Work, Energy, and Power - Unit 3

Work, Energy, and Power - Unit 3 When a force displaces an object, it does work on that object. Your calculator works using the amount of force that acts parallel to an object’s displacement: in other words, only force that contributes to displacement actually does any work. If there is a constant force acting only in the direction of displacement, then the work done by a force F when it displaces something d meters is: 𝑊 = 𝐹𝑑 Work W is in units of joules (1 J = 1 N/m). However: what happens when force isn’t only acting in the direction of displacement? Take this diagram, where red is the applied force and blue is the displacement. How do you calculate the work that the red force does? You need to consider only the amount of force acting in the direction of displacement. If we have force F and displacement r (these are vector quantities), then we get the work done by force F with a dot product. 𝑊 =𝐹⋅𝑟 However, if we only know the scalar magnitudes of these quantities along with the angle between them (which is theta here), then we can use the cosine definition of the dot product. 𝑊 = 𝐹𝑟𝑐𝑜𝑠 (𝜃 ) Work When Force Isn’t Constant Force isn’t always constant, though. What if force varies over displacement, time or something else? We need calculus to figure out how that would work. If we imagine taking the sum of all of the infinitesimal bits of force and displacement to find infinitesimal bits of work, we know that we will need an integral. The work done on an object by a force is calculated with the following integral: 𝑥2 𝑊=∫ 𝐹 (𝑟) ⋅ 𝑑𝑟 𝑥1 If you still want to think in terms of “areas under graphs”, then work is the area under a force vs. position graph. Work and Energy The net work done on a point mass is equal to the change in kinetic energy. 𝑊𝑛𝑒𝑡 = 𝛥𝐾 This is the work-energy theorem. You can model objects or systems in this way when objects can be described as point-like particles that don’t deform or otherwise internally store energy. Kinetic Energy Definition The definition of kinetic energy is: 1 𝐾 = 𝑚𝑣 2 2 Note: net work is the sum of all of the individual work provided by each of the forces acting on an object. Conservative Forces A force is a conservative force if the work done on objects by the force depends only on the initial and final positions of the objects. Therefore: the (net) work done by a conservative force is zero if the object takes a path that goes back to its original starting position. Forces that don’t conform to this definition are known as dissipative forces. Forces like friction, for example, are generally dissipative. Potential Energy with Conservative Forces Conservative forces that are internal to a system relate to the potential energy of the system through an integral relationship. 𝑏 𝛥𝑈 = − ∫ 𝑎 𝐹𝑐𝑜𝑛𝑠 ⋅ 𝑑𝑟 You can also use the one-dimensional differential relationship. 𝑑𝑈(𝑥 ) 𝐹𝑥 = − 𝑑𝑥 Now, let’s examine some common conservative forces and how they relate to potential energy. Elastic Potential Energy (Springs) An ideal spring that acts on an object exerts a conservative force within a system of the object and the spring. Recall Hooke’s Law, which gives us the general relationship between spring compression/stretch and the force they exert, given a particular spring constant. 𝐹𝑠 = −𝑘𝛥𝑥 Using the integral relationship on the previous page, we can find the potential energy in an ideal spring by integrating the force function. 1 𝑈𝑠 = 𝑘(𝛥𝑥 )2 2 Note that this is for linear springs. You may encounter springs that don’t follow this law on the exam. For example, I once had an exam question with a spring that had a different compressionforce relationship. 𝐹𝑠 = −𝑘𝛥𝑥 2 However, the process for solving the problem was the same as with a linear spring: I just had to integrate this new relationship to find the potential energy in relation to the compression. 𝑈𝑠 = − ∫ − 𝑘𝛥𝑥 2 𝑑𝑥 1 𝑈𝑠 = 𝑘𝛥𝑥 3 3 Gravitational Potential Energy (near-Earth) Recall that when you are close to Earth’s surface and your elevation change is small relative to Earth’s radius, force from gravity is constant. 𝐹𝑔 = 𝑚𝑔 Gravitational potential energy near Earth’s surface in the system of the Earth and the object in question is therefore: 𝑈𝑔 = 𝑚𝑔ℎ And the change in gravitational potential energy is: 𝛥 𝑈𝑔 = 𝑚𝑔𝛥ℎ Gravitational Potential Energy (non-trivial distances) When the distance between, say, Earth and a satellite are non-negligible in comparison to the radius of the Earth, we cannot approximate the force from gravity as constant anymore. We must use the actual force from gravity. 𝐹𝐺 = 𝐺𝑚1 𝑚2 𝑟2 Using this equation, we can get the potential energy of an object-Earth system. 𝑈𝐺 = − 𝐺𝑚1 𝑚2 𝑟 The potential energy of this system is zero at an infinite distance between the object and Earth. The Conservation of Energy This isn’t about saving energy, no no. This is about the idea that the energy in a system can change from one type of energy to another, without changing the total amount of energy. If the only forces acting on a system are internal to the system, and no external work is being done, then the total change in mechanical energy of the system MUST be ZERO. This is something called a conservative system. In essence: the sum of any potential energy and any kinetic energy at any time must ALWAYS be the same. For example, if we had a system of the Earth, a mass and a spring that the mass is resting on, then the sum of all the mechanical energy in the system would be 𝐸 = 𝑈𝑔 + 𝑈𝑠 + 𝐾 No matter what happens, this sum will be constant. If, for example, the object is oscillating on the spring, compressing the spring a little, then whatever potential energy that would be gained from the compression is matched by a loss in kinetic energy and gravitational potential energy. However, not everything is lost when external forces act on the system: if any nonconservative forces act on the system, then the work they do will change the total amount of energy in the system. 𝑊𝑛𝑜𝑛𝑐𝑜𝑛𝑠 = 𝛥𝐸 You can use this to your advantage when solving problems, where you may be able to still apply conservation of energy, if you account for any external, nonconservative work. Power Power is the rate of change of the energy in a system or object. It can be defined in 2 ways. Firstly, the differential relation: 𝑃= 𝑑𝐸 𝑑𝑡 There is also another relationship. Power is the rate of change of energy in a system, and work is the change in energy of a system. Therefore, the average power of a force is the amount of work it does divided by the time it takes. 𝑃= 𝑊 𝐹𝑑 ⟶𝑃= 𝑡 𝑡 𝑑 = 𝑣 ⟶ 𝑃 = 𝐹𝑣 𝑡 It also can be expressed with vector quantities: 𝑃 =𝐹⋅𝑣