Work and Energy. October 2012

Objectives. To define work by relating force and displacement. To calculate the net work done when many forces are applied to an object. To identify several forms of mechanical energy. To distinguish between kinetic and potential energy and calculate them. To solve problems using conservation of energy.

Work done by a constant force. In Physics, work is the meaning of what is accomplished when a force acts on an object, and the object moves through a distance. Work is the product of the magnitude of the displacement times the component of the force parallel to the displacement. W = 𝑭 || *d Force Mariano Rivera exerts a force on the ball over a displacement until he releases the ball. The total work done on the ball will be equal to the kinetic energy acquired by the ball. Displacement

Work is a scalar quantity, it has only magnitude which can be positive or negative. For example if you pull a loaded box a distance of 50 m by exerting a horizontal force of 30 N you do 30 N x 50 m = 1500 N*m of work on the box. A special name is given to this unit (N*m), the joule.

Example: Work on a backpack Find the work a hiker must do on a 15.0 kg backpack to carry it up a hill of height h=10.0 m as shown in the figure. For simplicity, assume the motion is smooth and at a constant velocity. Also find the work done by gravity on the backpack and the net work done on the backpack. Applying Newton’s laws ΣF=m 𝑎 𝑦 in the y axis we have: 𝐹 𝐻−𝑚𝑔=0 Hence, 𝐹 𝐻=𝑚𝑔 (15kg)(9.8m/ 𝑠 2 ) = 147N

Now we find the work done on the backpack by the hiker. 𝑊 𝐻 = 𝐹 𝐻 𝑑 𝑐𝑜𝑠𝜃 And we note from the figure that d cosθ = h so the work done by the hiker is: 𝑊 𝐻 =147𝑁 10𝑚 =1470 𝐽 We can conclude that the work done depends only on the change on elevation and not on the angle of the hill. The hiker will do the same work to lift the pack vertically the same height. The work done on the back pack by gravity is: 𝑊 𝐺 = 𝐹 𝐺 𝑑 cos⁡(180−𝜃) Since Cos(180-θ) =-Cosθ then: 𝑊 𝐺 = 𝐹 𝐺 𝑑 −𝑐𝑜𝑠𝜃 =−𝑚𝑔ℎ=−1470 𝐽 And finally the net work done on the backpack is 1470 J -1470 J = 0

Group Discussion: Does the Earth do work on the moon? In small groups discuss for a pair of minutes. The Moon revolves around the Earth in a nearly circular orbit, kept there by the gravitational force exerted by the Earth. Does gravity do positive work, negative work, or no work on the moon?

The gravitational force exerted by Earth on the Moon acts towards the Earth. The Moon displacement at any moment is tangent to the circle, in the direction of its velocity, perpendicular to the radius and thus perpendicular to the force of gravity. (90̊ angle) Cos90̊=0 So the work done on the Moon by Earth’s gravity is zero. This is why the Moon, as well as artificial satellites, can stay in orbit without the expenditure of fuel: No net work needs to be done against the force of gravity.

Work done by a varying force If the force acting on the object is not constant we can determine the work done graphically. The procedure is like that for determining displacement when the velocity is known as a function of time. To determine the work done by a variable force we plot the parallel component of the force to the direction of the motion as a function of distance. We divide the distance into small segments and find the area of the rectangles formed. As the value of the segments approaches zero the total area of the many rectangles approaches the area under the curve.

Are You Brave Enough? Will you release the ball and let it swing back to you?

Kinetic Energy Energy is one of the most important concepts in science. Energy as “the ability to do work” is a valid definition for mechanical energy. A moving object can do work on another object it strikes. A flying cannonball does work on a brick wall it knocks down, a hammer does work on a nail it drives through the wood. In either case, a moving object exerts a force on the second one and moves it.

The energy of motion is called kinetic energy from the Greek kinetikos meaning motion. Using Newton’s second law equation, and the equations for accelerated movement we get the following formula for net work done on an object: 𝑊 𝑛𝑒𝑡 = 1 2 m 𝑣 2 2 − 1 2 m 𝑣 2 1 We define the quantity 1 2 m 𝑣 2 as the translational kinetic energy (KE) 𝑲𝑬= 𝟏 𝟐 𝐦 𝒗 𝟐

Work – Energy Principle The net work done on an object is equal to the change in the object’s kinetic energy. So: 𝑾 𝒏𝒆𝒕 = 𝟏 𝟐 𝐦 𝒗 𝟐 𝟐 − 𝟏 𝟐 𝐦 𝒗 𝟐 𝟏 = 𝑲𝑬 𝟐 − 𝑲𝑬 𝟏 =∆𝑲𝑬 This means that if the net work on an object is positive, the kinetic energy increases. If it is negative, its kinetic energy decreases. Because of its direct connection, Energy is measured in the same units as Work. Joules for the SI units.

Potential Energy In addition to kinetic energy, it is also possible to have Potential Energy, which is the energy associated with forces that depend on the position or configuration of the object relative to its surroundings. Various types of potential energy exist, each associated with a particular kind of force. Note that there is a single universal formula for translational kinetic energy of an object, but there is no single formula for potential energy because it depends on the forces involved.

Gravitational Potential Energy Perhaps the most common example of potential energy. A heavy brick held high in the air has potential energy because of its position relative to Earth. It has the ability to do work, for if it’s released it will fall to the ground due to the gravitational force. It is defined as: 𝑷𝑬 𝑮𝒓𝒂𝒗 =𝒎𝒈𝒉

Activity: Pendulum of Peril With a polystyrene ball attached to a cord create a pendulum and attach it to the ceiling. Stand close to a wall holding the ball as shown on the picture and release it without pushing it. Will the ball come back and hit you? What kind of energies can you observe on the exercise? If energy cannot be created, explain why the ball did not hit you back.

Elastic Potential Energy Another sort of potential energy is associated with elastic materials. For example a spring has potential energy when compressed or stretched for it can then move an object when released. Elastic Potential Energy can be expressed as: Elastic PE = 1 2 𝑘𝑥 2 Where k is the spring stiffness constant and x the distance the spring is stretched or compressed.

Activity: Slinky Drop Watch the following video and try answering the question presented. www.youtube.com/watch?v=wGIZKETKKdw&feature=related Then perform the experiment with a real slinky. After discussing your observations watch the following videos. www.youtube.com/watch?v=QSoZlomFXlo&feature=related www.youtube.com/watch?v=oKb2tCtpvNU&feature=related

Conservative & Non-Conservative Forces Forces such as gravity, for which the work done does not depend on the path taken but only on the initial and final positions, are called Conservative Forces. Friction on the other hand, is a Non-conservative force since the work it does depends on the path.

Work-Energy Principle (Extended) Several forces act on an object which can undergo a translational motion. Suppose only some of this forces are conservative. Then the Net Work done can be defined as: 𝑊 𝑁𝑒𝑡 = 𝑊 𝐶 + 𝑊 𝑁𝐶 Work from conservative forces can be written as potential energy. And with this the Work-Energy Principle can be expressed as: 𝑊 𝑁𝐶 =∆𝐾𝐸+∆𝑃𝐸 Thus the work done by non conservative forces acting on an object is equal to the total change in kinetic and potential energies. www.youtube.com/watch?v=Jnj8mc04r9E&feature=related

Mechanical Energy We can define E, the total mechanical energy of a system as the sum of potential and kinetic energies at any moment. E = KE+PE If only conservative forces are acting, the total mechanical energy of a system neither increases nor decreases in any process. It stays constant, it is conserved. This is called the PRINCIPLE OF CONSERVATION OF MECHANICAL ENERGY.

Example: Falling Rock If the original height of the rock in the figure is 3.0 m, calculate the speed when it has fallen to 1.0 m above the ground. 1 2 𝑚 𝑣 2 1 +𝑚𝑔 𝑦 1 = 1 2 𝑚 𝑣 2 2 +𝑚𝑔 𝑦 2 In this case m cancel out. And 𝑉 1 =0. So solving for the final 𝑉 2 we find that: 𝑉 2 =6.3 m/s

Example: Toy Dart Gun A dart of mass 0.100 kg is pressed against a spring of a toy dart gun. The spring has a stiffness constant k=250 N/m and is compressed 6.0 cm and released. If the dart detaches from the spring when the spring reaches its natural length (x=0), what speed does the dart acquire? 0+ 1 2 𝑘 𝑥 2 1 = 1 2 𝑚 𝑣 2 2 +0 𝑉 2 =3.0 𝑚/𝑠

Law of conservation of Energy The total energy is neither increased nor decreased in any process. Energy can be transformed from one form to another, and transferred from one object to another, but the total amount remains constant. From this we can conclude that work is done when energy is transferred from one object to another.

Power Power is defined as the rate at which work is done. Average power equals the work done divided by the time needed to do it. We can define it also as the rate at which energy is transformed. 𝑃=𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑝𝑜𝑤𝑒𝑟= 𝑊𝑜𝑟𝑘 𝑇𝑖𝑚𝑒 = 𝐸𝑛𝑒𝑟𝑔𝑦 𝑡𝑟𝑎𝑛𝑠𝑓𝑜𝑟𝑚𝑒𝑑 𝑇𝑖𝑚𝑒 In SI power is measured in Joules per second and this special unit is called watt (W) Often a larger unit is used for practical purposes like motor engines’ power. The horsepower (hp) and is equivalent to 746 W.

Example: Stair climbing A 60.0 kg jogger runs up a set of stairs in 4 seconds. The vertical height of the stairs is 4.5m. Estimate the jogger power output in watts. 𝑃= 𝑊 𝑡 = 𝑚𝑔𝑦 𝑡 =661.5𝑊

Efficiency Efficiency in a system can be defined as: 𝐏 𝐨𝐮𝐭 𝐏 𝐢𝐧 =𝐞 𝐏 𝐨𝐮𝐭 𝐏 𝐢𝐧 =𝐞 It can never be greater than 1.0 because no engine can create energy, and can not transform one type of energy into another without some energy going to friction, thermal energy and other non useful forms of energy. For example in car engines efficiency is close to15% with the rest wasted as thermal energy and friction between parts.

References Giancoli , Douglas C. Physics Sixth Edition. USA Pearson 2005 Serway, Raymond A. Essentials of College Physics. USA Thomson 2007