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
x(t) = x(t + T)This is equivalent to
1 Hz = 1 oscillation per second = 1 s-1The period T and the frequency [nu] are related as follows
F = - k xComparing these last two equations we conclude that
k = m [omega]2and
" 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
Figure 15.1. A simple harmonic oscillator.The kinetic energy of the system is given by
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
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:
Figure 15.2. Classical simple pendulum.
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
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.
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
Figure 15.5. Problem 35PAssume that the spring constants are not the same. As the mass oscillates, spring 1 is stretched or compressed by a distance x1; the corresponding distance for the other spring is called x2. 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
15.2. Damped Simple Harmonic MotionUp 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
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
15.3. Driven Harmonic MotionThe 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]02 = 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
The general solution of the equation of motion is