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}(eta 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{max}) in terms of amplitude (A) and angular frequency (\omega) is ___.

v_{max} = A \omega

The equation for maximum acceleration (a{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{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{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{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 (FD) 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) = A0 e^{-\frac{b}{2m}t} cos(\omegad t + \phi)

The term A0 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 (\omega0) 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 (\omega0) 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 (\omega0) 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 (Fd) used in the model of forced oscillations is ___.

Fd = F0 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{F0/m}{\sqrt{(\omega0^2 - \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{max}) is given as ___, and the relationship between the two frequencies at this point is __.

A{max} = \frac{F0/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.