W = F Δx or W = F Δx cosθ

Transcription

1 WORK AND ENERGY When a force acts upon an object to cause a displacement of the object, it is said that work was done upon the object. In order for a force to qualify as having done work on an object, there must be a displacement and the force must cause the displacement. Mathematically, work can be expressed by the following equation: W = F Δx or W = F Δx cosθ where F is the force, d is the displacement, and the angle (theta) is defined as the angle between the force and the displacement vector.to gather an idea of it's meaning, consider the following three scenarios. Scenario A: A force acts rightward upon an object as it is and the displacement vector are in the same direction. Thus, the angle between F and d is 0 degrees. Scenario B: A force acts leftward upon an object that is and the displacement vector are in the opposite direction. Thus, the angle between F and d is 180 degrees. Scenario C: A force acts upward on an object as it is and the displacement vector are at right angles to each other. Thus, the angle between F and d is 90 degrees. Units of Work The Joule is the unit of work. 1 Joule = 1 Newton * 1 meter 1 J = 1 N * m

2 Diagram A Answer: W = (100 N) * (5 m)* cos(0 degrees) = 500 J The force and the displacement are given in the problem statement. It is said (or shown or implied) that the force and the displacement are both rightward. Since F and d are in the same direction,the angle is 0 degrees. Diagram B Answer: W = (100 N) * (5 m) * cos(30 degrees) = 433 J The force and the displacement are given in theproblem statement. It is said that the displacement is rightward. It is shown that the force is 30 degrees above the horizontal. Thus, the angle between F and d is 30 degrees. Diagram C Answer: W = (147 N) * (5 m) * cos(0 degrees) = 735 J The displacement is given in the problem statement. The applied force must be 147 N since the 15-kg mass (F grav =147 N) is lifted at constant speed. Since F and d are in the same direction, the angle is 0 degrees. The energy acquired by the objects upon which work is done is known as mechanical energy. Mechanical energy is the energy that is possessed by an object due to its motion or due to its position. Mechanical energy can be either kinetic energy (energy of motion) or potential energy (stored energy of position). Any object that possesses mechanical energy - whether it is in the form of potential energy or kinetic energy - is able to do work. Kinetic energy is the energy of motion. An object that has motion - whether it is vertical or horizontal motion - has kinetic energy. There are many forms of kinetic energy - vibrational (the energy due to vibrational motion), rotational (the energy due to rotational motion), and translational (the energy due to motion from one location to another). The following equation is used to represent the kinetic energy (KE) of an object: KE = ½ m v 2 where m = mass in kg and v = speed

3 Kinetic energy is a scalar quantity; it does not have a direction. Like work, the standard metric unit of measurement for kinetic energy is the Joule. As might be implied by the above equation, 1 Joule is equivalent to 1 kg*m 2 /s 2. Potential energy is the stored energy of position possessed by an object. For example, the heavy ball of a demolition machine is storing energy when it is held at an elevated position. Gravitational Potential Energy (GPE or PE grav ) Gravitational potential energy is the energy stored in an object as the result of its vertical position or height. The energy is stored as the result of the gravitational attraction of the Earth for the object. There is a direct relation between gravitational potential energy and the mass of an object. More massive objects have greater gravitational potential energy. There is also a direct relation between gravitational potential energy and the height of an object, the higher that an object is elevated, the greater the gravitational potential energy. These relationships are expressed by the following equation: PE grav = m g h In the above equation, m represents the mass of the object, h represents the height of the object and g represents the gravitational field strength (9.81 N/kg on Earth) - sometimes referred to as the acceleration of gravity. To determine the gravitational potential energy of an object, a zero height position must first be arbitrarily assigned. Typically, the ground is considered to be a position of zero height. But this is merely an arbitrarily assigned position that most people agree upon. Elastic Potential Energy Elastic potential energy is the energy stored in elastic materials as the result of their stretching or compressing. Elastic potential energy can be stored in rubber bands, bungee chords, trampolines, springs, an arrow drawn into a bow, etc. The amount of elastic potential energy stored in such a device is related to the amount of stretch of the device - the more stretch, the more stored energy. Springs are a special instance of a device that can store elastic potential energy due to either compression or stretching. A force is required to compress a spring; the more compression there is, the more force that is required to compress it further. For certain springs, the amount of force is directly proportional to the amount of stretch or compression (x); the

4 constant of proportionality is known as the spring constant (k). F = k Δx Such springs are said to follow Hooke's Law. If a spring is not stretched or compressed, then there is no elastic potential energy stored in it. The spring is said to be at its equilibrium position. The equilibrium position is the position that the spring naturally assumes when there is no force applied to it. In terms of potential energy, the equilibrium position could be called the zero-potential energy position. There is a special equation for springs that relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant. The equation is PE spring = ½ k x 2 k = spring constant, x = amount of compression (deformation) from the zero position The slope of an F vs. x graph will give you the spring constant (k) and the area under the curve will give you the elastic energy. The Total Mechanical Energy As already mentioned, the mechanical energy of an object can be the result of its motion and/or the result of its stored energy of position. The total amount of mechanical energy is merely the sum of the potential energy and the kinetic energy. This sum is simply referred to as the total mechanical energy (abbreviated TME). TME = PE + KE

5 The diagram below depicts the motion of Li Ping Phar (esteemed Chinese ski jumper) as she glides down the hill and makes one of her record-setting jumps. The total mechanical energy of Li Ping Phar is the sum of the potential and kinetic energies. The two forms of energy sum up to Joules. Notice also that the total mechanical energy of Li Ping Phar is a constant value throughout her motion. There are conditions under which the total mechanical energy will be a constant value and conditions under which it will be a changing value. Power Power is the rate at which work is done. It is the work/time ratio. Mathematically, it is computed using the following equation: P = work time The standard metric unit of power is the Watt. As is implied by the equation for power, a unit of power is equivalent to a unit of work divided by a unit of time. Thus, a Watt is equivalent to a Joule/second. For historical reasons, the horsepower is occasionally used to describe the power delivered by a machine. One horsepower is equivalent to approximately 750 Watts.