Lecture Notes on Springs, Pendulums, and Simple Harmonic Oscillators
Introduction
Discussion of springs, pendulums, and simple harmonic oscillators, key topics in physics.
Springs and Hooke's Law
Equilibrium Position: Relaxed length of the spring (x = 0).
Restoring Force: When a spring is stretched (x > 0), a restoring force acts to return it to equilibrium.
Hooke's Law: The restoring force (F) is proportional to the displacement (x).
Formula: F = -kx (where k is the spring constant, measured in N/m)
Negative sign indicates the force direction is opposite to the displacement.
Measuring the Spring Constant (k)
A practical method for measuring k is using gravity:
Hang a mass (m) on the spring and find the new equilibrium position.
Relationship: F = mg (force due to gravity) must equal the spring force at equilibrium.
Scatter plot of force vs displacement can help determine the spring constant:
Slope = k = ΔF / Δx
Behavior of Springs
Springs behave according to Hooke's Law under certain limitations.
Permanent deformation occurs when the spring is stretched beyond its elastic limit.
Dynamic Measurement of Spring Constant
Attach a mass (m) to a spring on a frictionless surface.
When extended and released, the mass oscillates back and forth:
The period of oscillation (T) is given by:
T = 2π√(m/k)
Key Point: The period is independent of the amplitude of oscillation (how far the mass is pulled).
Differential Equation of Motion
Newton's second law: mA = -kx
Formulate as a differential equation:
mx'' + kx = 0
General solution: x(t) = A cos(ωt + φ)
Where:
A = amplitude
ω = angular frequency ( ω² = k/m)
φ = phase constant
Period: T = 2π√(m/k)
Small Angle Approximation for Pendulums
Pendulum Setup:
Forces acting: tension (T) and weight (mg).
Decompose tension into x and y components:
For small angles, use approximations for sine and cosine.
Resulting equation after applying small angle approximation:
m x'' + (mg/L)x = 0 (similar to spring's equation)
Period of pendulum: T = 2π√(L/g)
Comparison of Springs and Pendulums
Springs: T = 2π√(m/k) (mass influences period)
Pendulums: T = 2π√(L/g) (mass does not influence period)
Summary of Key Points
Spring Constant (k): Higher k means shorter period.
Gravity (g): For pendulum, no gravity means infinite period.
Mass Influence:
Doubling mass affects springs (increases period), but not pendulums.
Experimental Observations:
Conducted experiments measuring periods of oscillation for both springs and pendulums, demonstrating periodic behavior and confirming theoretical predictions.
Conclusion
Emphasized the beauty and consistency of physical laws as demonstrated in oscillatory systems.
Reminder of the importance of understanding limits and conditions under which physical laws apply.