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Preparatory Program in Basic Science(PYSC001) PART (I) PYSICS(2) Coordinator: Prof.Dr.Hassan A.Mohammed

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UNIT II : MOMENTUM & ENERGY 3 - MOMENTUM & IMPULSE 3.1 Momentum :Impetus(im + petus):in motion Momentum is a vector quantity that reflects an object’s ability to do work or cause damage. This ability is directly proportional to both the object’s mass and velocity. Therefore, momentum is defined as: Momentum = mass × velocity, or, p = m × v The SI unit of momentum is: kg.m/s.

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Any object at rest has zero momentum. On the other hand, a moving object with either a large mass or large velocity has a large momentum. Examples: a supertanker (large mass), and a bullet (large velocity).

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3.2 Impulse: 3.2.1 Definition A force can be applied to an object only for a limited duration (∆t). The longer that ∆t is, the greater that the effect will be on the object (causing a change in its velocity and momentum). Therefore, we introduce impulse as the product of the applied force and the time interval during which the force acts: Impulse = force on object × time interval; or

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The adjacent graph shows the force’s behavior. From the above definition of impulse, impulse in this graph should be the area under the force curve. If we can determine the average impact force (F avg ), impulse would also be the area of the rectangle whose width is ∆t and height is F avg.

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3.2.2 Relationship of Impulse to Momentum: This means that the impulse on an object equals the change in its momentum.

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3.2.3 Important notes : 1. Whenever we exert a force on an object, we also exert an impulse on it. 2. When the force is not constant, we take its average to calculate the impulse. 3. We may say that an object has momentum, but may not say that it has impulse. 4. We may say that an impulse on an object causes a change in its momentum. Alternatively, we may say that a change in an object’s momentum causes an impulse.

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5. Bouncing results in a change of direction of v and p producing larger impulse and force a)If we drop two balls of equal mass, one made of play- dough and the other of highly elastic rubber, the first ball stops upon hitting the ground, producing an impulse ∆p = mv – 0 = mv. The second ball reverses direction, producing an impulse ∆p = mv – (– mv) = 2mv. b) Following the same principle, a karate expert breaks a stack of bricks by bouncing his hand off it.

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3.2.4 Car-Crash Example The figure shows a car whose driver lost control of the brakes while going at a constant speed, v. To stop the car, the driver must choose between three options: (a) Driving into a haystack, (b) Driving into a brick wall, or (c) Bouncing off a concrete wall. All options will cause the same change in momentum: I = ∆p = mv - 0 = mv.

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However, as is indicated in the figure, there are important differences between these options, as follows: a) The haystack option extends the impact time, which greatly decreases the impact force and reduces harm and damage. b) The brick-wall option shortens the impact time, which results in a very high impact force that would cause great harm and damage. c) The concrete-wall option is similar to the previous one, but with the added effect of bouncing. This doubles the change in momentum and, consequently, the impact force.

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3.3 Conservation of Momentum 3.3.1 Derivation From Newton’s 3rd law, we learned that the interaction forces between two objects (A and B) are given By: we can conclude from this that: The impulse from A on B cause a change in the momentum of object),B and vice versa. Thus, we have:

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Hence, the momentum of the A-B system is the same before and after the interaction. This is the law of “conservation of momentum : Momentum is never gained or lost in an interaction.

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3.3.2 Collisions 3.3.2.1 Definition Collision is a special kind of interaction in which the interacting objects come in contact with each other so as to exchange momentum and energy. A collision is distinguished by an impact that separates between what happens before and after the collision

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3.3.2.2 Types There are two types of collisions: a. Elastic Collisions in which the colliding objects are not permanently deformed and do not generate heat. Example:collision between the billiard balls in the figure. b. Inelastic Collisions in which the colliding objects become distorted and generate heat. This is usually associated with tangling, sticking, or coupling between the colliding objects. Example: collision between the two cars in the figure.

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3.3.2.3 Discussion 1. In a collision between two objects, momentum is exchanged. This exchange depends on the details of the collision, such as the velocity of the colliding objects, and whether the collision is elastic or not. 2. As was discussed earlier, momentum is conserved in a collision, which means that: before after. This is true for both elastic and inelastic collisions. 3. At impact, the impulses of the two colliding objects are equal and Opposite: which means that the momentum gained by one object equals the momentum lost by the other.

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3.3.3 Simple Examples 3.3.3.1 Zero Initial Speeds From momentum conservation, the total final momentum must be zero. This can only happen if the two objects move away along the same line(linear motion). Thus, we only need to consider scalar momenta and speeds. We have:

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As a specific example, consider a cannon of mass m c firing a cannonball of mass m b at a speed v b. To calculate the cannon’s recoil speed, v c, we substitute in the above equation: 3.3.3.2 Zero Final Speeds From momentum conservation, the total initial momentum must be zero. We have:

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3.3.3.3 Worked Exercise A 2-ton car going 40 km/h is hit at the rear by a 5-ton truck going 50 km/h.

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4- ENERGY, POWER & SIMPLE MACHINES: 4.1 Work, Energy, Power 4.1.1 Work: 4.1.1.1 Definition Work is the exertion of force through a distance. Work is a scalar quantity that is directly proportional to the applied force and to the distance the object moves because of the force. Thus, we say: Work = force × distance, or: The SI unit for work is the joule, defined as: [J ≡N.m].

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4.1.1.2 General Equation for Work The above equation is only true for linear motion: when the applied force and the object’s displacement are along the same line (θ = 0). If, on the other hand, the force applied to an object makes an angle θ with the object’s displacement, a more general equation for work is:

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4.1.2.1 Definition Energy is the capacity of an object to do work. Example: Muscles (energy source) enable creatures to move (work). On the other hand, doing work often results in stored energy. Example: Digesting food (work) produces (energy) for the body. Thus, we say that work and energy are interchangeable. Since energy and work are interchangeable, we use for energy measurements the same unit as for work, the joule[J]. 4.1.2 Energy

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4.1.2.2 Different Forms of Energy: Energy takes many forms: mechanical, chemical, electric, nuclear, etc. It is stored in plants, foods, batteries, and fuels. Specific examples: 1. Waves: All waves carry energy. Light and sound are two examples. Both light and sound waves carry energy that depends on the wave’s frequency and intensity. 2. Heat: Heat usually results from burning a substance. The amount of heat generated depends on the temperature, type, and amount of the substance.

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4.1.2.3 Observing and Using Energy: Examples: 1. A man can do work by exerting a force through a distance; but this requires food, so he converts the energy in food into work. 2. We burn coal to generate heat that can be converted into electricity and then into many modern forms of work:

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4.1.2.4 Conservation of Energy (and matter) A fundamental law of nature (as decreed by Allah) is: Thus, energy can only convert from one form to another. For a closed system, this law can be summarized as: Energy is conserved; it neither increases nor decreases.

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4.1.2.5 Common Energy Units: 1.A calorie is defined as the energy needed to raise the temperature of one gram of water by 1 ºC at normal temperature (18 ºC) and pressure (1 atmosphere). A calorie is related to the joule as: [1 cal = 4.184 J ≈4.2 J]. Although the calorie is not an SI unit, the SI permits using it in heat applications. 2. In food products, the energy available in a food item upon digestion is commonly expressed is the kilocalorie (sometimes written as Calorie, with a capital C), where [1 Cal = 1kcal = 1000 cal].

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3. A common unit of energy is the BTU (British thermal unit). 1 BTU is the energy needed to raise the temperature of 1 lb of water by 1 ºF (Fahrenheit). A BTU relates to the joule as: [1 BTU = 1.054 kJ]. 4. A common unit of energy is the kilowatt-hour (kWh), which is the standard unit of electricity consumption. It relates to the joule as: ( 1watt = 1J/s) 1 kWh = 10 3 J/s × 3600 s = 3.6×10 6 J or 3.6 MJ. Larger businesses and institutions sometimes use the megawatt-hour [MWh]. The energy outputs of large power plants over long periods of time, or the energy consumption of nations, can be expressed in gigawatt-hours [GWh]. 3. A common unit of energy is the BTU (British thermal unit). 1 BTU is the energy needed to raise the temperature of 1 lb of water by 1 ºF (Fahrenheit). A BTU relates to the joule as: [1 BTU = 1.054 kJ]. 4. A common unit of energy is the kilowatt-hour (kWh), which is the standard unit of electricity consumption. It relates to the joule as: ( 1watt = 1J/s) 1 kWh = 10 3 J/s × 3600 s = 3.6×10 6 J or 3.6 MJ. Larger businesses and institutions sometimes use the megawatt-hour [MWh]. The energy outputs of large power plants over long periods of time, or the energy consumption of nations, can be expressed in gigawatt-hours [GWh].

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4.1.3.2 Common Power Units: 1.A unit of power, commonly used in regard to motors, is the horsepower, which is defined as: 1 hp = 746 W ≈ 0.75 kW. 2. Another power unit, commonly used to describe an air conditioner’s cooling capacity, is the British Thermal Unit per hour, [BTU/h], where: 1 BTU/h = 0.293 W.

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4.2 Mechanical Energy: 4.2.1 Potential Energy 4.2.1.1 Definition: The work done on an object to change its position must equal the change in potential energy (by the law of conservation of energy). Therefore: Mechanical energy is the energy that arises from an object’s position or velocity, and is the sum of potential and kinetic energies. Potential energy Ep : is the energy that an object has because of its position under the influence of a certain force

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4.2.1.2 Forms of Potential Energy Potential energy can be mechanical or non-mechanical. Examples of mechanical potential energy: 1.Water trapped behind a dam has gravitational potential energy because of its height above the base of the dam. We can use this energy to run a hydroelectric station 2. A drawn bow has elastic potential energy stored in the bow and string. 3. A compressed or expanded spring has elastic potential energy.

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Examples of non-mechanical potential energy: 1. An electric circuit has electric potential energy stored in the battery or voltage source. 2. At the molecular level, chemical potential energy is stored in the relative position of atoms in molecules 3. There is potential energy in food, arising from molecular binding 4. Potential energy is stored in fuels, such as coal and natural gas. 5. Nuclear potential energy is stored in the atom’s nucleus.

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4.2.1.3 Gravitational Potential Energy To raise an object of mass (m) to height (h) requires work = force × distance = m·g·h. Once at that height, the object will possess a potential energy equal to the work done to place it at that location:

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4.2.1.4 Important Notes 1. Potential energy is relative, which means that its value depends on the point of reference. Example: If we raise a 1-kg book 1 m above a table that is 1 m above ground, the books potential energy relative to the table is 10 J, and is 20 J relative to the ground. 2. Potential energy can be positive or negative. Example: Consider the book in the previous example, and assume that the ground-ceiling (+J) distance is 3 m. This means that when we place the book 1 m above the table’s surface, its potential energy relative to the ceiling is -10 J. Thus, instead of doing work on the book to bring it down a distance of 1 m, the book will do the work (losing potential energy).

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3. An object’s potential energy at a certain location is the same regardless of how the object reaches that location. In the adjacent sketch, a man of mass (m) climbs to certain height (h) using three different routes. Regardless of the route, his potential energy at h will be the same in all three cases. 4. A more general form of gravitational potential energy between two objects of masses (m 1 ) and (m 2 ), separated by a distance (d), is given by: For Earth, M E = 5.98 × 10 24 kg, and R E = 6.37 × 10 6 m. Thus, we can use the above equation to calculate the value of g.

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4.2.2 Kinetic Energy: 4.2.2.1 Definition: An object’s energy of motion, called kinetic energy, E k depends on the object’s mass and speed. Your moving car has kinetic energy that keeps it going. Blowing wind has kinetic energy that can be converted into electric energy by means of windmills. Work is needed to change an object’s speed, and (by the law of conservation of energy ) this work must equal the kinetic energy gained by the object:

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4.2.2.2 Derivation: An object of mass m is subjected to a force (F) over a distance d during a time interval (t). It has acceleration (a), initial speed (v i ), and final speed (v f ). The work done on it to reach this speed must equal the kinetic energy it gained.

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4.2.2.3 Important Notes 1. Kinetic energy must always be either zero or positive. It cannot be negative. 2. Work can convert to potential energy which is then converted to kinetic energy, or vice versa. 3. Kinetic energy is directly proportional to mass. If a truck goes as fast as a car of half its mass, the truck’s kinetic energy is double the car’s. 4. Kinetic energy is proportional to the square of the speed, which means that if you double your car’s speed, you quadruple its kinetic energy. Since energy is the capacity to do work, for either good or bad, uncontrolled kinetic energy can be very dangerous. Furthermore, doubling the speed substantially increases fuel consumption

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4.2.3 Example of Mechanical Energy 4.2.3.1 Pile Driver If we let an object (like the pile driver fall, it starts gaining speed and losing height. This means that it gains kinetic energy and loses potential energy. By the conservation law, the potential energy it loses should equal the kinetic energy it gains. Thus, if it falls a distance (h) and gains speed (v), we have:

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4.2.3.2 Throwing a Ball If we throw a ball up with a speed vi, its speed becomes zero at the highest point it reaches, h. Using the same equations as above, we find that: h = v i 2 / 2g or v i =√ 2gh Example: The maximum height reached by a ball that is thrown up with v i = 30 m/s is: h = (900 m 2 / s 2 )/ 2 ×10 m/s 2 = 45 m.

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4.2.3.3 Bicycle up a Hill Assume that the bicycle in the figure is traveling over frictionless ground. At the bottom of the hill, all of its mechanical energy is kinetic. As it starts rising up the hill, it loses kinetic energy and gains potential energy (with the total energy remaining constant). Its potential energy is maximum at the peak, after which the bicycle starts regaining kinetic energy and losing potential energy.

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4.2.3.4 Pendulum An example similar to the previous one is the pendulum. Assume that we have a frictionless pendulum, with bob of mass (m) and string of length (l). At point (3), the pendulum’s total energy is all potential, whereas it is all kinetic at point (1). At any intermediate point (such as point (2)), the energy is a mixture of kinetic and potential energies. From the values in the figure, we have:

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