The Pendulum

Oscillatory MotionPeriodic motion is motion of an object that regularly repeatsthe object returns to a given position after a fixed time interval. A special kind of periodic motion occurs in mechanical systems when the force actingon an object is proportional to the position of the object relative to some equilibriumposition. If this force is always directed toward the equilibrium position, the motionis called simple harmonic motion, which is the primary focus of this chapter.

15.1 Motion of an Object Attached to a Spring

Hookes law&Newtons second law1) Fs = - kx Fs = restoring force(Newton) x = displacement ( meter) k = force constant (Newton/meter)2) Fx = max Fx = force (Newton) m = mass (Kg) ax = acceleration (m/s2)

So, -kx = max ax= -kx m That is the acceleration is propotional to the position of the block and its direction is opposite the direction ofthe displacement from equilibrium position. System that behave in this way are said to exhibit simple harmonic motion. 15.2 Mathematical representation of simple harmonic motion1) = = 2f = 2 T2) T= 2 3) T = 2 = angular frecuency (rad/s)f = frecuency (1/s)T = period (s)

4) Vmax = A= A5) amax = 2A = k A mVmax = maximum speed (m/s)

amax = maximum acceleration (m/s2)

3. Energy of the Simple Harmonic OscillatorKinetic energy of a simple harmonic oscillator

Potential energy of a simple harmonic oscillator

Total energy of a simple harmonic oscillator

The total mechanical energy of a simple harmonic oscillator is a constant of the motion and is proportional to the square of the amplitude.4. Plots of the kinetic and potential energies versus time

Kinetic energy and potential energy versus time for a simple harmonic oscillator with

Kinetic energy and potential energy versus position for a simple harmonic oscillator. In either plot, note that K + U = constant.we can use the principle of conservation of energy to obtain the velocity for an arbitrary position by expressing the total energy at some arbitrary position x as

Comparing Simple Harmonic Motionwith Uniform Circular Motion

Relationship between the uniform circular motion of a point P and the simple harmonic motion of a point Q. A particle at P moves in a circle of radius A with constant angular speed

(b) The x coordinates of points P and Q are equal and vary in time according to the expression x = A(c) The x component of the velocity of P equals the velocity of Q.(d) The x component of the acceleration of P equals the acceleration of Q.

Uniform circular motion can be considered a combination of two simple harmonic motions,one along the x axis and one along the y axis, with the two differing in phase by 90.5. The PendulumThe pendulum is another mechanical system that exhibits periodic motion.

6. Damped Oscillation systems that oscillate indefinitely under the action like nonconservative forces, such as friction, retard the motion.Consequently, the mechanical energy of the system diminishes in time, and the motionis said to be damped.

where the angular frequency of oscillation is

Figure 15.22 shows the position as a function of time for an object oscillating in thepresence of a retarding force. We see that when the retarding force is small, the oscillatorycharacter of the motion is preserved but the amplitude decreases intime, with the result that the motion ultimately ceases. Any system that behaves inthis way is known as a damped oscillator.

7. Forced Oscillationsforced oscillator is a damped oscillator driven by an external force that varies periodically

If an oscillator is subject to a sinusoidal driving force , it exhibits resonance, in which the amplitude is largest when the driving frequency matches then natural frequency of the oscillator.