Physics
Energy in Pendulum
In a pendulum, energy is constantly changing between potential energy (due to its height) and kinetic energy (due to its motion). At the highest point, the pendulum has maximum potential energy and minimum kinetic energy, while at the lowest point, it has maximum kinetic energy and minimum potential energy. This continuous exchange of energy allows the pendulum to swing back and forth.
Written by Perlego with AI-assistance
Related key terms
1 of 5
8 Key excerpts on "Energy in Pendulum"
eBook - PDFApplied Mathematics
Made Simple
- Patrick Murphy(Author)
- 2014(Publication Date)
- Butterworth-Heinemann(Publisher)
We can therefore summarize these last two results by what is called the principle of the conservation of energy: for any body moving under the force of gravity alone the sum of the potential and kinetic energies is constant. An equation such as $mv 2 + mgy = mu 2 above is called the energy equation for the body in motion. Since energy is defined as the capacity for doing work, any form of pent-up or sealed-in energy may be used to perform work; the potential energy due to the height and weight of a body is just one of many ways in which energy may be stored. For example, a watch spring when fully wound has a store of energy or potential energy which is used up in turning the hands to register the time. The old grandfather clocks have large masses suspended by cords which gradually unwind so that the work done by the weight as it descends is used to turn the hands of the clock. When the weight has descended to the bottom of the clock we wind it back up again to replace its potential energy. A piece of elastic also has potential energy when it is held in a stretched state. Thus a catapult releases the energy of the stretched elastic and in so doing imparts a velocity to a stone, i.e. the potential energy of the elastic is converted into kinetic energy for the stone. There are other forms of energy such as heat, chemical, and electrical energy. One very familiar example is the chemical action in a battery which can be converted to electrical energy to start the engine of a car. Considering the principle of the conservation of energy in the widest sense, we suggest that although energy can be converted from one form to another it can never be destroyed. This means that the sum of all the different energies in a system remains constant. To illustrate this point consider the energy stored in a gallon of petrol.
eBook - PDF- William Moebs, Samuel J. Ling, Jeff Sanny(Authors)
- 2016(Publication Date)
- Openstax(Publisher)
Figure 8.9 Bar graphs representing the total energy (E), potential energy (U), and kinetic energy (K) of the particle in different positions. (a) The total energy of the system equals the potential energy and the kinetic energy is zero, which is found at the highest point the particle reaches. (b) The particle is midway between the highest and lowest point, so the kinetic energy plus potential energy bar graphs equal the total energy. (c) The particle is at the lowest point of the swing, so the kinetic energy bar graph is the highest and equal to the total energy of the system. 374 Chapter 8 | Potential Energy and Conservation of Energy This OpenStax book is available for free at http://cnx.org/content/col12031/1.5 8.7 Check Your Understanding How high above the bottom of its arc is the particle in the simple pendulum above, when its speed is 0.81 m/s? Example 8.8 Air Resistance on a Falling Object A helicopter is hovering at an altitude of 1 km when a panel from its underside breaks loose and plummets to the ground (Figure 8.10). The mass of the panel is 15 kg, and it hits the ground with a speed of 45 m/s . How much mechanical energy was dissipated by air resistance during the panel’s descent? Figure 8.10 A helicopter loses a panel that falls until it reaches terminal velocity of 45 m/s. How much did air resistance contribute to the dissipation of energy in this problem? Strategy Step 1: Here only one body is being investigated. Step 2: Gravitational force is acting on the panel, as well as air resistance, which is stated in the problem. Step 3: Gravitational force is conservative; however, the non-conservative force of air resistance does negative work on the falling panel, so we can use the conservation of mechanical energy, in the form expressed by Equation 8.12, to find the energy dissipated. This energy is the magnitude of the work: ΔE diss = | W nc,if | = | Δ(K + U) if | . Step 4: The initial kinetic energy, at y i = 1 km, is zero.
No longer available |Learn morePhysics for Scientists and Engineers
Foundations and Connections, Extended Version with Modern Physics
- Debora Katz(Author)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
3. Kinetic energy K describes the motion of a system. It depends both on the mass and speed of the system: K 5 1 2 mv 2 (8.1) 4. Potential energy depends on the configuration of the system. For a system that consists of two particles, the change in potential energy is given by DU 5 U f 2 U i 5 2 3 x f x i F x dx (8.3) when one particle moves from x i to x f . ▲ Special Cases 1. Potential energy a. Gravity near surface of the Earth: Gravitational potential energy of the system is given by U g 1 y 2 5 mgy (8.5) where the reference configuration is at y 5 0. b. For a system that consists of two particles 1 and 2 with the reference configuration chosen such that lim r S` U 5 0, the universal gravitational potential energy is U G 1r 2 5 2G m 1 m 2 r (8.7) c. If the origin of a coordinate system is at the re- laxed position of a spring, which is set to the reference configuration, the elastic potential en- ergy of the particle–spring system is given by U s 5 1 2 kx 2 (8.9) 2. Conservation of mechanical energy and orbital motion a. For a circular orbit, there is a special relationship between the mechanical energy, the potential en- ergy, and the kinetic energy of the system, with E 5 1 2 U G (Eq. 8.21) and E 5 2K . b. For an elliptical orbit, the mechanical energy is E 5 2 1 2 GMm a (8.22) Tools 1. A bar chart helps visualize the conservation of me- chanical energy. The vertical height of the bar repre- sents the energy. A horizontal line is drawn to repre- sent the zero point of the energy. If the energy is positive, the bar is drawn above that horizontal line, and if the energy is negative, the bar is below the line. 2. Another pictorial representation is an energy graph, which displays a potential energy curve and a hori- zontal E line representing the constant mechanical energy. The kinetic energy is not displayed, but is represented by the space between the horizontal E line and the potential energy curve.
eBook - PDFThe Engineering Dynamics Course Companion, Part 1
ParticlesKinematics and Kinetics
- Edward Diehl(Author)
- 2022(Publication Date)
- Springer(Publisher)
We can see from the above 136 8. WORK-ENERGY METHOD AND THE CONSERVATION OF ENERGY (PART 1) δ 1 F 2 F 1 l 0 δ 2 0 2 1 Figure 8.7: Spring deflection is independent of direction. l y 3 y 1 2 3 1 v r θ θ Datum Figure 8.8: Pendulum demonstrating conservation of energy. equation that conservation of energy is a trade off between kinetic energy and potential energy just like the apple in Figure 8.1. Consequently, kinetic energy is at a maximum when potential energy is at a minimum and vice versa. A pendulum is an excellent example of this concept, as shown in Figure 8.8. In position 1, the ball is released from rest. Before it moves it has no kinetic energy, only potential. When it reaches the bottom, position 2, it is at its maximum velocity and therefore maximum kinetic energy but has reached the datum so has zero potential energy. When it rises to position 3, it again momentarily stops moving (no kinetic energy) but has regained the potential energy it had in position 1. This depends on there being no external losses such as air resistance 8.6. SOLVING WORK-ENERGY METHOD PROBLEMS 137 on the ball or friction at the pivot point. In equations we’d write this as: PE 1 D KE 2 D PE 3 mgy 1 D 1 2 mv 2 2 D mgy 3 : We’ll note that the masses will cancel which is an interesting aspect of pendula (their motion is independent of mass). We also conclude that the ball will return to the original height, y 1 D y 3 , as long as there are no energy losses. 8.6 SOLVING WORK-ENERGY METHOD PROBLEMS Applying the Work-Energy Method typically involves some typical steps. 1. If there is friction in the problem, an FBD/IBD should be drawn and the normal forces calculated in order to find the friction force. 2. If there is dependent motion in the problem, the relations (position changes and velocity) should be calculated. This step and step 1 can be performed later as an aside, but if we recognize they’re necessary it’s good to get them out of the way.
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- Vern Ostdiek, Donald Bord(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
At the lowest point in the arc, the child has only kinetic energy, which equals the original potential energy. The child then swings upward and converts the kinetic energy back into potential energy. This continues until the swing stops at a point nearly level with the starting point. This process is repeated over and over as the child swings. Air resistance takes away some of the kinetic energy. If the child is pushed each time, the work done puts energy into the system and counteracts the effect of the air resis- tance. Without air resistance or any other friction, the child would not have to be pushed each time and would continue swinging indefinitely. The maximum height that a pendulum reaches (at the turning points) depends on its total energy. The more energy a pendulum has, the higher the turning points. In Section 3.2 we described a way to measure the speed of a bullet or a thrown object (Figure 3.6). The law of conservation of linear momentum is used to relate the speed of the bullet before the collision to the speed of the block (and bullet) afterward. If the wood block is hanging from a string, the kinetic energy it gets from the impact causes it to swing up like a pendulum. The more energy it gets, the higher it will swing. We can determine the speed of the block after impact by measuring how high the block swings. PE PE KE KE v v d 1 2 3 4 Figure 3.30 As a child swings back and forth, gravitational potential energy is continually converted into kinetic energy and back again. The potential energy at the highest (turning) points equals the kinetic energy at the lowest point. Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300 3.5 The Conservation of Energyuni00A0 105 The potential energy of the block (and bullet) at the high point of the swing equals the kinetic energy of the block (and bullet) right after impact.
- Joseph C. Amato, Enrique J. Galvez(Authors)
- 2015(Publication Date)
- CRC Press(Publisher)
the body’s position . Force is a vector, whereas energy is a scalar; scalars are easier to work with than vectors, and often the energy approach is easier, or the only tractable way to study a physical system. s 346 Physics from Planet Earth - An Introduction to Mechanics 9.9 POTENTIAL ENERGY AND NONSINUSOIDAL OSCILLATIONS When the restoring force is nonlinear, the potential energy is not a parabolic function, and the motion of the body is nonsinusoidal. Solving Newton’s second law to find the position vs. time of the oscillating body may then be difficult or impossible, but with a potential energy diagram we can still learn a great deal about the motion. We can find the equilibrium point, the turning points, and the speed of the body as a function of position. We can also show that all oscillating systems undergo simple harmonic motion in the limit of small amplitudes. This is an exceedingly important result, because it allows us to derive the frequency of small oscillations in physical systems that are much more complicated and important than our ideal mass-spring system. To illustrate this important point, let’s take a closer look at the simple pendulum, a ball (mass m ) attached to a string (length ℓ ). See Figure 9.12a . The angular deflection of the string is φ , so the height of the ball above its lowest (equilibrium) position at φ = 0 is y = − C ( cos ) 1 ϕ . Choosing U ( φ ) = 0 at φ = 0, the potential energy is U ( φ ) = mgy = mgl (1 − cos φ ), which is decidedly nonparabolic in form ( Figure 9.12b ). Since U ( φ ) is a continuous function at φ = 0, it can be approximated using a polynomial or “power series” expression: U f c c c c c n o n n ( ) ( ) ϕ ϕ ϕ ϕ ϕ ϕ $ = + + + + + 1 2 2 3 3 (9.15) where the coefficients c 0 , c 1 , c 2 ,…, c n are constants to be determined. The polynomial f n ( φ ) is called a Taylor series expansion about the point φ = 0.
No longer available |Learn more- (Author)
- 2014(Publication Date)
- Library Press(Publisher)
________________________ WORLD TECHNOLOGIES ________________________ Chapter- 2 Kinetic and Internal Energy Kinetic Energy The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. When they start rising, the kinetic energy begins to be converted to gravitational potential energy. The sum of kinetic and potential energy in the system remains constant, ignoring losses to friction. ________________________ WORLD TECHNOLOGIES ________________________ The kinetic energy of an object is the energy which it possesses due to its motion. It is defined as the work needed to accelerate a body of the given mass from rest to its current velocity . Having gained this energy during its acceleration, the body maintains this kinetic energy unless its speed changes. The same amount of work would be done by the body in decelerating from its current speed to a state of rest. The speed, and thus the kinetic energy of a single object is completely frame-dependent (relative): it can take any non-negative value, by choosing a suitable inertial frame of reference. For example, a bullet racing past an observer has kinetic energy in the reference frame of this observer, but the same bullet is stationary, and so has zero kinetic energy, from the point of view of an observer moving with the same velocity as the bullet. By contrast, the total kinetic energy of a system of objects cannot be reduced to zero by a suitable choice of the inertial reference frame, unless all the objects have the same velocity. In any other case the total kinetic energy has a non-zero minimum, as no inertial reference frame can be chosen in which all the objects are stationery. This minimum kinetic energy contributes to the system's invariant mass, which is independent of the reference frame. According to classical mechanics (ie ignoring relativistic effects) the kinetic energy of a non-rotating object of mass m traveling at a speed v is mv 2 /2.
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
