Translational Kinetic Energy

7 Key excerpts on "Translational Kinetic Energy"

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  eBook - ePub

  MCAT Physics and Math Review 2024-2025

  Online + Book

  • (Author)
  • 2023(Publication Date)
  • Kaplan Test Prep

    (Publisher)

  This chapter reviews the fundamental concepts of energy and work. The work–energy theorem is a powerful expression of the relationship between energy and work that is often a simpler approach to kinematics questions on Test Day. Finally, we’ll discuss the topic of mechanical advantage, and we’ll examine how a pulley or ramp might be helpful in raising heavy objects. We hope to convince you throughout the Kaplan MCAT program that your preparation for Test Day is in no way a Sisyphean task.

  2.1 Energy

  LEARNING OBJECTIVES

  After Chapter 2.1, you will be able to:

  • Describe kinetic energy and potential energy
  • Compare and contrast conservative and nonconservative forces

  Energy refers to a system’s ability to do work or—more broadly—to make something happen. This broad definition helps us understand that different forms of energy have the capacity to perform different actions. For example, mechanical energy can cause objects to move or accelerate. An ice cube sitting on the kitchen counter at room temperature will absorb thermal energy through heat transfer and eventually melt into water, undergoing a phase transformation from solid to liquid. Nuclear binding energy can be released during fission reactions to run power plants. Let’s turn our attention to the different forms that energy can take. After that, we will discuss the two ways in which energy can be transferred from one system to another.

  Kinetic Energy

  Kinetic energy is the energy of motion. Objects that have mass and that are moving with some speed will have an associated amount of kinetic energy, calculated as follows:

  K =

  1 2

    m

  v 2

  Equation 2.1

  where K is kinetic energy, m is the mass in kilograms, and v is speed in meters per second. The SI unit for kinetic energy, as with all forms of energy, is the joule (J), which is equal to

  kg ⋅

  m 2

  s 2

  .

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  eBook - ePub

  The Handy Chemistry Answer Book

  • Justin P. Lomont, Ian C. Stewart(Authors)
  • 2013(Publication Date)
  • Visible Ink Press

    (Publisher)

  changes in potential energy in a meaningful way. A closed system can exchange potential energy for kinetic and vice versa, but the total energy must always remain constant. This is stated in the First Law of Thermodynamics, which we’ll get to soon.

  What is kinetic energy?

  Kinetic energy is the type of energy associated with the movement of an object. Faster-moving objects have more kinetic energy, and the kinetic energy of an object is related to its mass, m, and velocity, v, by the equation:

  E = ½mv2

  This tells us that, for example, if we have two objects of equal mass and one is moving twice as fast as the other, the faster-moving object will have four times the energy of the slower object.

  In this illustration, a horse pulls a pendulum into a position where it is about to be released to swing freely. Before it is released, the weight at the end of the pendulum has potential energy (A), and when the pendulum is in full swing, it has kinetic energy (B).

  Can molecules have any arbitrary energy?

  No, molecules actually have a discrete number of possible energy levels. Another way to say this is that their energies are quantized. To illustrate why this is so different from situations we’re used to in everyday life, consider what happens when you’re throwing a baseball. You could throw it at any speed between 0 meters per second (m/s) and however fast you are capable of throwing it. In molecules, though, only a discrete set of energies are possible. It’s as if you could throw the baseball either 2 m/s or 40 m/s, but not 20 m/s or any other speed in between. There aren’t many situations we encounter in everyday life in which the possible energies associated with objects come in a discrete set of values.

  What types of energy levels exist in molecules?

  There are three main types of energy levels that physical chemists are concerned with. These are electronic, vibrational, and rotational energies. Changes in electronic energy levels occur when an electron undergoes a transition from one molecular orbital to another. Vibrational energy levels are associated with vibrations of chemical bonds in the molecule, and rotational energy levels involve the molecule rotating in space. As you could probably guess, atoms don’t have vibrational energy levels since there aren’t chemical bonds present in single atoms. Physical chemists can often learn about the structure and reactivity of molecules by studying the transitions between these energy levels.

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  Available until 5 Jul |Learn more

  AP Physics C Premium, 2024: 4 Practice Tests + Comprehensive Review + Online Practice

  • Robert A. Pelcovits, Joshua Farkas(Authors)
  • 2023(Publication Date)
  • Barrons Educational Services

    (Publisher)

  ω). As with center of mass, this can be solved only by moving to a differential level. If we imagine dividing the object into infinitesimally small pieces, we can apply the KE equation to these pieces to obtain the total KE by summing the KE of each piece:

  We recognize an integral (the mass of the pieces shrinks to zero and becomes a differential, whereas the velocity does not):

  This can be expressed in angular terms using the relationship v = rω:

  Putting the constants in front (all the differential masses share the same angular velocity),

  This equation is very similar to the equation for translational KE, . The linear squared velocity v2 is analogous to the angular squared velocity ω2 . The integral, ∫r2 dm, is the angular analog of mass. Just as mass is a measure of translational inertia, this integral is a measure of rotational inertia (often called the moment of inertia).

  TIP Rotational inertia measures a body’s resistance to rotation.

  The next chapter will describe how to evaluate this integral for a variety of simple objects. We can now rewrite the rotational kinetic energy:

  The rotational KE is not a “new” type of kinetic energy: It is simply a useful way to sum the translational kinetic energies of the particles that make up a rotating object.

  Torque

  In words, torque is the ability of a force to cause an object to accelerate angularly (i.e., to rotate at a nonconstant angular velocity). Torque is the angular analog of force. Mathematically,

  Here r is defined as a relative position vector pointing from the axis of rotation to the point where the force is applied (which requires that torque be calculated about some specified axis). The angle θ

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  eBook - ePub

  Biomechanics of Human Motion

  Applications in the Martial Arts, Second Edition

  • Emeric Arus, Ph.D.(Authors)
  • 2017(Publication Date)
  • CRC Press

    (Publisher)

  In our daily life, we often hear talk about energy. Do you say to yourself “I have no energy to finish the job or I need more energy to be able to reach a destination when you are walking on a hill?” The energy is the ability to do work.

  It is difficult to define energy in precise terms of shape, mass, or size. Rather, the energy suggests a dynamic state related to a condition of change because the presence of energy is revealed only when a change has taken place.

  There are two kinds of energy: potential and kinetic energy. The potential energy is a stored energy in a body or system. The kinetic energy can be described as the capacity to do work by virtue of the body’s motion. Where is a motion, such as a wind blow, a waterfall, a karateka kick, or the chemical composition of a body with its atoms and molecules that are in motion, there is a kinetic energy (e.g., heat energy). The kinetic energy comes from the potential (stored) energy.

  We know that there are many different kinds of energy such as chemical, electrical, light, wind, thermonuclear, water, and mechanical energy. According to the law of thermodynamics (conservation of energy ), which states that in any system not involving nuclear reactions or velocities approaching the velocity of light, energy cannot be created or destroyed but can be lost. An athlete can lose or deplete his energy if he does not adequately supply it by means of meal, drink, or rest.

  Let us analyze the potential energy a little bit more, noted with PE or U . U also represents work. We can see or realize that when an object has been moved to a certain distance and it has the tendency to return to its original position due to a force, which is also often called a restoring force , it represents a stored energy called the potential energy. For example, a stretched spring has the tendency to return to its original position due to potential energy. Another example is that when a weight is lifted up to a certain height, the force of gravity will try to bring it back to its original position due to the gravitational potential energy.

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  eBook - ePub

  AP® Physics 1 Crash Course Book + Online

  • Amy Johnson(Author)
  • 2016(Publication Date)
  • Research & Education Association

    (Publisher)

       Rotational vs. Translational Energy

           a.     It is important to know when you will be using translational energy, when you’ll be using rotational energy, and when you would use both.

           b.     For example, a bicycle wheel, sliding along a frictionless table, only has Translational Kinetic Energy, since it is sliding, and not rotating.

           c.     If you had the same bicycle wheel suspended from the ceiling, and the wheel was rotating freely, you would use rotational energy.

           d.     If the bicycle wheel is rolling down the street, it has rotational and translational energy.

  E.	ANGULAR MOMENTUM

  1.     Angular momentum is the rotational analog of linear momentum. It can be calculated using the equation: L = Ιω, or L = mvR sin θ and is measured in kg · m2 /s.

  2.     Angular momentum is conserved if there is no net external torque, just as linear momentum is conserved if there is no external force.

  Angular momentum for an object rotating on its own axis can be calculated using L = Ιω. If an object is rotating around an external point, use L = mvR sin θ, where mv is the momentum of the object, R is the distance between the pivot point and the object, and θ is the angle between that R vector and the momentum vector.

  3.     If an ice skater spins with her arms stretched out and then pulls her arms in, you expect that her angular velocity will increase (she will spin faster), but why? Initially, with her arms stretched out, she has a large rotational inertia, and when she pulls her arms in, she has a smaller rotational inertia. There is no net external torque on her while she is pulling her arms in, since the force she is applying to bring her arms in is internal to the system. This means that her angular momentum is conserved, so her initial angular momentum and her final angular momentum are the same:

          

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  eBook - ePub

  Fluid Mechanics

  • Jean-Laurent Puebe(Author)
  • 2013(Publication Date)
  • Wiley-ISTE

    (Publisher)

  The quantity of acceleration is only the momentum's time derivative for systems of constant mass. A rocket which ejects a certain momentum in order to propel itself is not a closed system. In fluid mechanics, we will often reason in terms of open systems which exchange momentum with the exterior.

  3.1.6. Kinetic energy

  Every extensive quantity has a corresponding form of energy. The term for the energy differential (kinetic energy

  Ec

  ) associated with the extensive quantity p for the preceding system's motion is:

  (3.5 )

  which, taking account of expression [3.1] for a particle of constant mass m , is:

  (3.6 )

  Kinetic energy can thus be expressed by the relation:

  The kinetic energy theorem can be obtained by taking the scalar product of the two sides of equation [3.2] with the velocity .

  (3.7 )

  The quantity is the power of the force in the reference frame considered. As with the kinetic energy, it depends on the reference frame used for its evaluation.

  3.2. Mechanical material system

  3.2.1. Dynamic properties of a material system

  A mechanical material system will be constructed, as for any thermodynamic system, by decomposition into n sub-systems which are points (or nearly points) in separate equilibriums (here at uniform velocity). The sub-systems interact amongst themselves according to the principle of action and reaction which results from the extensive nature of momentum. This quantity and the kinetic energy associated with the movement for the whole system of points are additive:

  (3.8 )

  The conservative nature of extensive quantities entails that total momentum remains constant in an isolated system. The velocity, which is the corresponding intensive variable, is of course not defined for the complete system if this one is not in a uniform state.

  Quantity is called the total linear momentum or momentum of the system . We also define the

  total angular momentum or angular momentum

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  eBook - ePub

  Kinetic Theory of Gases

  • Walter Kauzmann(Author)
  • 2013(Publication Date)
  • Dover Publications

    (Publisher)

  Chapter 3

  THE MOLECULAR THEORY OFTHE THERMAL ENERGY ANDHEAT CAPACITY OF A GAS

  3-1   The translational energy of a gas; heat capacity of a monatomic gas

  In Eq. (2-17) it was suggested that for n moles of a dilute gas at temperature T it is possible to make the identification

  where R is the gas constant, N is the number of molecules present, m is the molecular mass, and is the mean square velocity of the molecules. If R is replaced by N 0 k , where N 0 is Avogadro’s number and k is Boltzmann’s constant [Eq. (2-42)], and if N is replaced by nN 0 , then Eq. (3-1) can be written

  The kinetic energy associated with the motion through space of the center of gravity of an object of mass M at a velocity V is . The energy associated with the motion (translation) of the center of gravity of a molecule is called the translational energy of the molecule, in order to distinguish it from other kinds of energy that the molecule might possess. For instance, later in this chapter we shall find that useful results can be obtained if a diatomic molecule such as hydrogen or oxygen is regarded as a pair of massive particles (the atomic nuclei) held together by a spring (the chemical bond). Such an object can undergo two kinds of internal motion: (1) rotation about axes passing through the center of gravity normal to the chemical bond and (2) oscillations of the nuclear masses when the spring-like chemical bond is compressed and stretched. Each of these internal motions makes a contribution to the total energy of the molecule—which will be discussed later in this chapter. For the present we shall confine our attention to the translational energy of gases.

  The translational energy of a molecule of mass m moving with velocity u is

  In a gas in which the N molecules are given the labels 1, 2, . . . , N and in which molecule i has the velocity u i the total translational energy is

  Introducing the mean square velocity we find From Eqs. (3-1) and (3-2) we see that

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