Overview
This lecture covers the concept of restoring force in springs, simple harmonic motion, the mathematical model for a mass-spring system, and how to interpret and apply the key equations for oscillatory motion.
Restoring Force and Hooke’s Law
- The restoring force from a spring is given by Hooke’s Law: ( F = -kx ), where ( k ) is the spring constant and ( x ) is the displacement from equilibrium.
- The force always points toward equilibrium, whether the spring is stretched or compressed.
- At equilibrium, ( x = 0 ).
Newton’s Second Law and Differential Equation
- For a mass on a spring, Newton’s Second Law gives ( ma_x = -kx ).
- Rearranged: ( a_x = -\frac{k}{m}x ).
- Acceleration relates to position by ( a_x = \frac{d^2x}{dt^2} ).
- The equation becomes ( \frac{d^2x}{dt^2} = -\frac{k}{m}x ), a second-order differential equation.
Solution to the Equation: Simple Harmonic Motion
- The general solution: ( x(t) = A \cos(\omega t + \phi) ).
- ( \omega ) (angular frequency) is ( \sqrt{\frac{k}{m}} ).
- ( A ) is the amplitude (maximum displacement).
- ( \phi ) is the phase constant, determining the initial position.
Period, Frequency, and Relationships
- Period ( T ) is the time for one complete cycle, ( T = \frac{2\pi}{\omega} ).
- Frequency ( f ) is ( 1/T ), measured in Hertz (Hz).
- ( f = \omega / 2\pi ).
- Key equation combinations: ( T = 2\pi\sqrt{m/k} = 1/f ).
Velocity and Acceleration in SHM
- Velocity as a function of time: ( v(t) = -A\omega \sin(\omega t + \phi) ).
- Maximum velocity: ( v_{max} = A\omega ).
- Acceleration as a function of time: ( a(t) = -A\omega^2 \cos(\omega t + \phi) ).
- Maximum acceleration: ( a_{max} = A\omega^2 ).
- Acceleration is always opposite in direction to the position (( a = -\omega^2 x )).
Example Application and Graph Analysis
- At a point with positive displacement and negative velocity, the system is moving toward equilibrium and speeding up if acceleration is also negative.
- If ( \phi = 0 ), the motion starts at maximum positive displacement.
- For given amplitude ( A ) and period ( T ), ( x(t) = A \cos\left(\frac{2\pi}{T}t\right) ).
Key Terms & Definitions
- Restoring Force — Force that acts to bring an object back to equilibrium.
- Hooke’s Law — ( F = -kx ), law describing the force in springs.
- Equilibrium — Position where net force is zero.
- Amplitude (A) — Maximum displacement from equilibrium.
- Angular Frequency ((\omega)) — ( \sqrt{k/m} ), rate of oscillation in radians per second.
- Period (T) — Time for one complete oscillation.
- Frequency (f) — Number of cycles per second (Hz).
- Phase Constant ((\phi)) — Initial angle that determines starting position and direction of motion.
- Simple Harmonic Motion (SHM) — Motion described by a sine or cosine function, characteristic of systems with restoring forces proportional to displacement.
Action Items / Next Steps
- Review and practice using equations for period, frequency, amplitude, and phase in SHM problems.
- Be comfortable identifying velocity and acceleration from position-time graphs.
- Bring questions about periodic motion for in-class discussion.