BAS 021 Physics I Chapter 4 Energy of a system v2 PDF

BAS 021 Physics I Energy of a system Prof. Taher Bazan Basic Sciences Department  Contents 1. Introduction 2. Work done by a constant force 3. Work done by a varying force 4. Kinetic energy and the work–kinetic energy theorem 5. Potential energy of a system 6....

BAS 021 Physics I Energy of a system Prof. Taher Bazan Basic Sciences Department  Contents 1. Introduction 2. Work done by a constant force 3. Work done by a varying force 4. Kinetic energy and the work–kinetic energy theorem 5. Potential energy of a system 6. Conservative and nonconservative forces 2 Energy of a system  Introduction Work Power Energy Work, Energy and Power are fundamental concepts of Physics and they are closed linked. Work is said to be done when a force (push or pull) applied to an object causes a displacement of the object. We define the capacity (ability) to do the work as energy. Power is the work done per unit of time (how fast to perform this work). 3 Energy of a system  Introduction Energy is defined as the ability to do work. Energy comes in various forms. Every physical process that occurs in the Universe involves energy and energy transfers or transformations. 4 Energy of a system  Work Done by a Constant Force we must consider not only the magnitude of the force but also its direction. Equal forces but with different directions The work W done on a system by an agent exerting a constant force on the system is the product of the magnitude F of the force, the magnitude r of the displacement of the point of application of the force, and cos , where  is the angle between the force and displacement vectors. 𝑊 ≡ 𝐹 ∆𝑟 𝑐𝑜𝑠𝜃 5 Energy of a system  Work Done by a Constant Force The normal force 𝑛 and the gravitational force 𝑚𝑔Ԧ do no work on the object. Forces are perpendicular to the displacement along x-axis The weightlifter does no work on the weights as he holds them for a while. The force exists but no displacement 6 Energy of a system  Work Done by a Constant Force Units of Work Work is a scalar quantity, and its units are force multiplied by length. SI Unit of Work is the Newton.meter N. 𝑚 = 𝑘𝑔. 𝑚2 /𝑠 2 ≡ joule (J) Work is an energy transfer. If W is the work done on a system and W is positive, energy is transferred to the system. If W is negative, energy is transferred from the system 7 Energy of a system  Work Done by a Constant Force The weightlifter lowers a weights to the floor When a weightlifter lowers a the weights, his hands and the weights move together with the same displacement. The weights exerts a force on his hands in the same direction as the hands’ displacement, so the work done by the weights on his hands is positive. But by Newton’s third law the weightlifter’s hands exert an equal and opposite force on the weights. This force, which keeps the weights from crashing to the floor, acts opposite to the weights' displacement. Thus, the work done by his hands on the weights is negative. 8 Energy of a system  Example for Work Done by a Constant Force 9 Energy of a system  Work Done by a Varying Force Consider a particle being displaced along the x axis under the action of a force that varies with position. ∆𝑥 is a very small displacement ❑ If more than one external force acts on a system 10 Energy of a system  Example for Work Done by a Varying Force Sol. 11 Energy of a system  Kinetic energy and the work–kinetic energy theorem According to newton’s third law: 𝑣𝑓2 − 𝑣𝑖2 𝑣𝑓2 = 𝑣𝑖2 + 2𝑎 ∆𝑥 𝑎 = 2∆𝑥 𝑣𝑓2 − 𝑣𝑖2 𝐹 = 𝑚𝑎 = 𝑚 2∆𝑥 1 1 𝐹∆𝑥 = 𝑊𝑒𝑥𝑡 = 𝑚𝑣𝑓2 − 𝑚𝑣𝑖2 2 2 Final Kinetic energy Initial Kinetic energy 1 1 𝑊𝑒𝑥𝑡 = 𝑚𝑣𝑓 − 𝑚𝑣𝑖2 = 𝐾𝑓 − 𝐾𝑖 2 𝑊𝑒𝑥𝑡 = ∆𝐾 = 𝐾𝑓 − 𝐾𝑖 2 2 12 Energy of a system  Kinetic energy and the work–kinetic energy theorem 1 1 𝑊𝑒𝑥𝑡 = 2 𝑚𝑣𝑓2 − 2 𝑚𝑣𝑖2 > 0 𝑣𝑓 > 𝑣𝑖 OR 𝐾𝑓 > 𝐾𝑖 The work–kinetic energy theorem indicates that the speed of a system increases if the net work done on it is positive because the final kinetic energy is greater than the initial kinetic energy. 1 1 𝑊𝑒𝑥𝑡 = 2 𝑚𝑣𝑓2 − 2 𝑚𝑣𝑖2 < 0 𝑣𝑓 < 𝑣𝑖 OR 𝐾𝑓 < 𝐾𝑖 The speed decreases if the net work is negative because the final kinetic energy is less than the initial kinetic energy. 13 Energy of a system  Example for Kinetic energy and the work–kinetic energy theorem In an electron microscope, there is an electron gun that contains two charged metallic plates 2.8 cm apart. An electric force accelerates each electron in the beam from rest to 9.6% of the speed of light over this distance in the same direction. For an electron passing between the plates in the electron gun, determine: (a) Determine the kinetic energy of the electron. (b) The magnitude of the constant electric force acting on the electron (c) The acceleration of the electron (d) The time interval the electron spends between the plates. Note that the speed of light is 3 × 108 m/s and the electron mass = 9.11 × 10−31 𝑘𝑔 Sol. 𝑣𝑖 = 0 𝑣𝑓 = 0.096 × 3 × 108 = 2.88 × 107 m/s 1 1 a) 𝐾𝑓 − 𝐾𝑖 = 𝐾𝑓 − 0 = 2 𝑚𝑣𝑓2 = 2 (9.11 × 10−31 )(2.88 × 107 )2 = 3.78 × 10−16 J b) 𝑊𝑒𝑥𝑡 = ∆𝐾 = 𝐾𝑓 − 𝐾𝑖 = 3.78 × 10−16 − 0 = 3.78 × 10−16 J 𝑊𝑒𝑥𝑡 = 𝐹∆𝑟 cos 𝜃 = F 0.028 cos 0 = 3.78 × 10−16 𝐹 = 1.35 × 10−14 𝑁 14 Energy of a system Example for Kinetic energy and the work–kinetic energy theorem c) a=? σ 𝐹 1.35 × 10−14 ෍ 𝐹 = 𝑚𝑎 𝑎= = = 1.48 × 1016 𝑚/𝑠 2 𝑚 9.11 × 10 −31 d) t=? 𝑣𝑥,𝑓 = 𝑣𝑥,𝑖 + 𝑎𝑥 𝑡 2.88 × 107 = 0 + 1.48 × 1016 t 𝑡 = 1.94 × 10−9 s 15 Energy of a system  Potential Energy of a System ❑ Let us imagine a system consisting of a book and the Earth, interacting via the gravitational force. ❑ We do some work on the system by lifting the book slowly from rest through a vertical displacement ❑ The book is at rest before we perform the work and is at rest after we perform the work. Therefore, there is no change in the kinetic energy of the system. The work-kinetic energy theorem does not apply here and the energy change must appear as some form of energy storage other than kinetic energy. 16 Energy of a system  Potential Energy of a System ❑ While the book was at the highest point, the system had the potential to possess kinetic energy, but it did not do so until the book was allowed to fall. ❑ Therefore, we call the energy storage mechanism before the book is released potential energy. ❑ Assume that lifting is done slowly without acceleration (constant velocity), the system at equilibrium: 17 Energy of a system  Potential Energy of a System Gravitational potential energy 𝑈𝑔 = 𝑚𝑔ℎ 𝑊𝑒𝑥𝑡 = ∆𝑈𝑔 The net external work done on the system appears as a change in the gravitational potential energy of the system. Work is required (positive value since 𝑦𝑓 > 𝑦𝑖 ) to raise objects against Earth’s gravity – this work is stored as gravitational potential energy. 18 Energy of a system  Potential Energy of a System If the book of mass m falls from height 𝑦𝑖 to 𝑦𝑓 𝑊𝑒𝑥𝑡 = 𝐹Ԧ𝑎𝑝𝑝. ∆𝑟Ԧ = 𝑚𝑔𝑗.Ƹ (𝑦𝑓 − 𝑦𝑖 )𝑗Ƹ 𝑊𝑒𝑥𝑡 = 𝑚𝑔𝑦𝑓 − 𝑚𝑔𝑦𝑖 Negative 𝑊𝑒𝑥𝑡 = ∆𝑈𝑔 sign 𝑦𝑓 < 𝑦𝑖 Work as gravitational potential energy is lost (negative value since 𝑦𝑓 < 𝑦𝑖 ) to fall objects in the same direction of Earth’s gravity. 19 Energy of a system  Potential Energy of a System ❑ Gravitational potential energy depends only on the vertical height of the object above the surface of the Earth. ❑ The same amount of work must be done on an object–Earth system whether the object is lifted vertically from the Earth or is pushed horizontally at the same height. where there is no term involving x in the final result because 𝑗.Ƹ 𝑖Ƹ = 0. 20 Energy of a system  Potential Energy of a System Example ❑ It doesn’t matter how the raise was done. ❑ The potential energy of the ball is the same at the top in all three cases. 21 Energy of a system  Example for Potential Energy of a System A book of 2 kg mass slips from a man’s hands at 1.4 m and drops on his foot (at 0.05 m from floor). Choosing floor level as the y = 0 point of your coordinate system. Estimate the change in gravitational potential energy of the book–Earth system as the book falls. Sol. Before the book is released When the book reaches the foot The change in gravitational potential energy 22 Energy of a system  Conservative and Nonconservative Forces The sum of the kinetic and potential energies of a system is the mechanical energy of the system. 𝑬𝒎𝒆𝒄𝒉 = 𝑲 + 𝑼 Conservative forces conserve the mechanical energy 𝑬𝒎𝒆𝒄𝒉 = 𝑲 + 𝑼 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 Nonconservative forces do NOT conserve the mechanical energy 𝑬𝒎𝒆𝒄𝒉 = 𝑲 + 𝑼 ≠ 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 23 Energy of a system  Conservative and Nonconservative Forces Conservative forces have these two equivalent properties: 1. The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle. 2. The work done by a conservative force on a particle moving through any closed path is zero. (A closed path is one for which the beginning point and the endpoint are identical.) 24 Energy of a system  Conservative and Nonconservative Forces Friction force is NOT conservative Gravitational force is a force conservative force (transform some energy into internal (no dependency on the path) energy – ME is lost and not conserved) 25 Energy of a system  Example for Conservative and Nonconservative Forces A 4 kg particle moves from the origin to position C, having coordinates x = 5 m and y= 5m. One force on the particle is the gravitational force acting in the negative y direction. Calculate the work done by the gravitational force on the particle as it goes from O to C along (a) the purple path, (b) the red path, and (c) the blue path. (d) Your results should all be identical. Why? Sol. 26 Energy of a system  THANK YOU 27 Energy of a system