6-Simple Harmonic Motion

Slideshow: SHM and oscillations notes

Textbook: Chapter 19 in Mastering Physics (get online code for registration on about page of google classroom)

Practice and reviews:

• Worksheet of practice problems with answers provided

• SHM Notes and Review with practice 

Objectives:

• Explain how restoring forces can cause oscillations; how linear restoring forces cause simple harmonic motion

  • predict, plan data collection, analyze data and explain oscillatory motion

• A system with internal structure can have internal energy (aka potential energy). Changes in a system's structure can change the energy

• Spring mass oscillators: 

  • T increases with m and decreases with k (spring stiffness)

  • Calculate force & acceleration for any location along an object oscillating along a spring. (Especially consider minima, maxima and zeroes for position, velocity and acceleration)

  • mass-spring systems convert kinetic energy into elastic potential and vice versa based on energy conservation. 

• Simple pendulums: 

  • T increases with L and decreases with the g (magnitude of gravitational field).

  • Potential energy exists within a system if the objects have conservative forces (springs & gravity)

  • Pendulums convert kinetic energy into gravitational potential  energy and vice versa based on energy conservation.

Helpful simulations

• • Horizontal Spring Oscillators: 

    • MyPhysicsLab: https://www.myphysicslab.com/springs/single-spring-en.html

    • OPhysics: https://ophysics.com/w1.html 

    • Javalab: https://javalab.org/en/spring_pendulum_en/ 

  • Pendulums:

    • MyPhysicsLab https://www.myphysicslab.com/pendulum/pendulum-en.html 

Common Misconceptions: (these statements all have flaws)

• • Misconception: The heavier a pendulum bob, the shorter its period. 

    • Physics Principles: For small angles the period of a pendulum can be determined by T=2π√ (L/g) and conservation of mechanical energy in this case is Ug(max) = K(max)

    • Reasoning: From a simple mathematical reasoning, since the mass is not part of the equation, it has no effect on the period. Similarly, this is true because the gravitational energy to kinetic energy conversion has mass cancel out (mgh=1/2 mv^2) meaning that the maximum (and therefore average) speed during the path is the same independent of the mass, taking the same time to travel the same distance. An additional conceptual consideration is that the acceleration is caused essentially by gravity (albeit with a factor based on the angle) and the additional mass resists acceleration as by the same factor that it has a greater gravitational force, so the acceleration is the same at any given point. 

  • All pendulum motion is perfect simple harmonic motion, for any initial angle.

    • Physics Principle: For small angles the period of a pendulum can be determined by T=2π√ (L/g)

    • Reasoning: As stated in the first misconception's response, the larger angles have additional factors that affect the timing and overall restoring force at greater angles. 

  • Harmonic oscillators go forever.

    • Physics Principle: Air resistance opposes motion through air; W=Fdcos(θ)=ΔE 

    • Reasoning: Even if frictional forces at the point of contact are ignored air resistance affects pendulums in all normal conditions and will remove energy from the system, causing smaller oscillations over time. Friction is also real and when we discuss assuming it is negligible this can only be a reasonable assumption if the time interval is short. In our lab, for example, I can swing a pendulum and see it come back to essentially the same position 1-2 times, but after 10 oscillations it always has an observable loss of energy, so air resistance cannot be negligible over long periods of time

  • A pendulum accelerates through lowest point of its swing.

    • Physics Principles: a_c =v^2/r; a=ΣF/m 

    • Reasoning: This one is partially right. The speed is not changing at this point because the net force for simple harmonic oscillators is equal to kx, where x is measured from the equilibrium position (where the pendulum would come to rest eventually even though it is technically not in equilibrium while in motion still). So, the net horizontal force is zero at the lowest position, and therefore the acceleration horizontally is zero. There is, however, centripetal acceleration at the lowest point because the pendulum is changing directions (v was slightly angled downward before and slightly upward after the lowest point) and the acceleration is equal to v^2/r.

  • Amplitude of oscillations is measured peak-to-peak.

    • Principle: Amplitude is the maximum displacement from the equilibrium position. 

    • Reasoning. Since the equilibrium position is at the center of the oscillating motion (assuming no damping/energy loss), the amplitude is half the distance between maximum and minimum, or can be measured as the distance between the sinusoidal axis and either a maximum or minimum.  

  • The acceleration is zero at the end points of the motion of a pendulum.

    • Physics principle: a=ΣF/m and ΣF=-kx for the net restoring force. 

    • Reasoning: At the end points of the motion the net restoring force is the greatest since the displacement from the equilibrium position is the greatest. Since the acceleration is directly proportional to restoring force, the acceleration is the greatest at the point where the x value has the greatest magnitude. 

  • The restoring force is constant at all points in the oscillation.

    • Physics Principle:  ΣF=-kx for the net restoring force

    • Reasoning: Simple harmonic motion experiences a force that varies directly with distance from the equilibrium position, so it changes throughout the motion.