Physics
Elastic Potential Energy
Elastic potential energy is the energy stored in an elastic material when it is stretched or compressed. It is directly proportional to the amount of deformation in the material and can be calculated using the formula 1/2 kx^2, where k is the material's stiffness and x is the amount of deformation. When the material returns to its original shape, the stored energy is released.
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5 Key excerpts on "Elastic Potential Energy"
- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2023(Publication Date)
- Wiley(Publisher)
(8.1.9) Elastic Potential Energy Elastic Potential Energy is the energy associated with the state of compression or extension of an elastic object. For a spring that exerts a spring force F = −kx when its free end has displacement x, the Elastic Potential Energy is U(x) = 1 _ 2 kx 2 . (8.1.11) REVIEW & SUMMARY The reference configuration has the spring at its relaxed length, at which x = 0 and U = 0. Mechanical Energy The mechanical energy E mec of a system is the sum of its kinetic energy K and potential energy U: E mec = K + U. (8.2.1) An isolated system is one in which no external force causes energy changes. If only conservative forces do work within an isolated system, then the mechanical energy E mec of the system cannot change. This principle of conservation of mechanical energy is written as K 2 + U 2 = K 1 + U 1 , (8.2.6) in which the subscripts refer to different instants during an energy transfer process. This conservation principle can also be written as ∆E mec = ∆K + ∆U = 0. (8.2.7) Potential Energy Curves If we know the potential energy function U(x) for a system in which a one-dimensional force F(x) acts on a particle, we can find the force as F(x) = − dU(x) _ dx . (8.3.2) If U(x) is given on a graph, then at any value of x, the force F(x) is the negative of the slope of the curve there and the kinetic energy of the particle is given by K(x) = E mec − U(x), (8.3.4) where E mec is the mechanical energy of the system. A turning point is a point x at which the particle reverses its motion (there, K = 0). The particle is in equilibrium at points where the slope of the U(x) curve is zero (there, F(x) = 0). Work Done on a System by an External Force Work W is energy transferred to or from a system by means of an external force acting on the system. When more than one force acts on 200 CHAPTER 8 Potential Energy and Conservation of Energy a system, their net work is the transferred energy.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
(8.1.7) If the reference point of the particle is set as y i = 0 and the cor- responding gravitational potential energy of the system is set as U i = 0, then the gravitational potential energy U when the par- ticle is at any height y is U( y) = mgy. (8.1.9) Elastic Potential Energy Elastic Potential Energy is the energy associated with the state of compression or extension of an elastic object. For a spring that exerts a spring force F = −kx when its free end has displacement x, the Elastic Potential Energy is U(x) = 1 _ 2 kx 2 . (8.1.11) Additional examples, video, and practice available at WileyPLUS 210 CHAPTER 8 POTENTIAL ENERGY AND CONSERVATION OF ENERGY is not involved, the work done on the system and the change ∆E mec in the mechanical energy of the system are equal: W = ∆E mec = ∆K + ∆U. (8.4.1, 8.4.2) When a kinetic frictional force acts within the system, then the thermal energy E th of the system changes. (This energy is asso- ciated with the random motion of atoms and molecules in the system.) The work done on the system is then W = ∆E mec + ∆E th . (8.4.9) The change ∆E th is related to the magnitude f k of the frictional force and the magnitude d of the displacement caused by the external force by ∆E th = f k d. (8.4.7) Conservation of Energy The total energy E of a system (the sum of its mechanical energy and its internal energies, including thermal energy) can change only by amounts of energy that are transferred to or from the system. This experimental fact is known as the law of conservation of energy. If work W is done on the system, then W = ∆E = ∆E mec + ∆E th + ∆E int . (8.5.1) If the system is isolated (W = 0), this gives ∆E mec + ∆E th + ∆E int = 0 (8.5.2) and E mec,2 = E mec,1 − ∆E th − ∆E int , (8.5.3) where the subscripts 1 and 2 refer to two different instants. Power The power due to a force is the rate at which that force transfers energy.
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2020(Publication Date)
- Wiley(Publisher)
● If the reference point of the particle is set as y i = 0 and the corresponding gravitational potential energy of the system is set as U i = 0, then the gravitational potential energy U when the particle is at any height y is U( y) = mgy. ● Elastic Potential Energy is the energy associated with the state of compression or extension of an elastic object. For a spring that exerts a spring force F = −kx when its free end has displacement x, the Elastic Potential Energy is U(x) = 1 2 kx 2 . ● The reference configuration has the spring at its relaxed length, at which x = 0 and U = 0. Key Ideas 8.01 Distinguish a conservative force from a nonconser- vative force. 8.02 For a particle moving between two points, identify that the work done by a conservative force does not depend on which path the particle takes. 8.03 Calculate the gravitational potential energy of a par- ticle (or, more properly, a particle–Earth system). 8.04 Calculate the Elastic Potential Energy of a block– spring system. 150 This is a pretty formal definition of something that is actually familiar to you. An example might help better than the definition: A bungee-cord jumper plunges from a staging platform (Fig. 8-1). The system of objects consists of Earth and the jumper. The force between the objects is the gravitational force. The configuration of the system changes (the separation between the jumper and Earth decreases — that is, of course, the thrill of the jump). We can account for the jumper’s motion and increase in kinetic energy by defining a gravitational potential energy U. This is the energy associated with the state of separation between two objects that attract each other by the gravitational force, here the jumper and Earth. When the jumper begins to stretch the bungee cord near the end of the plunge, the system of objects consists of the cord and the jumper. The force between the objects is an elastic (spring-like) force. The configuration of the system changes (the cord stretches).
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
● If the reference point of the particle is set as y i = 0 and the corresponding gravitational potential energy of the system is set as U i = 0, then the gravitational potential energy U when the particle is at any height y is U( y) = mgy. ● Elastic Potential Energy is the energy associated with the state of compression or extension of an elastic object. For a spring that exerts a spring force F = −kx when its free end has displacement x, the Elastic Potential Energy is U(x) = 1 2 kx 2 . ● The reference configuration has the spring at its relaxed length, at which x = 0 and U = 0. Key Ideas 8.01 Distinguish a conservative force from a nonconser- vative force. 8.02 For a particle moving between two points, identify that the work done by a conservative force does not depend on which path the particle takes. 8.03 Calculate the gravitational potential energy of a par- ticle (or, more properly, a particle–Earth system). 8.04 Calculate the Elastic Potential Energy of a block– spring system. 177 This is a pretty formal definition of something that is actually familiar to you. An example might help better than the definition: A bungee-cord jumper plunges from a staging platform (Fig. 8-1). The system of objects consists of Earth and the jumper. The force between the objects is the gravitational force. The configuration of the system changes (the separation between the jumper and Earth decreases — that is, of course, the thrill of the jump). We can account for the jumper’s motion and increase in kinetic energy by defining a gravitational potential energy U. This is the energy associated with the state of separation between two objects that attract each other by the gravitational force, here the jumper and Earth. When the jumper begins to stretch the bungee cord near the end of the plunge, the system of objects consists of the cord and the jumper. The force between the objects is an elastic (spring-like) force. The configuration of the system changes (the cord stretches).
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
It’s important to remember that the work done by gravity and springs in any given physical system is already included on the right-hand side of Equation 5.18 as potential energy and should not also be included on the left as work. Figure 5.21c shows how the stored Elastic Potential Energy can be recovered. When the block is released, the spring snaps back to its original length, and the stored Elastic Potential Energy is converted to kinetic energy of the block. The elas- tic potential energy stored in the spring is zero when the spring is in the equilib- rium position (x 5 0). As given by Equation 5.17, potential energy is also stored in the spring when it’s stretched. Further, the Elastic Potential Energy is a maximum Spring potential energy c Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-200-202 5.5 | Spring Potential Energy 139 Unless otherwise noted, all content on this page is © Cengage Learning. when the spring has reached its maximum compression or extension. Finally, because PE s s is proportional to is proportional to x 2 , the potential energy is always positive when the spring is not in the equilibrium position. In the absence of nonconservative forces, W nc nc 5 0, so the left-hand side of Equation 5.18 is zero, and an extended form for conservation of mechanical energy results: (KE 1 PE g g 1 PE s s ) i 5 (KE 1 PE g g 1 PE s s ) f [5.19] Problems involving springs, gravity, and other forces are handled in exactly the same way as described in the problem-solving strategy for conservation of mechani- cal energy, except that the equilibrium point of any spring in the problem must be defined in addition to the zero point for gravitational potential energy. Quick Quiz 5.5 Calculate the Elastic Potential Energy of a spring with spring constant k 5 225 N/m that is (a) compressed and (b) stretched by 1.00 3 10 22 m.
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