SAT Physics Simple Harmonic Motion - Oscillations Of Pendulums

SAT Physics Simple Harmonic Motion - Oscillations Of Pendulums

Pendulums are simple harmonic oscillators that take the form of a mass suspended at the end of a string. The period of a pendulum is the time for the mass at he end of the pendulum to oscillate from an initial position back to that initial position again.
Figure 14.3. The period of a pendulum

Pendulums are not perfect, simple harmonic oscillators. They approximate simple harmonic oscillators as long as the displacement angle, θ, is small (θ ≤ 10°). Keep in mind that diagrams are not drawn to scale on exams. Most questions on the SAT Subject Test will assume that pendulums are operating as simple harmonic oscillators regardless of how they are drawn. For pendulums with small displacements, the length of the string, L, and the acceleration of gravity, g, are the only variables contributing to the period of the pendulum. The period of a pendulum can be determined using the following formula.
As the length of the string increases, the period increases, causing the frequency to decrease. The amount of mass at the end of the string and the displacement from equilibrium (distance the mass is moved sideways) DO NOT affect the period of the pendulum at all.

When the displacement of an object in SHM is plotted on a graph of displacement versus time, the result is a sine wave, as shown in Figure 14.4.
Figure 14.4. Displacement versus time

One complete oscillation is represented by the time for the graph to go from point A to point E. The period is the time of one complete cycle.
T = tE - tA = 10 - 2 = 8 seconds
The frequency is the inverse of the period and is therefore 3-4 Hz. This means that one-eighth of an oscillation occurs every second.

In addition to being able to determine the period and frequency of a simple harmonic oscillator, the SAT Subject Test in Physics may ask questions involving the trends in key variables, such as speed, kinetic energy, displacement, potential energy, force, and acceleration. These trends are summarized in Figure 14.5.

Figure 14.5. Trends in oscillations

Energy in Oscillations
Chapter 7, “Energy, Work, and Power,” explains Hooke’s law and simple harmonic motion as it relates to energy. In an oscillation, energy continually changes form from potential energy to kinetic energy. Potential energy depends on displacement. When the oscillator reaches its maximum displacement, the potential energy will also reach its maximum value. At the equilibrium position, the oscillator has a displacement of zero and a potential energy of zero. Kinetic energy is the complete opposite. At maximum displacement, an oscillator has an instantaneous speed of zero and a kinetic energy of zero. When an oscillator passes through the equilibrium point, it attains its greatest speed and has maximum kinetic energy. How­ever, the total mechanical energy, ΣE = U+ K(potential energy plus kinetic energy), remains constant and is always conserved throughout a complete oscillation. These energy relation­ships are indicated in Figure 14.5, and they are graphed in Figure 14.6.

Figure 14.6. Energy graphs of oscillations

Conservation of energy is often needed to solve oscillation problems. Since the total energy is constant at every point in an oscillation, the total energy at any point 1 can be set equal to the total energy at any point 2.
E1 = E2
U1 + K1 = U2 + K2
For a spring, substitute the potential energy of a spring.
For a pendulum, substitute the potential energy of gravity.

Force and Acceleration
Acceleration is the result of the sum of the forces acting on an object. Figure 14.7 diagrams the key force and energy relationships for a mass-spring system that is oscillating on a hori­zontal frictionless surface. In an oscillation, the sum of forces is zero when the object is at the equilibrium position. When the sum of forces is zero, the acceleration is also zero, I.F= ma. This surprises many students since the acceleration is zero at the exact instant where veloc­ity is the greatest. However, if an oscillator is held at equilibrium and then released, it will not move at all. This confirms that acceleration is zero in this position. When the oscillator is moved out of equilibrium (the spring is stretched/compressed or the pendulum is moved to either side), the sum of the forces increases with displacement, reaching its highest value at maximum displacement. This is clearly evident when the oscillator is released. It moves rapidly toward equilibrium, gaining more and more speed. When it reaches equilibrium, the oscillator has so much speed that it coasts right through the equilibrium position, even though the acceleration and sum of forces are zero at this point. Then the oscillator begins building force and acceleration on the other side of equilibrium until these values max out at the point where the oscillator stops instantaneously and reverses direction. This concept often fools students. For an oscillator, the acceleration and sum of forces are greatest at maxi­mum displacement and are zero at equilibrium.
Figure 14.7. Force and energy trends in oscillations


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