6. Oscillations A.md

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Simple Harmonic Oscillations

Understanding and Use: Define the terms displacement, amplitude, period, frequency, angular frequency, and phase difference in the context of oscillations. How can the period be expressed in terms of both frequency and angular frequency? Answer: Displacement is the distance of an object from its equilibrium position. Amplitude is the maximum displacement. Period is the time taken for one complete cycle of oscillation. Frequency is the number of oscillations per unit of time. Angular frequency is the rate of change of the phase of a sinusoidal waveform (it is equal to 2π times the frequency). Phase difference is the difference in phase between two points in an oscillation. The period can be expressed as the inverse of the frequency (T = 1/f) and as the inverse of the angular frequency divided by 2π (T = 2π/ω).

Understanding: When does simple harmonic motion occur? Answer: Simple harmonic motion occurs when the acceleration of an object is directly proportional to its displacement from a fixed point and is always directed towards that fixed point. In other words, the object is always accelerated towards the equilibrium and the strength of the acceleration is proportional to the distance from the equilibrium.

Use: Use the equation a = -ω²x to calculate the acceleration of an object undergoing simple harmonic motion with an angular frequency of 3 rad/s and a displacement of 2 m. Answer: The acceleration of an object undergoing simple harmonic motion can be calculated using the equation a = -ω²x. Substituting the given values, a = -(3 rad/s)² 2 m = -18 m/s².

Use: Use the equations v = v₀ cos ωt and v = ± ω√(x₀² - x²) to calculate the velocity of an object undergoing simple harmonic motion with an initial velocity of 2 m/s, an angular frequency of 3 rad/s, and a displacement of 1 m at a time of 2 seconds. Answer: The velocity of an object undergoing simple harmonic motion can be calculated using the equation v = v₀ cos ωt. Substituting the given values, v = 2 m/s cos(3 rad/s 2 s) = -1.68 m/s. It can also be calculated using the equation v = ± ω√(x₀² - x²), but we would need to know the maximum displacement (x₀) for that.

Analysis and Interpretation: How would you interpret a graph showing the variation of displacement, velocity, and acceleration for simple harmonic motion? Answer: A graph showing the variation of displacement, velocity, and acceleration for simple harmonic motion would typically be sinusoidal. The displacement graph would show a sinusoidal variation with time, reaching a maximum at the amplitude of oscillation. The velocity graph would also be sinusoidal but shifted by a quarter of a period, reaching a maximum when the displacement is zero. The acceleration graph would be a negative sine function, reaching a maximum (in absolute value) when the displacement is at its maximum. All three graphs would have the same period.

Energy in Simple Harmonic Motion

Description: Can you describe the interchange between kinetic and potential energy during simple harmonic motion? Answer: In simple harmonic motion, the total energy of the system is conserved but continually interchanges between kinetic and potential energy. At the maximum displacement (amplitude), the velocity is zero, and all the energy is potential. As the object moves towards the equilibrium position, potential energy decreases while kinetic energy increases. At the equilibrium position, the speed is maximum, and all the energy is kinetic. This cycle of energy interchange repeats as the object oscillates.

Recall and Use: Recall the equation calculate the total energy of a system undergoing simple harmonic motion. Answer: The total energy E of the system can be calculated E = 1/2mω²x₀².

Damped and Forced Oscillations, Resonance

Understanding: What is the effect of a resistive force on an oscillating system? Answer: A resistive force on an oscillating system causes damping. Damping refers to the process in which the amplitude of an oscillation decreases over time due to the resistive force. The resistive force dissipates the energy of the system into other forms, such as heat, thereby reducing the amplitude of oscillation.

Understanding and Use: Define light, critical, and heavy damping and sketch displacement–time graphs for each. Answer:

Light damping: Light damping occurs when the damping force is relatively small. The system continues to oscillate with decreasing amplitude. The displacement-time graph shows a series of peaks that decrease gradually.

Critical damping: Critical damping is the minimal amount of damping at which the system returns to its equilibrium position without oscillating. The displacement-time graph shows a curve that decreases exponentially to zero without oscillating.

Heavy damping: Heavy damping occurs when the damping force is large. The system returns to equilibrium without oscillating, but more slowly than in critical damping. The displacement-time graph shows a curve that decreases slowly to zero.

Understanding: What is resonance and when does it occur in an oscillating system? Answer: Resonance is a phenomenon that occurs when an external force driving an oscillating system has a frequency equal to the natural frequency of the system. This matching of frequencies causes the amplitude of the oscillations to reach a maximum. In other words, the system absorbs maximum energy from the external force and oscillates with the greatest amplitude.