Table of Contents
ToggleSimple Harmonic Motion occurs when an object is pulled toward an equilibrium point by a force proportional to the displacement from that point. Common examples of SHM include:
A mass on a spring obeying Hooke’s Law.
A pendulum oscillating with a small angular displacement.
Newton’s second law (F = ma) is instrumental in solving harmonic oscillator problems. Here are the steps to apply it:
Identify Forces: List all forces acting on the object, such as gravity, elastic forces, and friction.
Free-Body Diagram: Create a diagram to visualize these forces.
Determine Mass and Acceleration: Note the mass and calculate the acceleration.
Write the Equation: Use Newton’s law to formulate: ΣF = ma.
Substitute Known Values: Insert forces and mass into the equation.
Solve for Motion: Use kinematic equations (e.g., v = at, x = at²/2) to determine velocity and position as functions of time.
Graph the Motion: Plot velocity and position over time to visualize oscillation.
Calculate Unknowns: Determine values like the spring constant or initial displacement.
Repeat for Additional Objects: If the system involves multiple objects, apply these steps for each.
Restoring forces are key to SHM. These forces act to return an object to its equilibrium position.
Definition: A restoring force is opposite to the displacement from equilibrium.
Examples: Pendulums, springs, and systems experiencing periodic motion.
Directionality: If a mass on a spring is displaced right, the restoring force points left.
Formula: The restoring force can be calculated as:
where is the spring constant, and is the displacement.
Spring Constant: A measure of stiffness; higher means a stronger restoring force.
Periodic Motion: Systems with restoring forces exhibit oscillatory behavior, such as masses on springs or swinging pendulums.
Note: For AP Physics 1, assume ideal springs. AP C: Mechanics involves non-ideal springs.
Amplitude (A): Maximum displacement from equilibrium.
Period (T): Time for one complete oscillation. Measured in seconds, it is the inverse of frequency (, in Hz).
For pendulums:
Answer: The period is 0.89 seconds.
Problem: A 2 kg mass is attached to a spring () and oscillates vertically without friction. Displacement is 0.5 m from equilibrium. Find the period.
Solution:
Answer: The period is 0.89 seconds.