simple harmonic motion

Definitions and Relationships

  • What are the definitions and SI units for period (T) and frequency (f)?

  • How are period and frequency related mathematically? (State the equation.)

  • What is the Hertz (Hz) defined as?

  • What is the maximum displacement from the equilibrium position called?

  • What is a phase shift (\phi) used for in the context of SHM equations?

Characteristics of SHM

  • What is the defining characteristic of Simple Harmonic Motion in terms of acceleration (or net force)?

  • For an object on a spring undergoing SHM, what two factors only affect the period and frequency?

  • In Simple Harmonic Motion, is the period and frequency dependent on the amplitude (A)?

  • What is the force that acts on a mass attached to a spring, and what law does it obey? (State the equation.)

Equations and Variables

  • Write down the generalized equation for position (x(t) ) as a function of time for a simple harmonic oscillator.

  • What is the relationship between angular frequency (\omega) and the period (T) and frequency (f)?

  • What are the equations for the maximum velocity (v{max}) and maximum acceleration (a{max}) in terms of amplitude (A) and angular frequency (\omega)?

  • How do the equations for the period (T) and angular frequency (\omega) for a mass on a spring relate to the mass (m) and force constant (k)? (State the equations.)

Vertical Springs

  • How does the force of gravity (mg) affect a mass oscillating on a vertical spring compared to a horizontal spring?

  • Does the angular frequency (\omega) or period (T) of a vertical spring system change due to gravity? Why or why why not?

💡 Energy in SHM Active Recall Questions
Definitions and Conservation

  • What are the two forms of energy that oscillate in a Simple Harmonic Oscillator (SHM) system like a block and spring?

  • Write the general equation for the total mechanical energy (E_{Total}) of an SHM system in terms of potential energy (U) and kinetic energy (K).

  • If there are no dissipative forces (like friction), what can be said about the total energy of the SHM system?

  • What is the equation for the potential energy (U) stored in a spring, and what position is generally defined as U=0?

Extreme Points and Equilibrium

  • At which points in the oscillation is the kinetic energy (K) maximum? What is the velocity (v) at these points?

  • At which points in the oscillation is the potential energy (U) maximum? What is the velocity (v) at these points?

  • How is the maximum total energy (E_{Total}) expressed in terms of the spring constant (k) and amplitude (A)? (State the equation.)

  • How is the force (F) related to the slope of the potential energy graph (U vs. x)?

Velocity and Stability

  • Write the equation for the magnitude of the velocity (|v|) at any position (x) during SHM, using energy conservation.

  • In the context of the potential energy curve (U vs. x), what characteristic defines a stable equilibrium point?

  • What happens if an object is slightly disturbed from an unstable equilibrium point?

  • If you were to decrease the amplitude (A) of a simple harmonic oscillator, how would this affect the total energy and the maximum velocity? (Refer to the Check Your Understanding 15.2 at the end of the section.)

Application

  • Explain the energy transformation (Potential \leftrightarrow Kinetic) that occurs in a block-spring system starting from the maximum positive displacement (x=+A) all the way to the equilibrium position (x=0).

The Model and Connection

  • What type of motion is used as an easy way to model Simple Harmonic Motion (SHM)?

  • Describe the physical setup (involving a peg, a disk, and a lamp) used to demonstrate the connection between circular motion and SHM.

  • What part of the uniform circular motion corresponds to the SHM of an oscillating block?

  • In this model, the disk must turn at a constant angular frequency (\omega) that is equal to what quantity from the oscillating system?

Key Components and Variables

  • If the disk has a radius r, what does r represent in the context of the SHM?

  • What is the equation for the position (x(t) ) of the shadow (and thus the SHM object) as a function of time, based on the rotating peg model?

  • In the rotating disk model, what does the tangential velocity of the peg around the circle equal for the block on the spring? (State the variable name.)

  • The velocity of the shadow is equal to which component of the peg's velocity?

Equations and Components

  • Write the equation for the velocity (v) of the shadow (or the SHM object) as a function of time, using the maximum velocity (v_{max}) and angular frequency (\omega).

  • What is the relationship between the maximum velocity (v_{max}) of the SHM object and the radius (A) and angular frequency (\omega) of the rotating disk? (State the equation.)

  • Write the equation for the acceleration (a) of the shadow (or the SHM object) as a function of time, using the maximum acceleration (a_{max}) and angular frequency (\omega).

  • Why does the position equation for the shadow use \cos(\omega t) when the peg starts at x=+A at t=0?

Application and Concept Check

  • Give an example of an object that undergoes uniform circular motion and explain how you could observe its resulting SHM. (Hint: Refer to the Check Your Understanding section.)

  • If the angular speed of the rotating disk is doubled, how would this change the period and frequency of the corresponding SHM?

⏳ Pendulums Active Recall Questions
The Simple Pendulum

  • What are the two forces that act on the bob of a simple pendulum?

  • What is the small angle approximation, and why is it necessary to apply it to a simple pendulum for the motion to be considered Simple Harmonic Motion (SHM)?

  • Write the equation for the period (T) of a simple pendulum.

  • The period of a simple pendulum is independent of what two factors?

  • How can a simple pendulum be used to measure the acceleration due to gravity (g)?

The Physical Pendulum

  • How does a physical pendulum differ from a simple pendulum?

  • Where does the force of gravity effectively act on a physical pendulum?

  • Write the equation for the period (T) of a physical pendulum in terms of its moment of inertia (I) and the distance (L) from the pivot to the center of mass.

  • In the physical pendulum equation, what does L represent?

  • Show how the period equation for a physical pendulum reduces to the period equation for a simple pendulum.

The Torsional Pendulum

  • What kind of restoring force/torque does a torsional pendulum use?

  • What is the variable \kappa (kappa) called, and what are its SI units?

  • Write the equation for the period (T) of a torsional pendulum.

  • How is the period of a torsional pendulum dependent on the moment of inertia (I)?

Concept Check

  • Describe how the motion of a 10 kg simple pendulum and a 100 kg simple pendulum, both displaced by 12^\circ, would differ. (Refer to the Check Your Understanding 15.4 on the page.)

📉 Damped Oscillations Active Recall Questions
Fundamental Concepts

  • What is the primary reason that real-world oscillations seldom follow true Simple Harmonic Motion (SHM)?

  • In damped harmonic motion, how does the amplitude (A) change over time?

  • What is the primary way that non-conservative damping forces remove energy from an oscillating system?

  • For a system with small damping, how do the period and frequency compare to those of SHM?

Equations and Forces

  • In the case of small velocity, what is the mathematical relationship for the damping force (F_D)? (State the equation and note the direction.)

  • Write the differential equation for the net force on a mass undergoing damped harmonic motion.

  • Write the equation for the position (x(t) ) of a damped harmonic oscillator.

  • In the position equation, what does the term A_0 e^{-\frac{b}{2m}t} represent?

  • Write the equation for the angular frequency (\omega) of a damped harmonic oscillator in terms of the natural angular frequency (\omega_0) and the damping constant (b).

Damping Regimes

  • Define the term natural angular frequency (\omega_0). (State the equation.)

  • What are the three damping regimes (types of damping)?

  • For a system to be considered underdamped, what must be the mathematical relationship between the damping constant (b), mass (m), and spring constant (k)?

  • Describe the motion of an underdamped system.

  • What is the condition for a system to be critically damped?

  • Describe the key characteristic of critically damped motion, and provide a common example where it is desirable.

  • Describe the motion of an overdamped system.

Critical Thinking

  • Why is there a maximum value for the damping constant (b) beyond which the angular frequency (\omega) becomes a complex number?

  • Explain why a critically damped system is often desired for car shock absorbers.

📢 Forced Oscillations and Resonance Active Recall Questions
Definitions and Concepts

  • Define forced oscillations.

  • What is the natural frequency (\omega_0) of a system?

  • Define the phenomenon of resonance.

  • What must be true about the driving frequency (\omega) and the natural frequency (\omega_0) for a system to resonate?

  • Give a real-world example (other than a piano or a swing) of resonance.

Equations and Amplitude

  • Write the equation for the periodic driving force (F_d) used in the model of forced oscillations.

  • Write the differential equation for the net force on a mass undergoing forced and damped harmonic motion.

  • What is the equation for the amplitude (A) of a forced oscillator's steady-state motion?

  • In the amplitude equation, how does the denominator change as the driving frequency (\omega) approaches the natural frequency ($\omega_0)?

  • What is the equation for the maximum amplitude (A_{max}), and what is the relationship between the two frequencies at this point?

Damping and Quality

  • How does the amount of damping (b) affect the maximum amplitude achieved at resonance?

  • Describe how the width of the resonance curve (amplitude vs. driving frequency) changes as damping decreases.

  • What is the term used for the narrowness of the resonance curve, and why is this property desirable in systems like a radio tuner?

  • Write the equation for the Quality Factor (Q) in terms of natural angular frequency ($\omega_0) and the spread of the angular frequency (\Delta\omega).

Concept Check

  • Explain why a singer must match the natural frequency of a crystal glass to make it shatter.

  • In the forced oscillation system, what is the motion called that occurs before the system reaches its steady-state periodic motion?