6.1 Period of Simple Harmonic Oscillators

6.1 Period of Simple Harmonic Oscillators

What is Simple Harmonic Motion (SHM)?

Simple 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 and SHM

Newton’s second law (F = ma) is instrumental in solving harmonic oscillator problems. Here are the steps to apply it:

1. Identify Forces: List all forces acting on the object, such as gravity, elastic forces, and friction.

2. Free-Body Diagram: Create a diagram to visualize these forces.

3. Determine Mass and Acceleration: Note the mass and calculate the acceleration.

4. Write the Equation: Use Newton’s law to formulate: ΣF = ma.

5. Substitute Known Values: Insert forces and mass into the equation.

6. Solve for Motion: Use kinematic equations (e.g., v = at, x = at²/2) to determine velocity and position as functions of time.

7. Graph the Motion: Plot velocity and position over time to visualize oscillation.

8. Calculate Unknowns: Determine values like the spring constant or initial displacement.

9. Repeat for Additional Objects: If the system involves multiple objects, apply these steps for each.

Understanding Restoring Forces

Restoring forces are key to SHM. These forces act to return an object to its equilibrium position.

Key Points:

• 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.

Periods of Pendulums and Springs

Definitions:

• 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.

Example 2: Period of a Spring (Different Values)

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.