation for separation
- Flywheels for energy storage
- Gyroscopes for navigation