Classical Mechanics : Work, energy, and power

Michael Kurniawan 1910403011 University of Cincinnati WORK, ENERGY, AND POWER 1. What is meant by kinetic energy? Kinetic energy is a fundamental concept in physics that describes the energy an object possesses due to its motion. When an object moves, it has the ability to do work because of its kinetic energy. This energy is directly proportional to the mass of the object and the square of its velocity. In simpler terms, the faster an object moves and the more massive it is, the greater its kinetic energy. Kinetic energy is a scalar quantity, meaning it has magnitude but not direction. It is expressed in the same units as work and energy, typically joules (J) in the International System of Units (SI). 2. What is the formula for calculating kinetic energy? The formula for kinetic energy is derived from the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy. Mathematically, the formula for calculating kinetic energy (KE) is: 𝐾𝐸 = (1/2) 𝑚𝑣2 Where: • 𝐾𝐸 is the kinetic energy (in joules, J), • 𝑚 is the mass of the object (in kilograms, kg), • 𝑣 is the velocity of the object (in meters per second, m/s). This formula illustrates that kinetic energy is directly proportional to the mass of the object and the square of its velocity. Thus, doubling the mass of an object doubles its kinetic energy, while doubling its velocity quadruples its kinetic energy. 3. What is gravitational potential energy? Gravitational potential energy is the energy an object possesses due to its position in a gravitational field. It arises from the gravitational attraction between the object and the Earth (or another massive body). When an object is raised to a certain height above the ground, it gains gravitational potential energy because it can potentially do work as it falls. The formula for calculating gravitational potential energy (PE) near the surface of the Earth is: 𝑃𝐸 = 𝑚𝑔ℎ Where: • 𝑃𝐸 is the gravitational potential energy (in joules, J), • 𝑚 is the mass of the object (in kilograms, kg), • 𝑔 is the acceleration due to gravity (in meters per second squared, m/s²), • ℎ is the height of the object above a reference point (in meters, m). Gravitational potential energy depends on the mass of the object, the acceleration due to gravity, and the height above the reference point. The higher the object is lifted, the greater its gravitational potential energy. Similarly, the more massive the object or the stronger the gravitational field, the greater its potential energy. 4. How do you calculate gravitational potential energy? Gravitational potential energy (GPE) is calculated using the formula: 𝑃𝐸 = 𝑚𝑔ℎ Where: • 𝑃𝐸 is the gravitational potential energy (in joules, J), • 𝑚 is the mass of the object (in kilograms, kg), • 𝑔 is the acceleration due to gravity (approximately 9.8 m/s2 on the surface of the Earth), • ℎ is the height of the object above a reference point (in meters, m). This formula tells us that gravitational potential energy is directly proportional to the mass of the object, the height of the object, and the gravitational acceleration. For example, if you lift a 1 kg object to a height of 10 meters, the gravitational potential energy would be: 𝑃𝐸 = 1kg × 9.8 m/s2 × 10 m = 98 J This means the object has 98 joules of potential energy due to its position in the gravitational field. 5. What are the differences between kinetic and potential energy? Kinetic energy and potential energy are two primary forms of mechanical energy, but they differ in several key ways: a. Nature: • Kinetic Energy: Energy of motion. An object has kinetic energy if it is moving. • Potential Energy: Energy of position or configuration. An object has potential energy if it is in a position where it can potentially move due to forces like gravity or tension. b. Dependence: • Kinetic Energy: Depends on the mass of the object and its velocity. The formula is 𝐾𝐸 = (1/2)𝑚𝑣2 • Potential Energy: Depends on the position of the object within a force field (like gravity or elastic force). For gravitational potential energy, the formula is 𝑃𝐸 = 𝑚𝑔ℎ c. Transformation: • Kinetic Energy: Can be transformed into potential energy. For example, when a moving car climbs a hill, its kinetic energy is converted into gravitational potential energy. • Potential Energy: Can be transformed into kinetic energy. For example, when an object falls from a height, its gravitational potential energy is converted into kinetic energy. d. Examples: • Kinetic Energy: A moving car, a running person, a flowing river. • Potential Energy: A rock perched on a hill, a stretched spring, a charged particle in an electric field. 6. What is work in the context of physics? In physics, work is defined as the process of energy transfer to an object via a force that causes the object to move. The amount of work done on an object is the product of the force applied to the object and the displacement of the object in the direction of the force. The formula for calculating work (W) is: 𝑊 = 𝐹 𝑑 cos (𝜃) Where: • 𝑊 is the work done (in joules, J), • 𝐹 is the force applied (in newtons, N), • 𝑑 is the displacement of the object (in meters, m), • 𝜃 is the angle between the force and the direction of displacement. Work is a scalar quantity, meaning it has magnitude but no direction. Positive work is done when the force and displacement are in the same direction, while negative work is done when they are in opposite directions. If the force is perpendicular to the displacement, no work is done. For example, if you push a box across the floor, you are doing work on the box. If you hold a heavy object stationary, even though you are exerting force, you are not doing any work in the physics sense because there is no displacement. 7. What is the formula for calculating work? The formula for calculating work in physics is: 𝑊 = 𝐹𝑑 cos (𝜃) Where: • 𝑊 is the work done (in joules, J), • 𝐹 is the magnitude of the force applied (in newtons, N), • 𝑑 is the displacement of the object (in meters, m), • 𝜃 is the angle between the force vector and the direction of displacement. This formula indicates that work is done when a force causes an object to move in the direction of the force. The cos (𝜃) term accounts for the direction of the force relative to the displacement. If the force is in the same direction as the displacement, 𝜃 = 0 and cos (0) = 1, resulting in maximum work. If the force is perpendicular to the displacement, 𝜃 = 90∘ and cos (90∘) = 0, resulting in no work done. If the force is in the opposite direction to the displacement, 𝜃 = 180∘ and cos (180∘) = −1, resulting in negative work. 8. What is the relationship between work and energy? The relationship between work and energy is that work is a mechanism of energy transfer. When work is done on an object, energy is transferred to or from that object. The amount of work done on an object is equal to the change in the object's energy. There are two primary forms of energy in this context: a. Kinetic Energy: The energy of motion. When work is done to accelerate an object, its kinetic energy increases. b. Potential Energy: The energy of position. When work is done to lift an object against gravity, its gravitational potential energy increases. The work-energy theorem formalizes this relationship and states that the net work done on an object is equal to its change in kinetic energy: 𝑊net = Δ𝐾𝐸 = (1/2)𝑚(𝑣2𝑓 – 𝑣2𝑖) Where 𝑣𝑓 is the final velocity and 𝑣𝑖 is the initial velocity. Additionally, work can change the potential energy of an object. For example, lifting an object increases its gravitational potential energy: 𝑊 = Δ𝑃𝐸 = 𝑚𝑔ℎ 9. What is power in physics? In physics, power is defined as the rate at which work is done or the rate at which energy is transferred or transformed. Power measures how quickly work is being performed or how quickly energy is being used or produced. The formula for calculating power (P) is: 𝑃 = 𝑊/𝑡 Where: • 𝑃 is the power (in watts, W), • 𝑊 is the work done or energy transferred (in joules, J), • 𝑡 is the time taken to do the work or transfer the energy (in seconds, s). Power can also be expressed in terms of force and velocity when dealing with constant motion: 𝑃 = 𝐹𝑣 Where : • 𝐹 is the constant force applied and • 𝑣 is the constant velocity. Power is a scalar quantity and is measured in watts (W), where 1 watt is equal to 1 joule per second (1 W = 1 J/s). High power means that a large amount of work is being done in a short amount of time, while low power means that work is being done more slowly. For example, a powerful engine can do more work in less time compared to a less powerful engine. 10. What is the formula for calculating power? The formula for calculating power is: 𝑃 = 𝑊/𝑡 Where: • 𝑃 is the power (in watts, W), • 𝑊 is the work done or energy transferred (in joules, J), • 𝑡 is the time taken to do the work or transfer the energy (in seconds, s). This formula tells us that power is the rate at which work is done or energy is transferred. If a large amount of work is done in a short amount of time, the power is high. Conversely, if the work is done slowly over a longer period, the power is low. Power can also be expressed in terms of force and velocity for an object moving at a constant speed: 𝑃 = 𝐹𝑣 Where: • 𝐹 is the constant force applied (in newtons, N), • 𝑣 is the constant velocity (in meters per second, m/s). This formula is particularly useful for understanding the power output of engines and motors. 11. How are power, work, and time related? Power, work, and time are related by the basic definition of power as the rate at which work is done. The relationship is given by the formula: 𝑃 = 𝑊𝑡 This means that power (P) is equal to the work done (W) divided by the time (t) it takes to do that work. Rearranging this formula, we can also express work in terms of power and time: 𝑊 = 𝑃𝑡 And time in terms of work and power: 𝑡 = 𝑊/𝑃 These relationships highlight that: a. To do a certain amount of work in less time, more power is required. b. If the power is fixed, doing more work will take more time. c. If a large amount of power is available, a large amount of work can be done in a short period. 12. If an object moves at a constant speed, does its kinetic energy change? If an object moves at a constant speed, its kinetic energy does not change. This is because kinetic energy is given by the formula: 𝐾𝐸 = (1/2) 𝑚𝑣2 Where: • 𝐾𝐸 is the kinetic energy (in joules, J), • 𝑚 is the mass of the object (in kilograms, kg), • 𝑣 is the velocity of the object (in meters per second, m/s). Kinetic energy depends on the mass of the object and the square of its velocity. If the speed (velocity) of the object remains constant, the kinetic energy remains constant as well, assuming the mass of the object does not change. For example, if a car is moving at a constant speed of 60 km/h, its kinetic energy remains the same as long as the speed does not change. If there are no external forces doing work on the car (ignoring friction and air resistance), the car's kinetic energy will stay constant. 13. How does the total energy of a system change when energy is absorbed by the system? When energy is absorbed by a system, the total energy of the system increases. Energy can be absorbed in various forms, such as thermal energy, electrical energy, or mechanical energy. The increase in total energy depends on the form and amount of energy absorbed. For instance: a. If a system absorbs heat, its thermal energy increases, which may result in a rise in temperature. b. If a system absorbs electrical energy, it can store this energy in batteries or capacitors, or convert it to other forms such as mechanical work. c. If a system absorbs mechanical energy, it can increase its kinetic or potential energy, depending on the nature of the work done on the system. The principle of conservation of energy states that energy cannot be created or destroyed, only transferred or transformed. Therefore, the energy absorbed by the system must come from the surroundings, leading to an increase in the system's total energy. 14. What is the principle of conservation of energy? The principle of conservation of energy is a fundamental concept in physics that states that the total energy of an isolated system remains constant over time. This means that energy cannot be created or destroyed; it can only be transformed from one form to another or transferred from one part of the system to another. Mathematically, the conservation of energy can be expressed as: 𝐸total = constant Where 𝐸 total represents the total energy of the system, which is the sum of all forms of energy within the system, such as kinetic energy, potential energy, thermal energy, chemical energy, etc. For example, in a closed system where only conservative forces (like gravity) act, the sum of kinetic and potential energy remains constant: 𝐾𝐸 + 𝑃𝐸 = constant This principle is crucial in solving many physical problems because it allows us to predict the behavior of systems without knowing all the details of the energy transformations involved. 15. How does energy transfer between different forms? Energy can transfer between different forms through various processes and mechanisms. Here are a few common examples: a. Mechanical to Thermal Energy: When you rub your hands together, mechanical energy from the motion is converted to thermal energy, warming your hands. b. Electrical to Mechanical Energy: In an electric motor, electrical energy is converted to mechanical energy, causing the motor's shaft to rotate. c. Chemical to Thermal Energy: In a combustion engine, chemical energy stored in fuel is converted to thermal energy through combustion, which then generates mechanical energy to move the vehicle. d. Gravitational Potential to Kinetic Energy: When an object falls, its gravitational potential energy is converted into kinetic energy as it accelerates towards the ground. e. Solar to Chemical Energy: Through photosynthesis, plants convert solar energy into chemical energy stored in glucose molecules. f. Nuclear to Thermal Energy: In a nuclear reactor, nuclear energy is converted to thermal energy, which is then used to produce electrical energy. Each of these processes follows the principle of conservation of energy, ensuring that the total energy remains constant while it changes from one form to another. Energy transfer mechanisms include work, heat transfer, and electromagnetic radiation, among others. The efficiency of these transformations varies, and some energy is often lost as waste heat, especially in real-world processes.