simple harmonic motion

Period ($T$) is defined as __, and its SI unit is __.

The time taken for one complete cycle of motion; seconds (s)

Frequency ($f$) is defined as __, and its SI unit is __.

The number of cycles per second; Hertz (Hz)

The mathematical relationship between period and frequency is given by the equation __.

$$f = \frac{1}{T}$$

The Hertz (Hz) is defined as __.

One cycle per second.

The maximum displacement from the equilibrium position is called __.

Amplitude ($A$)

A phase shift ($\phi$) in the context of SHM equations is used to __.

Indicate the initial angle or displacement in oscillation.

The defining characteristic of Simple Harmonic Motion is that the acceleration is __.

Proportional to the displacement and directed towards the equilibrium position.

For an object on a spring undergoing SHM, the two factors that only affect the period and frequency are and .

Mass ($m$) of the object and spring constant ($k$).

In Simple Harmonic Motion, the period and frequency are __ dependent on the amplitude ($A$).

Not.

The force that acts on a mass attached to a spring obeys and is given by the equation .

Hooke's Law; $F = -kx$

The generalized equation for position ($x(t)$) as a function of time for a simple harmonic oscillator is __.

$$x(t) = A imes ext{cos}(\beta t + \phi)$$

The relationship between angular frequency ($\omega$) and the period ($T$) is given by __.

$$\omega = \frac{2\pi}{T}$$

The equation for maximum velocity ($v<em>{max}$) in terms of amplitude ($A$) and angular frequency ($\omega$) is ___.

$$v_{max} = A \omega$$

The equation for maximum acceleration ($a<em>{max}$) in terms of amplitude ($A$) and angular frequency ($\omega$) is ___.

$$a_{max} = A \omega^2$$

The equations for the period ($T$) and angular frequency ($\omega$) for a mass on a spring relate to the mass ($m$) and force constant ($k$) as follows: and .

$T = 2\pi \sqrt{\frac{m}{k}}$ and $\omega = \sqrt{\frac{k}{m}}$

The force of gravity ($mg$) affects a mass oscillating on a vertical spring by __ compared to a horizontal spring.

Adding an additional static force on the spring, which modifies the equilibrium position.

The angular frequency ($\omega$) or period ($T$) of a vertical spring system due to gravity, because .

Does not change; the system's motion still follows Hooke's Law and the characteristics of SHM.

The two forms of energy that oscillate in an SHM system are and .

Potential energy ($U$) and kinetic energy ($K$).

The general equation for total mechanical energy ($E<em>{Total}$) of an SHM system is ___.

$$E_{Total} = U + K$$

Without dissipative forces, the total energy of the SHM system __.

Remains constant.

The equation for the potential energy ($U$) stored in a spring is __, and the position generally defined as $U=0$ is __.

$U = \frac{1}{2} k x^2$; at the equilibrium position ($x=0$).

In oscillation, the kinetic energy ($K$) is maximum at __. The velocity ($v$) at these points is __.

The equilibrium position; maximum velocity ($v_{max}$).

In oscillation, the potential energy ($U$) is maximum at __. The velocity ($v$) at these points is __.

The maximum displacement; velocity is zero.

The maximum total energy ($E<em>{Total}$) is expressed in terms of the spring constant ($k$) and amplitude ($A$) as follows: ___.

$$E_{Total} = \frac{1}{2} k A^2$$

The force ($F$) is related to the slope of the potential energy graph ($U$ vs. $x$) as __.

$$F = -\frac{dU}{dx}$$

The equation for the magnitude of the velocity ($|v|$) at any position ($x$) during SHM using energy conservation is __.

$$|v| = \sqrt{\frac{2}{m}(E_{Total} - U)}$$

In the context of the potential energy curve ($U$ vs. $x$), a stable equilibrium point is defined as __.

A point where a small disturbance results in forces that restore the system to equilibrium.

If disturbed from an unstable equilibrium point, an object __.

Will move further away from the equilibrium position.

If the amplitude ($A$) of a simple harmonic oscillator is decreased, the and will be affected.

Total energy decreases; maximum velocity decreases.

In the energy transformation of a block-spring system from maximum positive displacement ($x=+A$) to equilibrium position ($x=0$), the potential energy and kinetic energy .

Decreases; increases.

The type of motion used to model Simple Harmonic Motion (SHM) is __.

Uniform circular motion.

The physical setup to demonstrate the connection between circular motion and SHM involves __, __, and __.

A peg, a disk, and a lamp.

The part of the uniform circular motion that corresponds to the SHM of an oscillating block is __.

The projection of the point moving in circular motion onto the vertical axis.

In this model, the disk must turn at a constant angular frequency ($\omega$) that is equal to __ from the oscillating system.

The angular frequency ($\omega$).

If the disk has a radius $r$, $r$ represents __ in the context of SHM.

The amplitude ($A$) of oscillation.

The equation for the position ($x(t)$) of the shadow as a function of time in the rotating peg model is __.

$$x(t) = r \cos(\omega t)$$

The tangential velocity of the peg around the circle for a block on the spring equals to __.

The velocity ($v$) of the block.

The velocity of the shadow is equal to which component of the peg's velocity is __.

The horizontal component.

The equation for the velocity ($v$) of the shadow (or the SHM object) as a function of time is __.

$$v = -A \omega \sin(\omega t)$$

The relationship between the maximum velocity ($v<em>{max}$) of the SHM object and the radius ($A$) and angular frequency ($\omega$) is given by ___.

$$v_{max} = A \omega$$

The equation for the acceleration ($a$) of the shadow (or the SHM object) as a function of time is __.

$$a = -A \omega^2 \cos(\omega t)$$

The position equation for the shadow uses $\cos(\omega t)$ because __.

At $t=0$, the projection starts at maximum displacement ($x=+A$).

An example of an object that undergoes uniform circular motion is __.

A point on the edge of a rotating wheel.

If the angular speed of the rotating disk is doubled, the period will and the frequency will .

Halve; double.

The two forces acting on the bob of a simple pendulum are and .

Tension in the string; gravitational force.

The small angle approximation is necessary for a simple pendulum because __.

It allows the restoring force to be directly proportional to the displacement.

The equation for the period ($T$) of a simple pendulum is __.

$$T = 2\pi \sqrt{\frac{L}{g}}$$

The period of a simple pendulum is independent of and .

Mass of the bob; amplitude of the swing.

A simple pendulum can measure the acceleration due to gravity ($g$) by __.

Measuring the period and rearranging the period equation.

A physical pendulum differs from a simple pendulum by __.

Its mass is distributed and not concentrated at a point.

The force of gravity effectively acts on a physical pendulum at __.

The center of mass.

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

$$T = 2\pi \sqrt{\frac{I}{mgL}}$$

In the physical pendulum equation, $L$ represents __.

The distance from the pivot point to the center of mass.

The period equation for a physical pendulum reduces to the period equation for a simple pendulum by __.

Assuming that the moment of inertia simplifies under certain conditions.

A torsional pendulum uses __ as a restoring force/torque.

Torsional spring force.

The variable $\kappa$ (kappa) is called and its SI units are .

The torsional constant; N·m/radian.

The equation for the period ($T$) of a torsional pendulum is __.

$$T = 2\pi \sqrt{\frac{I}{\kappa}}$$

The period of a torsional pendulum is dependent on __.

The moment of inertia ($I$) of the system.

The primary reason real-world oscillations seldom follow true SHM is __.

Dissipative forces such as friction and air resistance.

In damped harmonic motion, the amplitude ($A$) __ over time.

Decreases.

Non-conservative damping forces remove energy from an oscillating system primarily by __.

Doing work against the motion.

For a system with small damping, the period and frequency __ compared to those of SHM.

Change only slightly.

For small velocity, the mathematical relationship for the damping force ($F<em>D$) is given by ___.

$F_D = -b v$, where $b$ is the damping constant.

The differential equation for the net force on a mass undergoing damped harmonic motion is __.

$$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = 0$$

The equation for the position ($x(t)$) of a damped harmonic oscillator is __.

$$x(t) = A<em>0 e^{-\frac{b}{2m}t} cos(\omega</em>d t + \phi)$$

The term $A<em>0 e^{-\frac{b}{2m}t}$ represents ___ in the position equation of a damped harmonic oscillator.

The exponentially decaying amplitude over time.

The equation for the angular frequency ($\omega$) of a damped harmonic oscillator is given as __.

$$\omega = \sqrt{\omega_0^2 - \left(\frac{b}{2m}\right)^2}$$

The natural angular frequency ($\omega<em>0$) is defined as ___.

$$\omega_0 = \sqrt{\frac{k}{m}}$$

The three damping regimes (types of damping) are __, __, and __.

Under damped, critically damped, and over damped.

For a system to be considered underdamped, the mathematical relationship between the damping constant ($b$), mass ($m$), and spring constant ($k$) must be __.

b < 2\sqrt{mk}.

The motion of an underdamped system is characterized by __.

Oscillation with gradually decreasing amplitude.

The condition for a system to be critically damped is when the relationship between damping constant ($b$), mass ($m$), and spring constant ($k$) is __.

$$b = 2\sqrt{mk}.$$

The key characteristic of critically damped motion is __, and a common example where it is desirable is __.

The system returns to equilibrium in the least time without oscillating; car shock absorbers.

The motion of an overdamped system is described as __.

Returning to equilibrium without oscillating more slowly than a critically damped system.

There is a maximum value for the damping constant ($b$) beyond which the angular frequency ($\omega$) becomes a complex number because __.

Excessive damping occurs, preventing oscillation.

A critically damped system is often desired for car shock absorbers because __.

It brings the system back to equilibrium quickly without oscillation.

Forced oscillations are defined as __.

Oscillations occurring when a periodic driving force is applied to a system.

The natural frequency ($\omega<em>0$) of a system is defined as ___.

The frequency at which a system oscillates when not subjected to a driving force.

The phenomenon of resonance is defined as __.

The increase in amplitude of oscillation when the driving frequency matches the natural frequency.

For a system to resonate, the driving frequency ($\omega$) and the natural frequency ($\omega<em>0$) must be ___.

Equal.

A real-world example of resonance is __.

A swing being pushed at the right time.

The equation for the periodic driving force ($F<em>d$) used in the model of forced oscillations is ___.

$$F<em>d = F</em>0 ext{cos}(\omega t)$$

The differential equation for the net force on a mass undergoing forced and damped harmonic motion is __.

$$m\frac{d^2x}{dt^2} + b\frac{dx}{dt} + kx = F_d$$

The equation for the amplitude ($A$) of a forced oscillator's steady-state motion is __.

$$A = \frac{F<em>0/m}{\sqrt{(\omega</em>0^2 - \omega^2)^2 + (\frac{b}{m}\omega)^2}}$$

As the driving frequency ($\omega$) approaches the natural frequency ($\omega0), the denominator in the amplitude equation ___.

Decreases, causing amplitude to increase significantly.

The equation for the maximum amplitude ($A<em>{max}$) is given as ___, and the relationship between the two frequencies at this point is __.

$A<em>{max} = \frac{F</em>0/m}{\frac{b}{m}}$; $\omega = \omega_0$.

The amount of damping ($b$) affects the maximum amplitude achieved at resonance by __.

Reducing the maximum amplitude as damping increases.

As damping decreases, the width of the resonance curve (amplitude vs. driving frequency) __.

Narrows.

The term used for the narrowness of the resonance curve is __, and this property is desirable in systems like a radio tuner because __.

Quality Factor ($Q$); it allows for precise tuning to a specific frequency.

The equation for the Quality Factor ($Q$) is given as __ in terms of natural angular frequency ($\omega_0) and the spread of angular frequency ($\Delta\omega$).

$$Q = \frac{\omega_0}{\Delta \omega}$$

A singer must match the natural frequency of a crystal glass to make it shatter because __.

The glass will resonate and amplify the vibrations until it breaks.

In the forced oscillation system, the motion that occurs before the system reaches its steady-state periodic motion is called __.

Transitional motion.