Work and Energy

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Work and Power

Work is done whenever energy is transferred. Usually you need a force to move something because you're having to overcome another force. The thing being moved has kinetic energy while it's moving. The kinetic energy is transferred to another form of energy when the movement stops. Work is the amount of energy transferred from one place to another when a force causes a movement of some sort. Work = Force x Distance. Work done (W) = force causing motion (F) x distance moved (s), where W is measured in joules (J), F is measured in newtons (N) and s is measured in metres (m). Work is the energy that's been changed from one form to another it's not necessarily the total energy. The distance has to be measured in metres. The force F will be a fixed value in any calculations, because it's constant or because it's the average force. The equation assumes the direction of force is the same as the direction of movement. The equation gives you the definition of the joule: One joule is the work done when a force of 1 newton moves an object through a distance of 1 metre. The force isn't always in the same direction as the movement. For these question you simply need to split the force into horizontal and vertical components, where the upward force is not causing any motion. This gives you W=Fs cos theta, where theta is the angle between the direction of the force and the direction of motion. Power = Work Done per Second. Power is the rate of doing work, or the amount of energy transformed from one form to another per second. Power (P) = work done (W)/time (t), where P is measured in watts (W), W is measured in joules (J) and t is measured in seconds (s). The watt is defined as a rate of energy transfer equal to 1 joule per second. Power is also Force x Velocity (P=Fv). You knowP =W/t, and you know that W =Fs, which gives P=Fs/t. But v=s/t, which you can substitute into the above equation to give P=Fv. It's easier to use this if speed is given in the question. This equation is a shortcut between power and speed. If the force and motion are in different direction, replace F with F cos theta to get: P=Fv cos theta.

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Conservation of Energy

Learn the principle of conservation of energy. The principle of conservation of energy says that: Energy cannot be created or destroyed, but can be transferred from one form to another, but the total amount of energy stays the same in a closed system. You need it for questions about kinetic and potential energy. Kinetic energy is energy of anything moving, which you work out from E=0.5x m x v squared, with v as velocity and m is it's mass. There are different types of potential energy. Gravitational potential energy is the energy something gains if you lift it up. You work it out using E=mgh, where m is the mass of the object, h is the height it is lifted and g is the gravitational field strength. Elastic potential energy is the energy you get in a stretched object like a spring. You work this out using E=0.5 x k x e squared, where e is the extension of the spring and k is the stiffness constant. Throwing a ball upwards would convert kinetic energy into gravitational potential energy. As it comes down, the opposite happens. By ignoring friction you can say that the sum of the kinetic and potential energies is constant. Use conservation of energy to solve problems. Change in potential energy = change in kinetic energy. The classic example is the simple pendulum. In a simple pendulum you assume all the mass is in the bob at the end. If the bob is at an angle you work out the height difference by using trig. Then use mgh for gravitational potential energy. Once you have this you can work out the maximum speed bu assuming no air resistance so mgh=0.5xmxv squared. You can either rearrange to find v=square root of 2 x the energy divided by the mass. Or you can cancel the m's in both equations to give v squared=2gh and solve for v. You could be asked to apply this stuff to almost any situation in the exam. Rollercoasters are a favorite.

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Efficiency and Sankey Diagrams

All energy transfers involve losses. Energy can never be created or destroyed, but can be lost as other forms of energy, often in a form you can't use. Most often this lost energy is heat. No device (except possibly heaters) are ever 100% efficient - because some energy is lost as heat. Energy can be lost in other forms like sound - the important thing is it's not in a useful form you can use. Efficiency is the ratio of useful energy output to total energy input. Efficiency is a measure of how well a device converts the energy you put in into the energy you want it to give out. Efficiency = useful output energy/total input energy x 100. Some questions may give you the useful output energy - others will tell you how much is wasted. In this case simply do total input energy - wasted energy. Sankey diagrams show energy input and output. Sankey diagrams are a way of showing how much of the input energy is being usefully employed compared with how much is being wasted. The input energy is shown on the left-hand side. The width of a square on a diagram represents a  certain amount of energy. The wasted output energy is shown at the bottom. It's split into different types, so you can see exactly how it's wasted. The thicker the arrow, the greater the energy. They are useful to see exactly where the energy is going. You can use these diagrams to work out efficiency, by reading off the total input energy and useful output energy. Drawing sankey diagrams is easy - if you take it step by step. You could be asked to interpret a sankey diagram or draw one. Find the total input energy, the useful output energy and the amount of energy wasted in each form. You may have to add or subtract to get the missing values. Choose your scale. It's easiest to represent all the energy values by a whole number of squares. You want your diagram to be a sensible size. Work out how many squares represent each energy value using energy divided by energy per square = number of sqaures. Draw the arrow for total input energy - make sure it's the right number of squares. Split the input energy into all the output ones, with useful energy at the top. Draw the output arrows useful energy points across, other energy points downwards. Label all the arrows to show what each bit represents.

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Hooke's Law

Hooke's Law says that extension is proportional to force. If a metal wire is supported at the top and then a weight attached to the bottom, it stretches. The weight pulls down with force F, producing an equal and opposite force at the support. The extension of a stretched wire, e, is proportional to the load or force F. This is called Hooke's Law. Hooke's LAw can be written as F=ke. Where k is a constant that depends on the materal being stretched. K is the stiffness constant. Hooke's Law also applies to springs. The extension or compression of a spring is proportional to the force applied - so Hooke's Law applies. For springs, K in the formula is the spring constant. Hooke's Law works for compressive and tensile forces, but not for all materials. Hooke's Law stops working when the load is great enough. There's a limit to the force you can aplly before Hooke's Law no longer applies. On a force against extension graph, a straight line shows Hooke's Law is being obeyed. When the load becomes great enough, the graph starts to curve. E is the elastic limit at which beyond it the material will be permanently stretched i.e when the force is removed it will be longer than at the start. Metals obey Hooke's Law up to the limit of proportionality, which is very near the elastic limit. Some materials like rubber only obey Hooke's Law for really small extensions. A stretch can be elastic or plastic. If a deformation is elastic, the material returns to its original shape once the forces are removed. The atoms in a material are pulled apart under a load. Atoms can move small distances relative to their equilibrium positions, without changing position in the material. Once the load is removed, the atoms return to their equilibrium position. For a metal elastic deformation happens as long as Hooke's Law is obeyed. If a deformation is plastic the material is permanently stretched. Some atoms in the material move position relative to each other. When the load is removed, the atoms don't return to their original positions.

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Stress and Strain

A stress causes a strain. Tensile stress is defined as the force applied , F, divided by the cross-sectional area, A. Stress = F/A, measured in N per m squared or Pascals. Tensile strain is defined as the change in length (the extension), divied by the original length of the material. Strain = e/l. Strain has no units - it's a number. The same equations apply if the forces are tensile or compressive - but tensile forces are positive, and compressive as negative. A stress big enough to break the material is called the breaking stress. The greater the tensile force applied to a material, the stress on it increases. The effect of the stress is to pull the atoms apart from one another. Eventually the stress becomes great enough to completely seperate the atoms in the material - it breaks. The stress at which this occurs is called the breaking stress. The point marked UTS on the graph is the ultimate tensile stress. This is the maximum stress the material can withstand. Elastic strain energy is the energy stored in a stretched material. When a material is stretched work has to be done in stretching the material. Before the elastic limit, all the work done is stored as potential energy in the material. The stored energy is called the elastic strain energy. On a graph of force against extension this is just the area under a graph. You can calulate the energy stored in a stretched wire. Provided a material obeys Hooke's Law, the potential energy stored inside it can be calculated quite easily. Work done = force x displacement. The force on the material is not constant, so to calculate work done, you use the average force. Work done = 0.5xFxe. Then the elastic strain energy, E, is: E=0.5xFxe. Because Hooke's Law is being obeyed, F=ke, so you can substitute that into the above equation for E to give E=0.5xkxe squared. If the material is stretched beyond the elastic limit, some work is done seperating atoms. This will not be stored as strain energy and so isn't available when the force is released.

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The Young Modulus

The Young modulus is stress/strain. When you apply a load to stretch a material, it experiences a tensile stress and a tensile strain. Up to a point called the limit of proportionality, the stress and strain are constant. So below this limit, for a particular material, stress divided by strain is constant. This constant is the Young modulus. E=tensile stress/tensile strain = F/a divided by e/l or Fl/eA. Where F is force in N, A is cross-sectional area in metres squared, l is intial length and e extension in m. The units for the Young modulus are the same as stress , since strain has no units. The Young modulus is used by engineers to make sure their materials can withstand sufficient forces. To find the Young Modulus, you need a very long wire. You have a wire fixed at one end, hung over a pulley at the other end of the bench, with a marker on a rule with mm markings, and you hang weights on the end of the wire to stretch the wire - measure the extension and the fixed end of the wire and the marker, and find out the cross-sectional area (by finding out diameter and using pi x r squared) and use F/A to find stress, and use e/l for strain. The longer and thinner the wire the more it extends for the same force. Increase the weight recording the marker reading each time - the extension is the difference between this reading and the unstretched length. Use a stress-strain graph to find E. You can plot a graph of stress against strain from your results. Area under graph = strain energy per unit volume. E=stress/strain = gradient. The gradient of the graph gives the Young modulus, E. The area under the graph gives the strain energy per unit volume i.e the energy stored in 1 metre cubed of wire. The stress-strain graph is a straight line provided that Hooke's Law is obeyed, so you can calulate the energy per unit volume as energy=0.5 x stress x strain.

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Interpreting Stress-Strain Graphs

Stress-strain graphs for ductile materials curve. Ductile means they keep their shape when drawn into wires or other shapes and keep their strength when deformed. Before point P, the graph is a straight line through the origin. This shows the material is obeying Hooke's Law. Point P is the limit of proportionality. After this, the graph starts to curve. At this point, the material stops obeying Hooke's Law, but would still return to its original shape if the stress was removed. Point E is the elastic limit - at this point the material starts to behave plastically. From this point the material will not return to its original shape when the stress is removed. Point Y is the yield point - here the material starts to stretch without any extra load. The yield point is the stress at which a large amount of plastic deformation takes place with a constant or reduced load. Stress-strain graphs for brittle materials don't curve. The graph starts the same as one for a ductile material, a straight line through the origin. So it obeys Hooke's Law. However with a large enough stress the material snaps - it doesn't deform plastically. Brittle fracture is where tiny cracks at the material's surface get bigger and bigger until the material breaks completely. Rubber and polythebe are polymeric materials. The molecules that make up polymeric materials are arranged in long chains. They have a range of properties, so the stress-strain graphs will be different. For rubber, the graph curves up and down, but when the stress is removed will return to its original shape. It behaves elastically. For polythene, the graph is straight until the yield point, and then undergoes plastic deformation with the same load - the graph is a horizontal line. Then when the stress is removed the graph is a steep straight line down to zero stress. It behaves plastically, so it is a ductile material.

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