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LESSON 3: WORK, ENERGY AND POWER
Introduction This lesson is devoted to the very important concepts of work and energy. These two quantities are scalars and so have no direction associated with them, which often makes them easier to work with than vectors. In this lesson, and the next we discuss an alternative analysis of the translational motion of objects in terms of the quantities energy and momentum. The significance of energy is that they are conserved. That is, in quite general circumstances they remain constant. Learning Outcomes After successful completion of this lesson, you should be able to:
- Calculate the work done by an applied force that moves an object through a certain displacement.
- Distinguished between kinetic and potential energy.
- Use the principle of conservation of energy to solve problems that involves moving objects.
- Determine the power output of an energy source. Discussion 3.1 Work The term work , commonly used in connection with widely physical or mental activities, is restricted in physics, in cases wherein there is a force and a displacement along the direction of the force. In general, work is defined as the product of the displacement and the component of the force along the displacement. Work is a scalar quantity. If the force and the displacement are in the same direction, as shown in the diagram above, then the work done by the force in moving the body is given by ,
If the force and the displacement are not in the same direction, as shown in the diagram below, then the work done by the applied force is given by NOTE There will be no work done if the displacement and the applied force are at right angles with each other, since cos90^0 = 0. If the applied force does work by moving the body in the direction of the force, the work done is positive. If the body moves in the opposite direction of the force, the work is done by the body and is negative. In General, There Are Three Ways On How Work Is Done. ✓ If the force is just to impart uniform motion on the body, the force of friction has done the same amount of work. ✓ In changing the position or configuration of the body system, as in the case of force applied on a body to raise the body on an inclined plane. ✓ In imparting acceleration to the body or system. Conversion: 1 Joule = 107 Ergs 1 foot-pound = 1.356 Joules Sample Problems with Solutions:
- A wooden box is being pulled 10 m from its original position along a horizontal surface by a constant force of 25 N. Calculate the work done on the box if a) the force is applied horizontally, and b) the force makes an angle of 370 above the horizontal
3.2.1 Potential Energy An object may store energy because of its position. The energy that is stored is called Potential Energy (PE), because in the stored state, it has the potential to do work. 3.2.1.1 Gravitational Potential Energy Work is required to lift objects against the earth’s gravity. The potential energy due to elevated positions is called Gravitational Potential Energy. The amount of gravitational potential energy possessed by an elevated object is equal to the work done against gravity on lifting it. If a mass m is raised from position 1 to position 2, a distance h , as shown in the diagram on the right, work is done on the body against gravity with the magnitude,
W = -mgh
Where: mg is the force and the negative sign signifies a force against gravity. If the body is allowed to fall, the weight of the body will do the same amount of work,
W = mgh
which in another way, is called the Potential Energy Of The Body. In other words, energy was stored in the body by virtue of its position relative to the surface. Therefore,
PE = mgh
Since : weight w = mg , PE = wh
Consider now the work done in dragging a body of mass m along a frictionless inclined plane, as shown. Since the component of the vertical force, (the weight = mg ) along the plane is ( mg sin), the work done against this component of the weight along the plane of length L is, Note: That the height h is the distance above some reference level, such as the ground or the floor of a building. The potential energy or the work done on a body raised to a height is independent of the path, or course, taken by the body. The potential energy is relative to some reference level and depends only on mg and the height h****. The potential energy of a body at high altitude with respect to the surface of the earth is given by Where: Universal Gravitational constant , M is the mass of the earth, m is the mass of the body, R is the radius of the earth, and r is the distance of the body from the center of the earth. Note: that r is not just the altitude above the earth’s surface but includes the radius of the earth as well.
3.2.3 Transformation and Conservation of Energy Energy is given to a body or system of bodies when work is done upon it. In this process, there is merely a transfer of energy from one body to another. “ In such transfer no energy is created nor destroyed: it merely changes from one form to another. “ This statement is known as the law of conservation of energy. An example of the law of the conservation of energy is the conservation of mechanical energy (potential and kinetic) in the case of a simple pendulum of mass m. If the pendulum is raised to a height h , it acquires potential energy. When it reaches the lowest point of the arc, its potential energy is minimum, but its velocity is maximum showing that the potential energy of the pendulum has been converted to kinetic energy. This conservation is 100% ; friction at the point of support and air resistance is neglected. The Kinetic Energy at the lowest point will carry the pendulum to the same height in the other side of the swing. The law of conservation of energy still holds even if friction and air resistance are taken into account, because in that case, when the body eventually stops swinging after some time, both its potential and kinetic energies, by then, will all have been dissipated into Heat Energy. Sample Problems with Solutions:
- Calculate the kinetic energy in joules of an 11.0 g rifle bullet travelling at 250 m/s. Solution:
- A 40-lb stone is hoisted to the top of a building 100 ft high. By how much does its potential energy increases?
- A body of mass m is thrown vertically upward with a velocity of 25 m/s. a) How high will it rise? b) What is its velocity at a height of 20 m?
- A volcanic ash flow is moving across horizontal ground when it encounters a 10^0 up slope. It is observed to travel 920 m on the upslope before coming to rest. The volcanic ash contains trapped gas, so the force of friction with the ground is very small and can be ignored. At what speed was the ash flow moving just before encountering the up slope? Consider an arbitrary mass m of the ash flow and see how it moves.
In the MKS system, the unit of power is in joule per second , also known as the Watt , named after James Watt. In the CGS system, the unit of power is in erg per second. In the English system, the standard unit of power is the Horsepower (Hp). Sample Problems with Solutions:
- How much power is expended by a man who can push a load with a force of 190 lbs to a distance of 100 ft in 4 min? Solution:
- An engine is needed to pump 10,000 gallons of water per hour into a reservoir 100 ft above the level ground. How many horsepower is required?
- Water is pumped from a river with a depth of 80 m to a reservoir on the surface at a rate of. What is the minimum power in watts required in pumping the water up? Solution: Lesson 3: Work, Energy and Power Assessment Instruction; Show your complete and neat solution. Identify your final answer in the solution. See answer sheet format.
- A cord is used to lower vertically a block of mass M a distance d at a constant downward acceleration of g/4. (a) Find the work done by the cord on the block. (b) Find the work done by the work of gravity.
- A child pulls a 5.6 kg box a distance of 12 m along a horizontal surface at a constant speed. What work does the child do on the box if the coefficient of kinetic friction is 0.20 and the cord makes an angle of 45^0 with the horizontal?
- To push a 25 kg crate up a 27^0 incline, a worker exerts a force of 120 N, parallel to the incline. As the crate slides 3.6 m, how much work is done on the crate by: (a) the worker, (b) the force of gravity, and (c) the normal force due to the incline?
- A projectile with a mass of 2.4 kg is fired from a cliff 125 m high with an initial velocity of 150 m/s, directed 41.0º above the horizontal. What are (a) the kinetic energy of the projectile just after firing and (b) its potential energy? (c) Use conservation of energy to find the speed of the projectile just before it strikes the ground.
- A 220-lb man jumps out a window into a fire net 36 ft below. The net stretches 4.4 ft before bringing him to rest and tossing him back into the air. What is the potential energy of the stretched net if no energy is dissipated?
- A body of mass m starts down from the top of an inclined plane 20 ft long and 10 ft high. What is its velocity at a point 12 ft from the top if coefficient of friction is 0.1?
