# 15.1. Simple Harmonic Motion

**Any motion that repeats itself at regular intervals is called**

**harmonic motion**. A particle experiences a

**simple harmonics motion**if its displacement from the origin as function of time is given by

_{ }

_{m}, [omega] and [phi] are constants, independent of time. The quantity x

_{m}is called the

**amplitude of the motion**and is the maximum displacement of the mass. The time-varying quantity ([omega]t + [phi]) is called the

**phase of the motion**and [phi] is called the

**phase constant**. The phase constant is determined by the initial conditions. The

**angular frequency**[omega] is a characteristic of the system, and does not depend on the initial conditions. The unit of angular frequency is rad/s. The

**period**T of the motion is defined as the time required to complete one oscillation. Therefore, the displacement x(t) must return to its initial value after one period

x(t) = x(t + T)

This is equivalent to_{ }

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**frequency of the oscillation**. The symbol for frequency is [nu] and its unit is the Hertz (Hz):

1 Hz = 1 oscillation per second = 1 s

The period T and the frequency [nu] are related as follows^{-1}_{ }

_{ }

_{m}is called the

**velocity amplitude**and is the maximum velocity of the object. Note that the phase of the velocity and displacement differ by 90deg.. This means that

**the velocity is greatest when the displacement is zero and vice versa**. The acceleration of an object carrying out simple harmonic motion is given by

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^{2}x

_{m}is the

**acceleration amplitude**a

_{m}. Using the expression for x(t), the expression for a(t) can be rewritten as

_{ }

_{ }

F = - k x

Comparing these last two equations we conclude that
k = m [omega]

and^{2}"

**Simple harmonic motion is the motion executed by a particle of mass m, subject to a force F that is proportional to the displacement of the particle, but opposite in sign.**"

The system shown in Figure 15.1 forms a simple harmonic oscillator. It will oscillate with an angular frequency [omega] given by

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Figure 15.1. A simple harmonic oscillator.

The kinetic energy of the system is given by_{ }

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**total mechanical energy of the simple harmonic oscillator is constant**(independent of time). However, the kinetic and potential energies are functions of time.

**Example: The torsion pendulum**

The operation of a torsion pendulum is associated with twisting a suspension wire. The motion described by the torsion pendulum is called

**angular simple harmonic motion**. The restoring torque is given by

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**Example: Classical simple pendulum**

The classical simple pendulum is shown in Figure 15.2. It consists out of a mass m suspended from a massless string of length L. The forces acting on the mass are the gravitational force m g and the tension T in the string. The radial component of the gravitational force, m g cos([theta]), determines the tension in the wire, but will not alter the motion of the mass. The tangential component of the gravitational force, m g sin([theta]), is always directed towards the rest position of the pendulum. This component of the gravitational force is called the restoring force:

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**the restoring force is proportional to the displacement, and of opposite sign**. The motion is therefore that of a harmonic oscillator. The acceleration of the mass is related to the displacement s

Figure 15.2. Classical simple pendulum.

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Figure 15.3. The physical pendulum.

**Example: The Physical Pendulum**

In the real world pendulums are far from simple. In general, the mass of the pendulum is not concentrated in one point, but will be distributed. Figure 15.3 shows a physical pendulum. The physical pendulum is suspended through point O. The effect of the force of gravity can be replaced by the effect of a single force, whose magnitude is m g, acting on the center of gravity of the pendulum (which is equal to the center of mass if the gravitational acceleration is constant). The resulting torque (with respect to O) is given by

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^{2}. The period of the oscillation is then given by

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**Note**: The equations of motion that describe harmonic motion all have the following form:

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**Example: Problem 33P**

Two springs are attached to a block of mass m and to fixed supports as shown in Figure 15.4. Show that the frequency of oscillation on the frictionless surface is given by

Figure 15.4. Problem 33P.

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**Example: Problem 35P**

Two springs are joined and connected to a mass m as shown in Figure 15.5. The surfaces are frictionless. If the springs each have a force constant k, show that the frequency of oscillation of m is

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Figure 15.5. Problem 35P

Assume that the spring constants are not the same. As the mass
oscillates, spring 1 is stretched or compressed by a distance x_{1}; the corresponding distance for the other spring is called x

_{2}. By Newton's third law, the forces exerted by the springs on each other are equal in magnitude but pointed in opposite directions. The force exerted by spring 1 on spring 2 is given by

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_{1}> 0) the force exerted by spring 1 on spring 2 is pointed in the negative direction. The force exerted by spring 2 on spring 1 is given by

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_{2}> 0) the force exerted by spring 2 on spring 1 is pointed in the positive direction. Applying Newton's third law we conclude that

_{ }

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_{1}is the only force acting on the mass, and F

_{1}is equal to k

_{1}x

_{1}. The previous relation can now be used to express the force F

_{1}in terms of the displacement x:

_{ }

_{1}and k

_{2}and joined in the way shown in Figure 15.5, act like a single spring with spring constant k, where k is given by

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## 15.2. Damped Simple Harmonic Motion

**Up to now we have discussed systems in which the force is proportional to the displacement, but pointed in an opposite direction. In these cases, the motion of the system can be described by simple harmonic motion. However, if we include the friction force, the motion will not be simple harmonic anymore. The system will still oscillate, but its amplitude will slowly decrease over time.**

Suppose the total force acting on a mass is not only proportional to its displacement, but also to its velocity. The total force can be represented in the following way

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**damping constant**. Substituting the expression for the force in terms of the acceleration we obtain the following differential equation

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**Example: Problem 87P**

A damped harmonic oscillator involves a block (m = 2 kg), a spring (k = 10 N/m), and a damping force F = - b v. Initially it oscillates with an amplitude of 0.25 m; because of the damping, the amplitude falls to three-fourths of its initial value after four complete cycles. (a) What is the value of b ? (b). How much energy is lost during these four cycles ?

The time dependence of the amplitude of the oscillation is given by

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## 15.3. Driven Harmonic Motion

**The case of a harmonic oscillator driven by a sinusoidal varying force is an extremely important one in many branches of physics. In the previous sections we have discussed several examples of harmonic oscillators, and for each system we have been able to calculate the natural frequency [omega]**

_{0}, (for example, for the spring [omega]

_{0}

^{2}= k/m). The equation of motion for an oscillator on which no damping force is working, and no external force is applied is given by

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_{0}and an angular frequency [omega]. The equation of motion describing the system is now given by

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**steady state**(the state of the system after any transient effects have died down) response of the system will be precisely at the driving frequency. Otherwise the relative phase between force an response would change with time. Thus,

**the steady-state response of a harmonic oscillator is at the driving frequency [omega] and not at the natural frequency [omega]**.

_{0}The general solution of the equation of motion is

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_{0}. The first condition than shows that

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_{0}. The system is said to be

**in resonance**when this happens.

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