Harmonic motion forced simple

The hydrogen atom attached to an alkane molecule vibrates along the bond axis at a frequency of about 3000 cm. What wavelength of electromagnetic radiation is resonant with this vibration What is the frequency in hertz What is the force constant of the C II bond if the alkane is taken to be a stationary mass because of its size and the H atom is assumed to execute simple harmonic motion ... 

For small displacements molecular vibrations obey Hooke s law for simple harmonic motion of a system that vibrates about an equilibrium configuration. In this case the restoring force on a particle of mass m is proportional to the displacement x of the particle from its equilibrium position, and acts in the opposite direction. In terms of Newton s second law ... 

Mathematically, the movement of vibrating atoms at either end of a bond can be approximated to simple-harmonic motion (SHM), like two balls separated by a spring. From classical mechanics, the force necessary to shift an atom or group away from its equilibrium position is given by... 

Consider the situation shown in Figure 2.4 where a mass m is caused to oscillate by an initial displacement up to an amount oq at t = 0. The amplitude a would have to be smaller than shown for simple harmonic motion as a real spring would only obey Hooke s law over a limited strain amplitude. However the assumption is that Hooke s law is obeyed and the restoring force from both spring displacements is — IJcoq where k is the force constant or elastic modulus of the spring. So we may write the force at any position as... 

For negative values of , a molecule thus experiences a restoring force towards the axis, and the molecule can execute simple harmonic motion about the axis. Newton s equations predict that molecules entering the field with no radial component of velocity will be focused to a point on the axis when the voltage is ... 

There is an equivalence between the differential equations describing a mechanical system which oscillates with damped simple harmonic motion and driven by a sinusoidal force, and the series L, C, R arm of the circuit driven by a sinusoidal e.m.f. The inductance Li is equivalent to the mass (inertia) of the mechanical system, the capacitance C to the mechanical stiffness and the resistance Ri accounts for the energy losses Cc is the electrical capacitance of the specimen. Fig. 6.3(b) is the equivalent series circuit representing the impedance of the parallel circuit. 

The Vibration of Diatomic Molecules.—In addition to their rotation, we have seen that diatomic molecules can vibrate with simple harmonic motion if the amplitude is small enough. We shall use only this approximation of small amplitude, and our first stop will be to calculate the frequency of vibration. To do this, we must first find the linear restoring force when the interatomic distance is displaced slightly from its equilibrium value / ,. We can get this from Eq. (1.2) by expanding the force in Taylor s series in (r — rt). We have... 

Simple harmonic motion, such as the (undamped by frictional forces) sinusoidal oscillation of a weight suspended by a spring can also be thought of in terms of the projection of a vector traveling in a circular path. This is something you should have covered in your elementary mechanics classes, of course, but we will reexamine it here, first because it is important in infrared spectroscopy, and second because it provides some illumination concerning resonance. 

We can ascribe these frequency differences to the effect of the different reduced masses, fi, and force constants, / From considerations of simple harmonic motion [3]. 

The principle of classical equation solving, which is at the heart of MM, can be appreciated by imagining two objects, connected by a spring, executing simple harmonic motion collinear with the spring. If the force constant k of the spring is known, it is possible to calculate the potential energy V of the system at any separation x of the objects as... 

This value is quite close to the accurately determined force constant of 1902 N m" (see Banwell and McCash ). The reason why this simple calculation does not give the true value is that the vibration of the CO molecule deviates slightly from simple harmonic motion, and an anharmonic factor has to be included in an accurate calculation to allow for this. 

Simple harmonic motion n. Periodic oscillatory motion in a straight Une in which the restoring force is proportional to the displacement. If a point moves uniformly in a circle, the motion of its projection on the diameter (or any straight hne in the same plane) is simple harmonic motion. If r is the radius of the reference circle, (o the angular velocity of the point in the circle, 0 the angular displacement at the time t after the particle passes the mid-point of its path, the linear displacement... 

The simplest possible assumption about the form of the vibrations in a diatomic molecule is that each atom moves toward or away from the other in a simple harmonic motion (2-5). Such a motion of the two atoms can be reduced to the harmonic vibration of a single mass point about an equilibrium position. In classic mechanics a harmonic oscillator can be defined as a mass point of mass m that is acted upon by a force F proportional to the distance Q from the equilibrium position and directed toward the equilibrium position ... 

To a reasonable approximation, at least for small displacements, the vibrations in a polyatomic molecule can be described as a kind of simple harmonic motion. This is essentially equivalent to considering a chemical bond between two atoms as a weightless spring that obeys Hooke s law (i.e., the force is proportional to the displacement). The simplest vibration is that in a diatomic molecule, and we can refer back to the potential energy curve for He (O Fig. 10-1) as an example. At the minimum, near the bottom of the weU, the potential energy curve is indeed parabolic to a very good approximation. 

The quadrupolar field of the ion trap exerts a restoring force on ions near its center that is linear with respect to displacement from the trap s center. This results in simple harmonic motion. 

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