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Rotational kinetic energy and angular momentum

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Intros
Lessons
  2. Rotational kinetic energy
  3. Angular momentum
  4. Newton's 2nd law for rotation
  5. The law of conservation of angular momentum
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Examples
Lessons
  1. A bowling ball with a mass on 5.0kg and radius of 12.0cm is rolling down an inclined surface with a negligible friction without slipping. If it starts from rest at a vertical distance of 1.5m, what will be the speed of the ball when it reaches the bottom of the inclined surface?
    1. A sphere of radius 24.0 cm and mass of 1.60 kg, starts from the rest and rolls without slipping down a 30.0° incline that is 12.0m long.
      1. Calculate the translational and rotational speed when it reaches the bottom.
      2. What is the total kinetic energy as it gets to the bottom?
    2. Two masses m1m_{1} = 16.0kg and m2m_{2} = 24.6kg, are connected by a rope that hangs over a uniform pulley with a mass of 6.50kg and radius of 0.24m. Initially, m1m_{1} is on the ground and m2m_{2} rests 2.00m above the ground. If the system is released what will be the speed of m2m_{2}, as it hits the ground?
      1. A 2.6kg uniform cylindrical grinding wheel of radius 16cm makes 1600rpm.
        1. What is the angular momentum?
        2. How much torque is required to stop it in 4.0s?
      2. The speed of a skater increases from 2.0 rev every 4.0s to a final rate of 4.0 rev/s. If her moment of inertia was 4.4kg.m2, what is her final moment of inertia?
        1. A potter's wheel is rotating around a vertical axis through its center at a frequency of 1.2 rev/s. The wheel can be modeled as a uniform disk of mass 4.0kg and diameter of 0.60m. The potter the throws a 2.1 kg chunk of clay, shaped as a flat disk of radius 6.0cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?
          Topic Notes
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          Introduction to Rotational Kinetic Energy and Angular Momentum

          Welcome to our exploration of rotational kinetic energy and angular momentum! These fascinating concepts are crucial in understanding the motion of rotating objects. Rotational kinetic energy is the energy possessed by an object due to its rotational motion, while angular momentum describes the object's tendency to continue rotating. Our introduction video serves as an excellent starting point, providing clear visuals and explanations to help you grasp these ideas. As your math tutor, I'm excited to guide you through these concepts, which have wide-ranging applications in physics and engineering. From spinning ice skaters to orbiting planets, rotational kinetic energy and angular momentum play key roles in explaining the world around us. By the end of this lesson, you'll have a solid foundation in these principles, enabling you to tackle more complex problems with confidence. Let's dive in and unravel the mysteries of rotational motion together!

          Rotational Kinetic Energy

          Understanding Rotational Kinetic Energy

          Rotational kinetic energy is a fundamental concept in physics that describes the energy possessed by an object due to its rotational motion. It is a crucial component in understanding the dynamics of rotating bodies and plays a significant role in various mechanical systems. Unlike translational kinetic energy, which is associated with linear motion, rotational kinetic energy is specifically related to the rotation of an object around an axis.

          Definition and Equation

          Rotational kinetic energy is defined as the energy of rotation of a rigid body. The equation for calculating rotational kinetic energy is:

          KErotational = ½ I ω²

          Where:

          • KErotational is the rotational kinetic energy
          • I is the moment of inertia of the object
          • ω (omega) is the angular velocity of the object

          Comparison with Translational Kinetic Energy

          While rotational kinetic energy deals with rotating objects, translational kinetic energy is associated with objects moving in a straight line. The equation for translational kinetic energy is:

          KEtranslational = ½ mv²

          Where m is the mass of the object and v is its velocity. The key difference lies in the use of moment of inertia (I) and angular velocity (ω) for rotational motion, as opposed to mass (m) and linear velocity (v) for translational motion.

          The Basketball Example

          Consider a basketball spinning on a player's finger. The ball possesses rotational kinetic energy due to its rotation. The faster it spins (higher angular velocity), the more rotational kinetic energy it has. The distribution of the ball's mass (affecting its moment of inertia) also influences its rotational kinetic energy. In contrast, when the same basketball is thrown in a straight line, it has translational kinetic energy based on its mass and linear velocity.

          Derivation of the Rotational Kinetic Energy Formula

          To understand the derivation of the rotational kinetic energy formula, let's consider a rotating ball. Imagine the ball as composed of many small particles. Each particle has a mass mi and is at a distance ri from the axis of rotation. The velocity of each particle is vi = ω ri, where ω is the angular velocity of the ball.

          The kinetic energy of each particle is:

          KEi = ½ mivi² = ½ mi(ω ri)² = ½ (miri²)ω²

          The total rotational kinetic energy is the sum of the kinetic energies of all particles:

          KErotational = Σ ½ (miri²)ω² = ½ (Σ miri²)ω²

          The term Σ miri² is defined as the moment of inertia (I) of the object. Thus, we arrive at the final equation:

          KErotational = ½ I ω²

          Combined Rotational and Translational Kinetic Energy

          In many real-world scenarios, objects possess both rotational and translational kinetic energy

          Angular Momentum: Understanding Rotational Motion

          Definition and Significance

          Angular momentum is a fundamental concept in physics that describes the rotational motion of an object. It is a measure of the amount of rotational motion an object possesses and is crucial in understanding various phenomena in classical mechanics, quantum mechanics, and astrophysics. Angular momentum plays a significant role in maintaining the stability of rotating systems, from spinning tops to planetary orbits.

          Equation and Components

          The equation for angular momentum (L) is expressed as:

          L = I × ω

          Where:

          • I is the moment of inertia, which represents the object's resistance to rotational acceleration
          • ω (omega) is the angular velocity, which measures the rate of rotation

          The moment of inertia depends on the object's mass distribution relative to its axis of rotation. Angular velocity is a vector quantity that describes the speed and direction of rotation.

          Comparison with Linear Momentum

          While angular momentum relates to rotational motion, linear momentum pertains to translational motion. The key differences include:

          • Linear momentum (p) is calculated as mass (m) times velocity (v): p = mv
          • Angular momentum involves moment of inertia instead of mass
          • Angular momentum is conserved in rotational systems, while linear momentum is conserved in linear systems

          Units of Angular Momentum

          In the International System of Units (SI), angular momentum is measured in kilogram-square meters per second (kg·m²/s). This unit is derived from the components of its equation:

          • Moment of inertia: kg·m²
          • Angular velocity: radians per second (rad/s)

          Examples Illustrating Angular Momentum

          1. Figure Skater's Spin: When a figure skater pulls their arms close to their body during a spin, their angular velocity increases. This demonstrates the conservation of angular momentum, as the decrease in moment of inertia leads to an increase in angular velocity.

          2. Gyroscope: The stability of a gyroscope is due to its high angular momentum. As it spins rapidly, it resists changes to its axis of rotation, making it useful in navigation systems and stabilization devices.

          3. Earth's Rotation: The Earth's angular momentum plays a crucial role in maintaining its day-night cycle and influences climate patterns. Any changes in the Earth's moment of inertia, such as the redistribution of mass due to melting ice caps, can affect its rotation rate.

          Conservation of Angular Momentum

          One of the most important principles related to angular momentum is its conservation of angular momentum in closed systems. This law states that the total angular momentum of a system remains constant unless acted upon by external torques. This principle explains phenomena such as the precession of a spinning top and the formation of spiral galaxies.

          Applications in Physics and Engineering

          Understanding angular momentum is crucial in various fields:

          • Astrophysics: Explaining the formation and behavior of celestial bodies
          • Quantum Mechanics: Describing the intrinsic angular momentum (spin) of particles
          • Engineering: Designing rotating machinery, gyroscopes, and stabilization systems
          • Sports Science: Analyzing and improving rotational movements in athletics

          In conclusion, angular momentum is a vital concept in physics that provides insights into rotational motion. Its conservation principle and relationship with moment of inertia and angular velocity make it an essential tool for understanding and predicting the behavior of rotating systems across various scales, from subatomic particles to galaxies.

          Newton's Second Law for Rotational Motion

          Understanding Newton's Second Law in Rotational Context

          Newton's Second Law, a fundamental principle in physics, extends beyond linear motion to encompass rotational dynamics. In the realm of rotational motion, this law takes on a new form while maintaining its core essence. Just as force influences linear acceleration, torque becomes the driving factor behind angular acceleration in rotating objects.

          Comparing Translational and Rotational Equations

          The parallel between translational and rotational motion is evident in their respective equations. For linear motion, Newton's Second Law states that F = ma, where F is force, m is mass, and a is acceleration. In rotational motion, this transforms into τ = Iα, where τ represents torque, I is the moment of inertia, and α denotes angular acceleration. This comparison highlights the analogous roles of force and torque, mass and moment of inertia, and linear and angular acceleration in their respective domains.

          Defining Torque and Its Significance

          Torque, often described as the rotational equivalent of force, is crucial in understanding rotational dynamics. It's defined as the product of force and the perpendicular distance from the axis of rotation to the line of action of the force. Mathematically expressed as τ = r × F, torque measures the tendency of a force to rotate an object around an axis. The greater the torque applied, the higher the angular acceleration produced, assuming a constant moment of inertia.

          The Role of Moment of Inertia

          Moment of inertia, symbolized by I, plays a role similar to mass in linear motion. It quantifies an object's resistance to rotational acceleration and depends on both the object's mass and its distribution around the axis of rotation. Objects with larger moments of inertia require more torque to achieve the same angular acceleration compared to objects with smaller moments of inertia.

          Practical Examples of Rotational Motion

          To illustrate these concepts, consider a figure skater performing a spin. As they pull their arms closer to their body, their moment of inertia decreases, resulting in increased angular velocity due to the conservation of angular momentum. Another example is a merry-go-round; applying a force at its edge creates a torque that causes it to rotate. The further from the center the force is applied, the greater the torque and the resulting angular acceleration.

          Applications in Engineering and Everyday Life

          Understanding rotational motion is crucial in various fields. In automotive engineering, it's essential for designing efficient engines and transmissions. In sports, it helps athletes optimize their performance in activities involving rotation, such as diving or gymnastics. Even in everyday scenarios like opening a door, we unconsciously apply the principles of torque and rotational motion.

          Challenges in Rotational Dynamics

          While the principles of rotational motion parallel those of linear motion, they often present more complex scenarios. Calculating moments of inertia for irregularly shaped objects or understanding the behavior of rotating systems with multiple components can be challenging. These complexities make rotational dynamics a rich and intriguing area of study in physics and engineering.

          Conclusion: The Importance of Rotational Motion in Physics

          Newton's Second Law for rotational motion provides a powerful framework for understanding and predicting the behavior of rotating objects. By drawing parallels with linear motion while introducing concepts like torque and moment of inertia, it offers a comprehensive approach to analyzing rotational dynamics. Whether in advanced scientific research or everyday applications, grasping these principles enhances our understanding of the physical world and our ability to interact with it effectively.

          Conservation of Angular Momentum

          Understanding the Law of Conservation of Angular Momentum

          The law of conservation of angular momentum is a fundamental principle in physics that describes the behavior of rotating objects. It states that the total angular momentum of a closed system remains constant if no external torques act on the system. This principle plays a crucial role in understanding various phenomena in nature and has significant applications in engineering and space exploration.

          The Equation and Its Implications

          The conservation of angular momentum can be expressed mathematically as:

          L = I * ω

          Where:

          • L is the angular momentum
          • I is the moment of inertia
          • ω (omega) is the angular velocity

          This equation implies that if the moment of inertia changes, the angular velocity must change proportionally to maintain a constant angular momentum. This relationship has profound implications for rotational motion in various scenarios.

          The Ice Skater Example: A Perfect Illustration

          One of the most vivid demonstrations of the conservation of angular momentum is the spinning ice skater. When an ice skater begins to spin with their arms outstretched, they have a large moment of inertia due to the mass distribution away from their axis of rotation. As they pull their arms close to their body, their moment of inertia decreases significantly.

          According to the conservation of angular momentum, this decrease in moment of inertia must be compensated by an increase in angular velocity. As a result, the skater spins much faster when their arms are pulled in close to their body. This dramatic increase in rotational speed is a direct consequence of the conservation of angular momentum.

          Changes in Moment of Inertia and Angular Velocity

          The relationship between moment of inertia and angular velocity is inverse. When the moment of inertia decreases, the angular velocity increases, and vice versa. This principle extends beyond the ice skater example and applies to various scenarios in physics and engineering.

          For instance:

          • A diver tucking into a ball to increase rotation speed during a dive
          • A figure skater extending their arms to slow down a spin
          • The formation of compact, rapidly rotating neutron stars from larger, slower-rotating stars

          Applications in Science and Technology

          The conservation of angular momentum has numerous practical applications:

          • Spacecraft attitude control: Satellites use reaction wheels to adjust their orientation based on this principle
          • Gyroscopes: These devices maintain their orientation due to the conservation of angular momentum
          • Weather systems: The rotation of the Earth influences large-scale atmospheric circulation patterns
          • Planetary formation: The conservation of angular momentum plays a role in the formation and rotation of planets and stars

          Conclusion

          The law of conservation of angular momentum is a powerful principle that governs rotational motion in the universe. From the graceful movements of ice skaters to the formation of celestial bodies, this law provides insights into a wide range of phenomena. Understanding the relationship between moment of inertia and angular velocity allows us to explain and predict complex rotational behaviors in both natural and engineered systems.

          By grasping this fundamental concept, we gain a deeper appreciation for the elegant simplicity underlying the complex motions we observe in the world around us. Whether you're a physics enthusiast, an engineer, or simply curious about the principles governing motion, the conservation of angular momentum offers a fascinating glimpse into the fundamental laws that shape our universe.

          Applications and Examples of Rotational Kinetic Energy and Angular Momentum

          Sports Applications

          Rotational kinetic energy and angular momentum play crucial roles in various sports, enhancing performance and technique. In figure skating, athletes utilize the conservation of angular momentum to increase their spinning speed by pulling their arms close to their body. This principle is also evident in diving, where divers adjust their body position to control rotation. In gymnastics, understanding these concepts helps athletes perform intricate flips and twists with precision.

          Baseball pitchers leverage rotational kinetic energy to increase ball speed. By rotating their body and arm, they transfer energy to the ball, resulting in faster pitches. Similarly, in golf, players use rotational motion to generate power in their swings, maximizing the distance the ball travels. Tennis players also rely on rotational kinetic energy when serving, using their entire body to create a powerful, spinning serve.

          Machinery and Industrial Applications

          Rotational kinetic energy is fundamental in the design and operation of various machines. Flywheels, for instance, store rotational kinetic energy and help maintain constant angular velocity in engines, smoothing out power delivery. This principle is utilized in automotive engines, industrial machinery, and even in some renewable energy systems for energy storage.

          Gyroscopes, which rely on the conservation of angular momentum, are used in navigation systems for ships, aircraft, and spacecraft. They help maintain orientation and stability in these vehicles. In manufacturing, rotational kinetic energy is harnessed in centrifuges for separating materials of different densities, a process crucial in medical laboratories and industrial settings.

          Everyday Life Examples

          Rotational motion and angular momentum are present in many everyday objects and activities. A simple example is a spinning top or a fidget spinner, where the conservation of angular momentum keeps the object rotating. When opening or closing a door, we unconsciously apply torque, demonstrating rotational motion. The operation of a washing machine's spin cycle is another practical application, using centrifugal force (a consequence of rotational motion) to remove water from clothes.

          In transportation, the wheels of vehicles embody rotational kinetic energy. The gyroscopic effect of a bicycle wheel helps maintain balance while riding. Even in cooking, whisking eggs or using a blender involves principles of rotational motion to mix ingredients effectively.

          Solving Practical Problems

          Understanding rotational kinetic energy and angular momentum is crucial for solving various real-world problems. In engineering, these concepts are essential for designing more efficient wind turbines, optimizing the performance of electric motors, and improving the stability of rotating machinery. In automotive design, knowledge of rotational dynamics helps in creating safer vehicles with better handling characteristics.

          In sports science, these principles are applied to enhance athletic performance and develop better training techniques. Coaches and athletes use this understanding to refine movements in activities like discus throwing, ice skating, and gymnastics. In space exploration, angular momentum conservation is vital for controlling the orientation of satellites and spacecraft.

          Environmental scientists and engineers apply these concepts in developing more efficient renewable energy technologies. For instance, the design of tidal turbines and improvements in wind turbine efficiency rely heavily on understanding rotational dynamics. In medical technology, these principles are used in the development of centrifuges for blood separation and in the design of artificial joints that mimic natural rotational movements.

          By grasping these fundamental concepts, professionals across various fields can innovate and solve complex problems. From improving sports equipment to designing more energy-efficient machines, the applications of rotational kinetic energy and angular momentum continue to shape our technological advancements and daily lives.

          Conclusion

          In this article, we've explored the fundamental physics concepts of rotational kinetic energy and angular momentum. Understanding these principles is crucial for grasping the behavior of rotating objects in various real-world applications. We've discussed how rotational kinetic energy depends on an object's moment of inertia and angular velocity, while angular momentum relates to mass distribution and rotational speed. The conservation of angular momentum explains phenomena like figure skaters' spins and the stability of bicycle wheels. To enhance your comprehension, we encourage you to watch the introduction video, which provides a visual representation of these concepts. By mastering rotational kinetic energy and angular momentum, you'll gain valuable insights into the mechanics of rotating systems. We invite you to explore further resources on our website and engage in discussions to deepen your understanding of these fascinating physics principles. Remember, a solid grasp of these concepts can open doors to advanced topics in physics and engineering.

          Rotational Kinetic Energy

          In this section, we will explore the concept of rotational kinetic energy, derive the equation for calculating it, and understand its relationship with translational kinetic energy.

          Step 1: Introduction to Rotational Kinetic Energy

          Objects rotating about an axis possess rotational kinetic energy. For example, consider a basketball. As it rotates, it exhibits a form of kinetic energy known as rotational kinetic energy. This is distinct from translational kinetic energy, which is the energy of objects moving along a straight line.

          Step 2: Translational vs. Rotational Kinetic Energy

          When an object like a basketball is both rotating and moving forward, it possesses both translational and rotational kinetic energy. Translational kinetic energy is calculated using the formula 12mv2 \frac{1}{2}mv^2 , where m m is the mass and v v is the velocity. To convert this formula for rotational kinetic energy, we use analogs from rotational motion.

          Step 3: Rotational Motion Analogs

          In translational motion, speed is represented by v v , while in rotational motion, angular velocity is represented by ω \omega . Similarly, mass m m in translational motion is analogous to the moment of inertia I I in rotational motion. The moment of inertia is a measure of an object's resistance to changes in its rotation.

          Step 4: Deriving the Rotational Kinetic Energy Formula

          The formula for translational kinetic energy is 12mv2 \frac{1}{2}mv^2 . To derive the formula for rotational kinetic energy, we substitute m m with I I (moment of inertia) and v v with ω \omega (angular velocity). This gives us the formula 12Iω2 \frac{1}{2}I\omega^2 .

          Step 5: Proving the Equation

          Consider a rotating ball made up of many tiny particles, each with mass m m and distance r r from the center of rotation. The kinetic energy of each particle is 12mv2 \frac{1}{2}mv^2 . Using the relationship v=rω v = r\omega , we substitute v v with rω r\omega to get 12mr2ω2 \frac{1}{2}mr^2\omega^2 . Summing the kinetic energies of all particles, we get 12∑mr2ω2 \frac{1}{2}\sum mr^2\omega^2 . Recognizing that ∑mr2 \sum mr^2 is the moment of inertia I I , we arrive at the formula 12Iω2 \frac{1}{2}I\omega^2 .

          Step 6: Combining Rotational and Translational Kinetic Energy

          In some cases, an object may possess both rotational and translational kinetic energy. For example, a sphere rolling down an inclined surface has both types of kinetic energy. The total kinetic energy is the sum of translational kinetic energy 12mv2 \frac{1}{2}mv^2 and rotational kinetic energy 12Iω2 \frac{1}{2}I\omega^2 . Here, m m is the total mass, I I is the moment of inertia about the center of mass, v v is the translational speed, and ω \omega is the angular speed.

          FAQs

          1. What is rotational kinetic energy?

            Rotational kinetic energy is the energy possessed by an object due to its rotational motion. It is defined by the equation KErotational = ½ I ω², where I is the moment of inertia and ω is the angular velocity. This energy form is distinct from translational kinetic energy and is crucial in understanding the behavior of rotating objects.

          2. How do you find angular kinetic energy?

            To find angular kinetic energy, use the formula KErotational = ½ I ω². First, determine the object's moment of inertia (I) based on its shape and mass distribution. Then, measure or calculate its angular velocity (ω). Multiply half of the moment of inertia by the square of the angular velocity to obtain the rotational kinetic energy.

          3. What is the angular equation for kinetic energy?

            The angular equation for kinetic energy is KErotational = ½ I ω². This equation is analogous to the linear kinetic energy equation (KE = ½ mv²), with moment of inertia (I) replacing mass (m) and angular velocity (ω) replacing linear velocity (v).

          4. What is the kinetic energy of a rotating object?

            The kinetic energy of a rotating object is the sum of its translational and rotational kinetic energies. If the object is purely rotating (not moving linearly), its kinetic energy is entirely rotational, given by KErotational = ½ I ω². For objects with both rotational and translational motion, the total kinetic energy is KEtotal = ½ mv² + ½ I ω².

          5. How does angular momentum relate to rotational kinetic energy?

            Angular momentum (L) and rotational kinetic energy are closely related but distinct concepts. Angular momentum is given by L = I ω, while rotational kinetic energy is KErotational = ½ I ω². Both involve moment of inertia (I) and angular velocity (ω), but angular momentum is a vector quantity conserved in closed systems, whereas rotational kinetic energy is a scalar quantity that can be transferred or converted to other forms of energy.

          Prerequisite Topics

          Understanding rotational kinetic energy and angular momentum is crucial in physics, but to truly grasp these concepts, it's essential to have a solid foundation in prerequisite topics. Two key areas that form the bedrock of this understanding are Newton's Second Law and the principle of conservation of angular momentum.

          Newton's Second Law is fundamental to understanding rotational motion. This law, which states that the acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass, forms the basis for analyzing rotational dynamics. When we extend this principle to rotating objects, we derive equations for rotational motion that parallel those for linear motion. The concept of force in linear motion translates to torque in rotational motion, while mass becomes moment of inertia. This connection allows us to calculate rotational kinetic energy, which is analogous to linear kinetic energy but involves angular velocity and moment of inertia instead of linear velocity and mass.

          The principle of conservation of angular momentum is another critical prerequisite for understanding rotational motion. While the link provided focuses on linear momentum, the concept extends beautifully to rotational systems. Just as linear momentum is conserved in the absence of external forces, angular momentum is conserved in the absence of external torques. This principle is crucial for analyzing spinning objects and explains phenomena like the speeding up of figure skaters when they pull in their arms. Understanding this conservation law is essential for solving problems involving rotating bodies and for grasping the behavior of systems like gyroscopes and planets.

          By mastering these prerequisite topics, students can build a strong foundation for understanding more complex concepts in rotational dynamics. Newton's Second Law provides the framework for analyzing forces and motion, which directly applies to rotational systems. The conservation of angular momentum principle, an extension of linear momentum conservation, offers powerful insights into the behavior of rotating objects. Together, these concepts enable students to tackle problems involving rotational kinetic energy and angular momentum with confidence and clarity.

          In conclusion, a thorough understanding of these prerequisite topics is not just helpful but essential for mastering rotational kinetic energy and angular momentum. By building on these fundamental principles, students can develop a deeper, more intuitive grasp of rotational dynamics, preparing them for more advanced topics in physics and engineering.

          In this lesson, we will learn:

          • How to calculate rotational kinetic energy?
          • Definition of angular momentum
          • Newton’s 2nd law for rotation
          • The law of conservation of angular momentum

          Notes:


          Rotational Kinetic Energy

          • Objects rotating about an axis possess “Rotational Kinetic Energy”
          • Objects moving along a straight line possess “Translational Kinetic Energy”.
          • Translational kinetic energy is calculated using, 12mv2\frac{1}{2}mv^{2} , we can convert this formula to rotational kinetic energy question using the rotational motion analogues:

            • Rotational Kinetic Energy and Angular Momentum


            If we consider a rotating ball, every point on the ball is rotating with some speed.
            The ball is made up many tiny particles, each of mass “mm”. let’s take “rr” to be the distance of any one particle from the axis of rotation (O);

            Rotational Kinetic Energy and Angular Momentum


            ⇒ \Rightarrow \, rotational KE=12I ω2KE = \frac{1}{2}I\, \omega^{2}

          • The center of mass of a rotating object might undergo translational motion (a sphere rolling down an incline), in this case we have to consider both rotational and translational kinetic energy.


            • KE=12Mv2 + 12ICM  ω2KE = \frac{1}{2}M v^{2} \, + \, \frac{1}{2}I_{CM \;} \omega^{2}


            MM : total mass of the object
            ICMI_{CM} : moment of inertia about the axis through center of mass 
            vv : translational speed 
            ω\omega : angular speed 


            Angular Momentum
            In the previous section, we used the rotational analogue to translate the translational kinetic energy to rotational kinetic energy.
            In like manner, the linear momentum can be changed to angular momentum, using the rotational analogues;

            Rotational Kinetic Energy and Angular Momentum

            LL: is the angular momentum with a standard unit of kg.m2/s


            Newton’s 2nd Law for Rotation


            The Newton’s 2nd law also can be written in terms of rotational analogues;

            ∑F= ma =\sum F = \, ma \, = mΔvΔt\large \frac{m \Delta v}{\Delta t} == ΔpΔt\large \frac{\Delta p}{\Delta t} \quad , in translational motion “Force” causes linear acceleration

            \qquad \quad Similarly for rotational motion , ∑=\sum = ΔLΔt \large \frac{\Delta L }{\Delta t}

            ∑τ= \sum \tau = \, ΔLΔt\large \frac{\Delta L}{\Delta t} = = IΔωΔt\large \frac{I \Delta \omega}{\Delta t}  =I∝\, = I \propto \quad , in rotational motion “Torque” causes rotational acceleration

            ∑F=ma\sum F = ma \quad Newton’s 2nd Law in Translational Motion
            ∑τ=I∝\sum \tau = I \propto \quad Newton’s 2nd Law in Rotational Motion


            The law of Conservation of Angular Momentum

            The law of conservation of angular moment states that;

            “The total angular momentum of a rotating object remains constant if the net torque acting on it is zero.”

            ∑τ=0  ⇒  Li=Lf  ⇒  Iiωi=Ifωf\sum \tau = 0 \; \Rightarrow \; L_{i} = L_{f} \; \Rightarrow \; I_{i \omega i} = I_{f \omega f}

            Example: A skater doing a spin on ice, illustrating conservation of angular momentum.

            Rotational Kinetic Energy and Angular Momentum


            Open arms: Li=Iωi=mri2ωi \qquad L_{i} = I \omega_{i} = mr^{2}_{i} \omega_{i}
            Closed arms:   Lf=Iωf=mrf2ωf\qquad L_{f} = I \omega_{f} = mr^{2}_{f} \omega_{f}

            When the arms of the skater are tucked in, the mass is not changing, but the radius of rotation decreases; rfr_{f} < rir_{i} , therefore;  mrf2mr^{2}_{f} < mri2mr^{2}_{i}

            According to the law of conservation of angular momentum; Li=LfL_{i} = L_{f}

            Li=LfL_{i} = L_{f}

            ⇒ωf\qquad \Rightarrow \quad \omega_{f} > ωi  \omega_{i} \; “the result would be higher angular velocity”

            mrf2 mr^{2}_{f} < mri2 mr^{2}_{i}