# B.1 Rigid Bodies and Rotational Dynamics

### Torque

• Torque measures the rotational effect of a force acting on an object.

• Calculation: τ=r×F, where r is the moment arm and F is the force.

• When force is applied perpendicular to the object (θ=90∘), sinθ is 1.

• Directional considerations are crucial for torque calculations.

### Moment of Inertia

• Moment of inertia (I) resists rotational acceleration.

• Calculation: I=∑miri2

• Shapes have different moments of inertia due to mass distribution.

• Moment of inertia (I) is a property of an object that describes its resistance to rotational motion.

• It depends on how mass is distributed relative to the axis of rotation.

• Objects with larger moments of inertia require more torque to achieve the same angular acceleration.

• It is akin to mass in linear motion, representing the object's inertia in rotation.

• Moment of inertia varies based on the shape and distribution of mass within the object.

• It is a fundamental concept in rotational dynamics, essential for understanding the behavior of rotating bodies.

### Rotational and Translational Equilibrium

• Rotational equilibrium occurs when the net torque acting on an object is zero, meaning it remains stationary or rotates at a constant angular velocity.

• Translational equilibrium happens when the net force acting on an object is zero, resulting in no change in its velocity, whether stationary or moving with constant velocity.

• In rotational equilibrium, the sum of all torques acting on the object must balance out to zero.

• In translational equilibrium, the vector sum of all forces acting on the object must cancel out, resulting in no acceleration.

• Objects in rotational equilibrium may still be in translational motion if their center of mass is moving at a constant velocity.

• Both types of equilibrium are crucial concepts in understanding the stability and motion of objects, whether static or dynamic.

### Angular Acceleration

• Angular acceleration (α) describes how quickly the rotational speed of an object changes over time.

• It indicates whether the rotation is speeding up or slowing down.

• It is influenced by the net torque acting on the object.

• Objects with larger moments of inertia require more torque to achieve the same angular acceleration.

• Angular acceleration is a fundamental concept in understanding rotational dynamics and engineering applications involving rotating systems.

• Calculation: ω=2πf or ω= 2π/T.

• Angular acceleration: α= Δω/Δt

### Equations of Rotational Motion for Uniform Angular Acceleration

• Equations analogous to linear motion equations.

• Equations of rotational motion for uniform angular acceleration describe the relationship between angular displacement, initial and final angular velocities, angular acceleration, and time.

• These equations are analogous to the equations of motion in linear kinematics.

• They provide a framework for predicting the behavior of rotating objects undergoing constant angular acceleration.

• The equations allow for the calculation of various parameters such as final angular velocity, angular displacement, and time taken for rotation under uniform acceleration.

• They are essential tools in analyzing rotational dynamics problems and designing mechanisms involving rotating components.

• Understanding these equations enables engineers and physicists to predict and control the motion of rotating systems accurately.

### Newton’s Second Law Applied to Angular Motion

• Newton's Second Law applied to angular motion states that the net torque acting on an object is proportional to the rate of change of its angular momentum.

• Mathematically, it can be expressed as τ=Iα, where τ is the net torque, I is the moment of inertia, and α is the angular acceleration.

• This law illustrates that a torque applied to an object will cause it to undergo angular acceleration, similar to how a force applied to an object causes linear acceleration.

• It implies that the change in angular momentum of an object is directly proportional to the torque applied and occurs in the direction of the torque.

• Newton's Second Law for angular motion provides a fundamental principle for analyzing rotational dynamics problems and designing systems involving rotating components.

• It is crucial for understanding the behavior of rotating objects under the influence of external torques and for predicting their motion accurately.

### Conservation of Angular Momentum

• Total angular momentum is conserved when no external torques act.

• Conservation of angular momentum states that the total angular momentum of a system remains constant when no external torques act upon it.

• Mathematically, it can be expressed as Linitial = Lfinal, where L represents angular momentum.

• This principle applies to isolated systems where external torques are absent, allowing angular momentum to be conserved.

• When external torques are present, angular momentum can still be conserved if the net external torque acting on the system is zero.

• Conservation of angular momentum plays a crucial role in various physical phenomena, such as the motion of celestial bodies, spinning objects, and collisions involving rotational motion.

• Understanding this principle helps in predicting the behavior of rotating systems and designing mechanisms to utilize or conserve angular momentum efficiently.

### Rotational Kinetic Energy

• Rotational kinetic energy refers to the energy associated with the rotational motion of an object.

• It is analogous to linear kinetic energy but involves rotational motion instead of linear motion.

• Rotational kinetic energy depends on both the moment of inertia of the object and its angular velocity.

• Objects with larger moments of inertia or higher angular velocities possess greater rotational kinetic energy.

• Rotational kinetic energy is an important concept in understanding the behavior of rotating systems and is often used in engineering applications, such as designing rotating machinery and analyzing the stability of rotating objects.

• Rotational kinetic energy calculation: KErot = (½)where I is the moment of inertia and ω is the angular velocity.

# B.2 – Thermodynamics

### The First Law of Thermodynamics

• The First Law of Thermodynamics, also known as the law of conservation of energy, states that energy cannot be created or destroyed in an isolated system; it can only change forms.

• This law is based on the principle of energy conservation and applies to all forms of energy, including thermal, mechanical, chemical, and nuclear energy.

• QU+W where Q is heat, ΔU is internal energy change, and W is work.

• The equation implies that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system.

• The First Law of Thermodynamics has various practical applications, such as understanding heat engines, refrigerators, and chemical reactions, and it forms the basis for the study of thermodynamics and energy conservation principles.

### The Second Law of Thermodynamics

• The Second Law of Thermodynamics states that the total entropy of an isolated system tends to increase over time.

• Entropy is a measure of the disorder or randomness of a system.

• The Second Law implies that natural processes tend to move towards states of higher entropy, leading to an increase in overall disorder within the system.

• One consequence of the Second Law is the concept of irreversibility; many natural processes are irreversible and result in an overall increase in entropy.

• The Second Law also predicts the direction of heat flow, stating that heat energy spontaneously flows from hotter regions to colder regions, but not vice versa, without external intervention.

• The law imposes limitations on the efficiency of heat engines and refrigeration systems, indicating that perfect conversion of heat into work is impossible.

• Understanding the Second Law of Thermodynamics is crucial in various fields, including physics, chemistry, engineering, and environmental science, as it governs the behavior of energy and entropy in natural systems.

• Entropy increases or remains the same in any cyclic process.

• Heat cannot flow from colder to hotter without work.

### Entropy

• Entropy is a thermodynamic quantity that represents the amount of disorder or randomness in a system.

• It is a measure of the number of possible microscopic arrangements or states that a system can have.

• Entropy tends to increase in natural processes, leading to a spontaneous move towards greater disorder.

• Systems with higher entropy are associated with greater randomness or unpredictability in the distribution of their constituent particles or energy.

• Entropy is closely related to the concept of probability; systems are more likely to be found in states with higher entropy.

• The Second Law of Thermodynamics states that the total entropy of an isolated system tends to increase over time.

• Entropy plays a crucial role in various fields, including thermodynamics, statistical mechanics, information theory, and ecology.

• Understanding entropy helps in analyzing the behavior of systems undergoing transformations and in designing efficient processes in engineering and science.

• Calculation: ΔS = Qrev / T

### Cyclic Processes and pV Diagrams

• Heat engines operate in cycles converting heat to work.

• pV diagrams represent cyclic processes.

• Cyclic processes occur when a system returns to its initial state after a series of thermodynamic changes.

• These processes are common in heat engines, refrigerators, and other cyclic systems.

• pV diagrams, also known as pressure-volume diagrams, graphically represent the changes in pressure and volume of a system during a thermodynamic process.

• In cyclic processes, pV diagrams form closed loops, indicating that the system returns to its initial state after completing a cycle.

• The area enclosed by the loop on a pV diagram represents the net work done by or on the system during the cycle.

• Heat engines, such as car engines, operate in a cyclic manner by taking in heat, converting some of it into work, and expelling the rest.

• pV diagrams provide valuable insights into the efficiency and performance of heat engine cycles, such as comparing them to idealized cycles like the Carnot cycle.

• Understanding cyclic processes and pV diagrams is essential for analyzing and optimizing the operation of various thermodynamic systems, including engines, refrigerators, and heat pumps.

### Isovolumetric, Isobaric, Isothermal, and Adiabatic Processes

• Isovolumetric process: Volume remains constant, resulting in changes in internal energy only.

• Isobaric process: Pressure stays constant, allowing for both work done and changes in internal energy.

• Isothermal process: Temperature remains constant, with energy transfer occurring as work done and changes in internal energy.

• Adiabatic process: No heat exchange with surroundings, leading to changes in temperature and pressure through work done alone.

• These processes are essential in thermodynamics and find applications in various systems such as engines, refrigerators, and heat exchangers.

### Carnot Cycle

• Most efficient heat engine cycle.

• Comprises two isothermal and two adiabatic processes.

• The Carnot cycle is a theoretical model representing the most efficient heat engine cycle.

• It consists of four reversible processes: two isothermal and two adiabatic processes.

• The cycle operates between two temperature reservoirs, a high-temperature source and a low-temperature sink.

• Heat is absorbed from the source during isothermal expansion and rejected to the sink during isothermal compression.

• The Carnot cycle's efficiency depends solely on the temperatures of the two reservoirs and is the maximum achievable efficiency for a heat engine.

• While an idealized concept, the Carnot cycle provides a standard for assessing the performance of real-world heat engines and refrigeration systems.

### Thermal Efficiency

• Efficiency: η= W/Qin

• Maximum efficiency in Carnot cycle:

• ηmax=1−(Tcold/Thot)

• Thermal efficiency is a measure of how effectively a device converts heat into useful work or energy.

• It is calculated as the ratio of the useful work output to the heat input.

• The efficiency of heat engines, refrigerators, and other thermal systems is often expressed in terms of thermal efficiency.

• For example, in heat engines, thermal efficiency is given by the ratio of the work done by the engine to the heat input from the high-temperature reservoir.

• The Carnot cycle sets the maximum theoretical efficiency for heat engines operating between two temperature reservoirs.

• Improving thermal efficiency is a key goal in engineering to optimize the performance and energy consumption of thermal systems.