🎡AP Physics 1 Unit 6 – Simple Harmonic Motion

Key Concepts

• Simple harmonic motion (SHM) periodic motion where restoring force is directly proportional to displacement
• Oscillation repetitive variation, typically in time, of some measure about a central value (equilibrium) or between two or more different states
• Period ($T$) time required for one complete cycle of vibration to pass a given point
• Frequency ($f$) number of occurrences of a repeating event per unit of time
  • Measured in hertz (Hz), where 1 Hz equals 1 cycle per second
• Angular frequency ($\omega$) rate of change of the phase of a sinusoidal waveform (angular velocity)
  • Measured in radians per second
• Amplitude ($A$) maximum displacement from the equilibrium position
• Phase shift ($\phi$) represents the difference in position of the wave cycle relative to the origin

Equations and Formulas

• Hooke's Law: $F = -kx$, where $F$ is the restoring force, $k$ is the spring constant, and $x$ is the displacement from equilibrium
• Period of a mass-spring system: $T = 2\pi\sqrt{\frac{m}{k}}$, where $m$ is the mass and $k$ is the spring constant
• Frequency of a mass-spring system: $f = \frac{1}{2\pi}\sqrt{\frac{k}{m}}$
• Angular frequency of a mass-spring system: $\omega = \sqrt{\frac{k}{m}}$
• Position as a function of time: $x(t) = A\cos(\omega t + \phi)$, where $A$ is the amplitude, $\omega$ is the angular frequency, $t$ is time, and $\phi$ is the phase shift
• Velocity as a function of time: $v(t) = -A\omega\sin(\omega t + \phi)$
• Acceleration as a function of time: $a(t) = -A\omega^2\cos(\omega t + \phi)$
• Relationship between period and frequency: $T = \frac{1}{f}$

Real-World Applications

• Pendulums used in clocks to keep time
• Vibrations in musical instruments (guitar strings, drumheads)
• Suspension systems in vehicles to provide a smooth ride
  • Shock absorbers and springs work together to minimize the effect of bumps on the road
• Seismic waves during earthquakes exhibit SHM
• Molecular vibrations in chemistry and physics
• Electrical oscillations in LC circuits
• Mechanical resonance in bridges and buildings
  • Tacoma Narrows Bridge collapse in 1940 due to wind-induced resonance

Experimental Setup

• Mass-spring system consists of a mass attached to a spring suspended vertically or horizontally
  • The mass oscillates when displaced from its equilibrium position
• Pendulum consists of a mass (bob) suspended from a fixed point by a string or rod
  • The pendulum swings back and forth under the influence of gravity
• Oscilloscope used to visualize waveforms and measure period, frequency, and amplitude
• Motion sensors or photogates used to record position and velocity data
• Force sensors measure the restoring force acting on the oscillating mass
• Data loggers collect and store experimental data for analysis
• High-speed cameras capture the motion of the oscillating system

Problem-Solving Strategies

• Identify the type of oscillating system (mass-spring, pendulum, etc.)
• Determine the given variables and the quantity to be calculated
• Select the appropriate equation or formula based on the given information
  • For example, if given mass and spring constant, use $T = 2\pi\sqrt{\frac{m}{k}}$ to find the period
• Substitute the given values into the equation and solve for the unknown variable
• Check the units of the solution to ensure they are consistent with the quantity being calculated
• Analyze the result to see if it makes sense in the context of the problem
• If necessary, sketch a diagram of the oscillating system to visualize the motion and identify relevant variables

Common Misconceptions

• Confusing period and frequency
  • Period is the time for one complete cycle, while frequency is the number of cycles per unit time
• Assuming that the period depends on the amplitude
  • The period of a simple harmonic oscillator is independent of the amplitude
• Neglecting the negative sign in the restoring force equation ($F = -kx$)
  • The negative sign indicates that the restoring force acts in the opposite direction of the displacement
• Thinking that the velocity is always positive
  • The velocity alternates between positive and negative values during an oscillation cycle
• Believing that the acceleration is constant throughout the motion
  • The acceleration in SHM varies sinusoidally and is proportional to the displacement
• Assuming that the total energy of the system changes over time
  • In the absence of friction or damping, the total energy (kinetic + potential) remains constant

Connections to Other Topics

• Conservation of energy
  • In SHM, energy is continuously converted between kinetic and potential energy
• Rotational motion
  • The angular frequency in SHM is analogous to the angular velocity in rotational motion
• Waves and oscillations
  • SHM is a fundamental concept in understanding wave motion and oscillations
• Differential equations
  • The equation of motion for SHM is a second-order linear differential equation
• Fourier analysis
  • Complex oscillations can be decomposed into a sum of simple harmonic motions using Fourier analysis
• Resonance and damping
  • Resonance occurs when the frequency of an external force matches the natural frequency of the oscillating system
  • Damping reduces the amplitude of oscillations over time due to friction or other dissipative forces

Practice Problems

1. A 0.5 kg mass is attached to a spring with a spring constant of 20 N/m. Calculate the period and frequency of the resulting oscillations.
2. A pendulum has a length of 1.2 m and is displaced by an angle of 15° from the vertical. If the mass of the pendulum bob is 0.3 kg, find the restoring force acting on the bob at this displacement.
3. The position of a particle undergoing SHM is given by $x(t) = 0.1\cos(2\pi t + \frac{\pi}{4})$, where $x$ is in meters and $t$ is in seconds. Determine the amplitude, angular frequency, and phase shift of the motion.
4. A 2 kg block oscillates on a horizontal spring with a period of 0.5 s. Calculate the spring constant and the maximum velocity of the block if the amplitude of the motion is 0.1 m.
5. A mass-spring system has a natural frequency of 3 Hz. If the mass is doubled, what will be the new frequency of the system?