Lecture 11

Work, Kinetic Energy, and Work-Energy Theorem Introduction to the Concept of Energy The concept of energy is very important in physics, because energy is a conserved quantity. We can take advantage of the fact that energy is conserved to find easy solutions to otherwise complex problems of mechanics. Furthermore, energy is easy to use because it is a scalar quantity, that is a number, not a vector. Although energy is conserved, it can be transformed from one form to another, for example from chemical energy to mechanical energy in our muscles, or from kinetic energy of falling water into mechanical energy by a turbine and then into electrical energy by a generator, in a hydropower plant. Energy is also always partly wasted, due to drag forces and friction forces that tend to disperse kinetic energy into heat and sound (other forms of energy), especially when transforming one type of energy into another for practical use (the combustion engine of your car turns the chemical energy of the oil into mechanical energy, but some of the heat generated by the combustion is wasted). Therefore, when we want to use energy conservation to solve problems in mechanics, we must be very careful to account for all possible ways in which energy can be transformed or may be wasted. Over the next five lectures we will learn how to use the concept of energy accounting for its various forms. Work In order to use the concept of energy in mechanics, we must find a way to relate this concept to that of force, because forces are the cause of the motion we study. The connection between forces and energy is provided by the concept of work. IMPORTANT: By work, we always mean the work done by a force on an object. Meaning that work alone is meaningless, we must always refer to a force that does the work, and to an object on which the work is done by that force. You can think of the work as the energy given to an object by a force. Because we have seen that a force is always an interaction of two objects (Newton’s third law), then when we talk of work done by a force on an object we know there must be a second object imparting that force. So you could also think of the work as the energy used by that second object to impart that force. And in summary, we can think of the work as the energy exchanged by one object (imparting that force) to anther object (on which the force acts).

Definition for constant force in the same direction as the displacement: The work W done by a force F on an object is given by the product of the magnitude of the force and the magnitude of the displacement: W = F Δx The work, and the energy, are measured in Joule. Since the work is the product of a force and a displacement, the Joule must be the product of Newton and meter: Joule = N m = kg m2/s2 This definition of work is not general enough, because the force and the displacement may point in different directions. The general definition, still assuming the force is constant, is the following: The work is the dot (or scalar) product of the force and the displacement: W = F ⋅Δx = F Δx cos(θ) where θ is the angle between the two vectors F and Δx. The dot product of one vector times the other is the length (magnitude) of the projection on one vector on the other (hence the cosine). It is a scalar quantity, a number, not another vector. Examples: Positive work: Zero work: Negative work: F F F θ<90° θ=90° θ>90° Δx Δx Δx When we walk holding a heavy suitcase we therefore do zero work on the suitcase. On the other hand, if we walk upstairs or uphill holding the same suitcase we do some work on the suitcase (it is important to specify on which object the work is done). As explained above, we can see the work has energy going from one object to another, because the work is done by a force, and the force is always the interaction of two objects. In other words, work is always done by something to something else. For example we do work on the suitcase. If we walk on flat terrain holding the suitcase we are not increasing the energy of the suitcase, so we do not transfer any energy to the

suitcase (zero work). But if we walk uphill we are taking the suitcase to a higher level, increasing its gravitational potential energy (we will discuss this form of energy later this week), so we have transferred energy to the suitcase with a positive work (the energy that the suitcase could acquire by falling from the higher level under the force of gravity). Positive work done on an object means that the force increases the energy of that object, negative work means that the energy of the object is decreased. For example, if we walk downstairs holding the suitcase, we do negative work on the suitcase, based on the scalar product of force and displacement. In fact, by going to a lower level, we reduce the gravitational potential energy of the suitcase (this will be explained better later on). Friction Forces and Work Since the work is done by a force on an object, friction forces also do work. Of course the static friction does zero work, because the object does not move, but the kinetic friction certainly does a non-zero work, because the object is moving. Remember that the friction force is always in the opposite direction as the motion, and therefore the work done by friction is always negative. Negative work means that the object subject to the friction force is losing energy. And in fact that is precisely what happens: The object moving without friction keeps a constant velocity, while the object subject to friction will decelerate and eventually will stop, so it has clearly lost energy (due to the negative work done by friction). Since the energy is conserved, where did it go? We said that work is the transfer of energy from an object to another, so where did the energy of the object motion go? It went primarily into heat, because friction causes heat on the surfaces. Part of that heat may go into the object itself, part on the surface the object was skidding through. Since the friction also causes some sound, part of the energy has also gone into sound, which is mechanical energy of waves in the air around the object. Kinetic Energy Remember that we said that energy is useful because it is conserved, and we can take advantage of that in problems, and we introduced the work to connect energy and forces. However, the work is the energy transferred from one object to another via a force. But what is the energy of an object? We will now introduce the concept of kinetic energy of an object, and we will justify its definition based on the concept of work in the following way: Let’s consider a constant net force (some of all forces) acting on an object. We will call it the force, but keep in mind it could be the sum of many forces. Let’s assume the displacement is in the same direction, so W = F Δx. Using Newton’s second law we know the acceleration must also be constant and we can write:

W = F Δx = m a Δx We have also learned the following formula related velocity and displacement in the case of constant acceleration: v2 = v0

2 + 2 a Δx which implies: a Δx = ( v2 - v0

2 ) / 2 There fore the work of the net force on the object is: W = m ( v2 - v0

2 ) / 2 ⇒ W = m v2 / 2 – m v02 / 2

So we can see that the work done on the object is equal to the variation of the quantity (m v2 / 2). But since we said that the work is the energy given to the object, the variation of that quantity must be the variation of the energy of the object, and so the quantity (m v2 / 2) can be considered as the energy of the object. Since it is an energy associated to the motion of the object, we call it kinetic energy. Definition: The kinetic energy KE of an object of mass m moving with speed v is defined by: KE = m v2 / 2 As the work, the energy is measured in Joule. Work-Energy Theorem As we said above, the work plays two role: i) it connects forces to energy; ii) it gives the amount of energy transferred between two objects. From the way we have justified the definition of kinetic energy, it is now totally clear that the work done on an object gives the change in energy of the object. For now, since we have defined only one type of energy, the kinetic energy, we will introduce the following simple work-energy theorem: The net work done on an object is equal to the change in the object’s kinetic energy: Wnet = KEf – KEi = ΔKE where by net work we mean the work done by the net force (the sum of all forces acting on that body).

Application to road accidents: How much further would a car go before stopping if the speed at the moment of braking is doubled, from v to 2v? Solution: The car stops eventually because of the force of friction, while it skids with the wheels locked by the breaks. In order to stop the car, the friction force must do a work on the car equal to the initial kinetic energy of the car (by initial we mean the moment when the breaking started). Since the work done by friction if Ffriciton Δx, and the initial kinetic energy is m v2 / 2, then we can write: Ffriciton Δx = m v2 / 2 ⇒ Δx = m v2 / ( 2 Ffriciton ) If the force of friction is assumed to be constant (independent of velocity), and since the mass of the car is constant, then the distance traveled after breaking is proportional to the squared of the velocity. In summary, if you double your speed, your car will go four times further before stopping. This relation allows one to estimate the speed of the car before breaking, from the length of the skid marks (all you need to know is the approximate mass of the car and the coefficient of kinetic friction, to compute the friction force), which is very useful for reconstructing what happened in a road accident, for example.