SAT Physics Simple Harmonic Motion - Oscillations Of Pendulums

SAT Physics Simple Harmonic Motion - Oscillations Of Pendulums

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.
3-1
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.
3-2
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.

GRAPHICAL REPRESENTATIONS OF SHM
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.
3-3
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.

TRENDS IN OSCILLATIONS
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.

3-5
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.

3-6
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.
3-7
For a pendulum, substitute the potential energy of gravity.
3-8

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.
3-9
Figure 14.7. Force and energy trends in oscillations
Location: Texas, USA

Comments

Post a Comment

Popular posts from this blog

SAT Chemistry The Laboratory - Laboratory Safety Rules

Image
SAT Chemistry The Laboratory -   Laboratory Safety Rules The Ten Commandments of Lab Safety The following is a summary of rules you should be well aware of in your own chemistry lab. Dress appropriately for the lab. Wear safety goggles and a lab apron or coat. Tie back long hair. Do not wear open-toed shoes. Know what safety equipment is available and howto use it. This includes the eyewash fountain, fire blanket, fire extinguisher, and emergency shower. Know the dangers of the chemicals in use, and read labels carefully. Do not taste or sniff chemicals. Dispose of chemicals according to instructions. Use designated disposal sites, and follow the rules. Never return unneeded chemicals to the original containers. Always add acids and bases to water slowly to avoid splattering. This is especially important when using strong acids and bases that can generate significant heat, form steam, and splash out of the container. Never point heating test tubes at yourself or others. Be
Read more

SAT Chemistry Atomic Structure and the Periodic Table of the Elements - Atomic Spectra

Image
SAT Chemistry Atomic Structure and the Periodic Table of the Elements - Atomic Spectra The Bohr model was based on a simple postulate. Bohr applied to the hydrogen atom the concept that the electron can exist only in certain energy levels without an energy change but that, when the electron changes its state, it must absorb or emit the exact amount of energy that will bring it from the initial state to the final state. The ground state is the lowest energy state available to the electron. The excited state is any level higher than the ground state. The formula for a change in energy (ΔE) is: When an electron moves from the ground state to an excited state, it must absorb energy. When it moves from an excited state to the ground state, it emits energy. This release of energy is the basis for atomic spectra. (See Figure 6.) The energy values shown were calculated from Bohr’s equation. When energy is released in the “allowed” values, it is released in the form of discrete radi
Read more

SAT Physics Historical Figures and Contemporary Physics - Historical Figures

Image
SAT Physics Historical Figures and Contemporary Physics - Historical Figures HISTORICAL FIGURES Newtonian Mechanics GALILEO GALILEI (1564-1642) Bodies dropped from the same height will all fall with the same acceleration, g. The distance they travel is proportional to the square of time, y = Principle of inertia: The natural state of motion is uniform constant velocity. ISAAC NEWTON (1642-1727) 1st law of motion (law of inertia): Modifies Galileo’s principle of inertia. The natural state of motion is constant velocity unless acted upon by an unbalanced force. 2nd law of motion (ΣF = ma): The acceleration of an object is directly proportional to the net force acting on the object. Acceleration is inversely proportional to the mass of the object. 3rd law of motion: When two objects interact (action-reaction pair), an equal and oppo­site force acts on each object. This causes an opposite reaction. However, the actual motion depends on the masses involved. Law of gravity
Read more