Potential Energy and Conservation of Energy Chapter 8 Potential Energy and Conservation of Energy

Potential Energy Potential energy is energy determined by the configuration of a system in which the components of the system interact by forces. The forces are internal to the system. Can be associated with only specific types of forces acting between members of a system. If the arrangement of the system changes, then the potential energy of the system changes Section 7.6 Prof Dr Ahmet ATAÇ

Gravitational Potential Energy As an object falls toward the Earth, the Earth exerts a gravitational force mg on the object, with the direction of the force being the same as the direction of the object’s motion. For example; The system is the Earth and the book. Do work on the book by lifting it slowly through a vertical displacement. The work done on the system must appear as an increase in the energy of the system. The energy storage mechanism is called potential energy. Section 7.6 Prof Dr Ahmet ATAÇ

Gravitational Potential Energy, cont Assume the book in fig. is allowed to fall. There is no change in kinetic energy since the book starts and ends at rest. Gravitational potential energy is the energy associated with an object at a given location above the surface of the Earth. Section 7.6 Prof Dr Ahmet ATAÇ

Gravitational Potential Energy, final The quantity mgy is identified as the gravitational potential energy, Ug. Ug = mgy Units are joules (J) Is a scalar Work may change the gravitational potential energy of the system. Wext = Dug Potential energy is always associated with a system of two or more interacting objects. Section 7.6 Prof Dr Ahmet ATAÇ

Gravitational Potential Energy, Problem Solving The gravitational potential energy depends only on the vertical height of the object above Earth’s surface. In solving problems, you must choose a reference configuration for which the gravitational potential energy is set equal to some reference value, normally zero. The choice is arbitrary because you normally need the difference in potential energy, which is independent of the choice of reference configuration. Often having the object on the surface of the Earth is a convenient zero gravitational potential energy configuration. The problem may suggest a convenient configuration to use. Section 7.6 Prof Dr Ahmet ATAÇ

Elastic Potential Energy Elastic Potential Energy is associated with a spring. The force the spring exerts (on a block, for example) is Fs = - kx The work done by an external applied force on a spring-block system is W = ½ kxf2 – ½ kxi2 The work is equal to the difference between the initial and final values of an expression related to the configuration of the system. Section 7.6 Prof Dr Ahmet ATAÇ

Elastic Potential Energy, cont. This expression is the elastic potential energy: Us = ½ kx2 The elastic potential energy can be thought of as the energy stored in the deformed spring. The stored potential energy can be converted into kinetic energy. Observe the effects of different amounts of compression of the spring. Section 7.6 Prof Dr Ahmet ATAÇ

Elastic Potential Energy, final The elastic potential energy stored in a spring is zero whenever the spring is not deformed (U = 0 when x = 0). The energy is stored in the spring only when the spring is stretched or compressed. The elastic potential energy is a maximum when the spring has reached its maximum extension or compression. The elastic potential energy is always positive. x2 will always be positive. Section 7.6 Prof Dr Ahmet ATAÇ

Prof Dr Ahmet ATAÇ

Conservatıve and Nonconservatıve Forces Conservative Forces The work done by a conservative force on a particle moving between any two points is independent of the path taken by the particle. The work done by a conservative force on a particle moving through any closed path is zero. A closed path is one in which the beginning and ending points are the same. Examples of conservative forces: Gravity Spring force Prof Dr Ahmet ATAÇ

Conservative Forces, cont We can associate a potential energy for a system with any conservative force acting between members of the system. This can be done only for conservative forces. In general: Wint = - DU Wint is used as a reminder that the work is done by one member of the system on another member and is internal to the system. Positive work done by an outside agent on a system causes an increase in the potential energy of the system. Work done on a component of a system by a conservative force internal to an isolated system causes a decrease in the potential energy of the system. Section 7.7 Prof Dr Ahmet ATAÇ

Non-conservative Forces A non-conservative force does not satisfy the conditions of conservative forces. Non-conservative forces acting in a system cause a change in the mechanical energy of the system. Emech = K + U K includes the kinetic energy of all moving members of the system. U includes all types of potential energy in the system. Section 7.7 Prof Dr Ahmet ATAÇ

Non-conservative Forces, cont. The work done against friction is greater along the brown path than along the blue path. Because the work done depends on the path, friction is a non-conservative force. Section 7.7 Prof Dr Ahmet ATAÇ

Example 1 A 40.0-N child is in a swing that is attached to ropes 2.00 m long. Find the gravitational potential energy of the child – Earth system relative to the child’s lowest position when (a) the ropes are horizontal, (b) the ropes make a 30.0° angle with the vertical, and (c) the child is at the bottom of the circular arc Prof Dr Ahmet ATAÇ

Conservative Forces and Potential Energy Define a potential energy function, U, such that the work done by a conservative force equals the decrease in the potential energy of the system. The work done by such a force, F, is DU is negative when F and x are in the same direction. Section 7.8 Prof Dr Ahmet ATAÇ

Conservative Forces and Potential Energy The conservative force is related to the potential energy function through. The x component of a conservative force acting on an object within a system equals the negative of the potential energy of the system with respect to x. Can be extended to three dimensions Section 7.8 Prof Dr Ahmet ATAÇ

Conservative Forces and Potential Energy – Check Look at the case of a deformed spring: Us=1/2kx2 This is Hooke’s Law and confirms the equation for U U is an important function because a conservative force can be derived from it. Prof Dr Ahmet ATAÇ

Solution 1 Prof Dr Ahmet ATAÇ

Example 2 A 4.00-kg particle moves from the origin to position C, which has coordinates x=5.00 m and y=5.00 m (Fig.). One force on it is the force of gravity acting in the negative y direction. Calculate the work done by gravity as the particle moves from O to C along (a) OAC, (b) OBC, and (c) OC. Your results should all be identical. Why? Prof Dr Ahmet ATAÇ

Solution 2 The results should all be the same, since gravitational forces are conservative. Prof Dr Ahmet ATAÇ

Conservatıon of Mechanıcal Energy An object in free fall, this principle tells us that any increase (or decrease) in potential energy is accompanied by an equal de crease (or increase) in kinetic energy. Note that the total mechanical energy of a system remains constant in any isolated system of objects that interact only through conservative forces. Total mechanical energy E of a system is defined as the sum of the kinetic and potential energies, we can write; E K+ U We can state the principle of conservation of energy Ei=Ef as and so we have Ki + Ui = Uf + Kf Prof Dr Ahmet ATAÇ

Example At time ti, the kinetic energy of a particle is 30.0 J and the potential energy of the system to which it belongs is 10.0 J. At some later time tf, the kinetic energy of the particle is 18.0 J. (a) If only conservative forces act on the particle, what are the potential energy and the total energy at time tf ? (b) If the potential energy of the system at time tf is 5.00 J, are there any non-conservative forces acting on the particle? Explain Prof Dr Ahmet ATAÇ

Solution Prof Dr Ahmet ATAÇ

Example A single conservative force acts on a 5.00-kg particle. The equation Fx=(2x+4) N, where x is in meters, describes this force. As the particle moves along the x axis from x=1.00m to x=5.00m, calculate (a) the work done by this force, (b) the change in the potential energy of the system, and (c) the kinetic energy of the particle at x=5.00m if its speed at x=1.00m is 3.00 m/s. Prof Dr Ahmet ATAÇ

Solution Prof Dr Ahmet ATAÇ

Work done by Nonconservatıve Forces if some of the forces acting on objects within the system are not conservative, then the mechanical energy of the system does not remain constant. There are two types of non-conservative forces: An applied force and The force of kinetic friction.

Work Done by an Applied Force The force you apply does work Wapp on the book, while the gravitational force does work Wg on the book. net work done on the book is related to the change in its kinetic energy as described by the work – kinetic energy theorem given by Equation; Wapp + Wg=K Because the gravitational force is conservative Wapp = K + U We conclude that if an object is part of a system, then an applied force can transfer energy into or out of the system.

Situations Involving Kinetic Friction As the book moves through a distance d, the only force that does work is the force of kinetic friction. This force causes a decrease in the kinetic energy of the book; Kfriction = fkd If the book moves on an incline that is not frictionless, a change in the gravitational potential energy of the book – Earth system also occurs, and –fkd is the amount by which the mechanical energy of the system changes because of the force of kinetic friction. In such cases, E = K + U= –fkd Ei + E = Ef

Example A 70.0-kg diver steps off a 10.0-m tower and drops straight down into the water. If he comes to rest 5.00 m beneath the surface of the water, determine the average resistance force exerted by the water on the diver.

Example The coefficient of friction between the 3.00-kg block and the surface in Figure is 0.400. The system starts from rest. What is the speed of the 5.00-kg ball when it has fallen 1.50 m?

Solution

Solutions

Energy Diagrams and Equilibrium Motion in a system can be observed in terms of a graph of its position and energy. In a spring-mass system example, the block oscillates between the turning points, x = ±xmax. The block will always accelerate back toward x = 0. Consider the potential energy function for a block – spring system, given by Us=1/2kx2 Prof Dr Ahmet ATAÇ Section 7.9

Energy Diagrams and Stable Equilibrium The x = 0 position is one of stable equilibrium. Any movement away from this position results in a force directed back toward x = 0. Configurations of stable equilibrium correspond to those for which U(x) is a minimum. x = xmax and x = -xmax are called the turning points. Section 7.9 Prof Dr Ahmet ATAÇ

Energy Diagrams and Unstable Equilibrium Fx = 0 at x = 0, so the particle is in equilibrium. For any other value of x, the particle moves away from the equilibrium position. This is an example of unstable equilibrium. Configurations of unstable equilibrium correspond to those for which U(x) is a maximum. Section 7.9 Prof Dr Ahmet ATAÇ

Potential Energy in Molecules There is potential energy associated with the force between two neutral atoms in a molecule which can be modeled by the Lennard-Jones function. Find the minimum of the function (take the derivative and set it equal to 0) to find the separation for stable equilibrium. The graph of the Lennard-Jones function shows the most likely separation between the atoms in the molecule (at minimum energy). Section 7.9 Prof Dr Ahmet ATAÇ