Chapter 14: Periodic Motion

When the object is displaced from its equilibrium position at x = 0, the spring exerts a _________ back toward the equilibrium position.

restoring force

An oscillating system undergoes simple harmonic motion (SHM) only if the restoring force is __________ to the displacement.

directly proportional

The ________ of the motion, denoted by A, is the maximum magnitude of displacement from equilibrium (i.e. the max. value of |x| and is always positive). A complete vibration, or ________, is one complete round trip (e.g. A to -A to A).

amplitude , cycle

The ________, T, is the time to complete one cycle. It is always positive. (units: seconds per cycle)

period

The ________, ƒ, is the number of cycles in a unit of time. It is always positive. (units: Hz; 1 Hz = 1 cycle/s = 1 s^-1)

frequency

The _________, ω, is 2π times the frequency: _________ (units: rad/s). It is not necessarily related to rotational motion.

angular frequency , ω = 2πƒ

For a spring mass system, ω = _________.

ω = √(k / m)

Period and frequency are _________ of each other: _________. Also from the definition of ω, __________.

reciprocals , ƒ = 1/T and T = 1/ƒ , ω = 2πƒ = 2π/T

State the equation for the restoring force.

F_x = -kx

• F_x = restoring force exerted by an ideal spring (x-component of force)

• k = spring constant (units: N/m or kg/s²)

• x = displacement

An idealized spring exerts a restoring force that obeys _________, F_x = -kx. Oscillation with such a restoring force is called __________.

Hooke’s law , simple harmonic motion

The acceleration, a_x, of an object in SHM is:

a_x = d²x/dt² = -(k/m)x = -ω²x

• a_x = equation for SHM (x-component of acceleration)

• d²x/dt² = second derivative of displacement

• k = spring constant of restoring force

• m = mass of object

• ω = angular frequency (2πƒ)

• x = displacement

NOTE: the minus sign means that in SHM, acceleration and displacement always have opposite signs (acceleration is not constant)

In ________ real oscillations, Hooke’s law applies provided the object doesn’t ________. In such a case, _________ are approximately simple harmonic.

most , move too far from equilibrium , small-amplitude oscillations

State the general equation for the position of a harmonic oscillator as a function of a time.

x(t) = Acos(ωt + φ)

• x(t) = position of a harmonic oscillator at time t

• A = amplitude (maximum magnitude of displacement on either side of x = 0)

• ω = angular frequency (2πƒ or √(k/m)

• φ = phase constant (represents a horizontal shift of the graph; determined by initial displacement and velocity)

By taking the derivative of the general equation for x(t), the equations for the velocity and acceleration of an SHM are:

v(t) = dx/dt = -ωAsin(ωt + φ) = ±ω√(A² - x²)

• |v_max| = ωA

a(t) = d²x/dt² = -ω²Acos(ωt + φ) = -ω²x

When an object is at either its most positive displacement (x = +A) or its most negative displacement (x = -A), the velocity is _________ and the object is instantaneously at _________. Hence, the restoring force (F_x) and the acceleration of the object have their __________ magnitudes. At x = +A, a_x = _________ and at x = -A, a_x = __________.

zero , rest , maximum , -a_max , +a_max

Given an SHM’s initial displacement (x₀) and initial velocity (v₀) where t = 0, the equation to solve for φ is: ___________. φ should be in ___________.

φ = arctan(-v_0x / ωx₀)

radians

Given an SHM’s initial displacement (x₀) and initial velocity (v₀) where t = 0, the equation to solve for A is: ___________.

A = √(x₀² + (v_0x² / ω²))

State the equation for calculating the total mechanical energy (E) of SHM.

E = ½mv_x² + ½kx² = ½kA² = constant

• E = total mechanical energy in SHM

• m = mass

• v_x = velocity

• k = force constant of restoring force

• x = displacement

• A = amplitude

For angular SHM, a coil spring that exerts a restoring torque (τ_z) is represented by the equation ___________.

τ_z = -κθ = Iα

OR

d²θ / dt² = -(κ / I)θ

• τ_z = restoring torque

• κ = torsion constant

• θ = angular displacement

• I = moment of inertia

• α = angular acceleration

For describing angular SHM, the angular frequency (ω) and the frequency (ƒ) are given by the following equations _____________.

ω = √(κ / I)

• ω = angular frequency

• κ = torsion constant (analogous to spring constant)

• I = moment of inertia (analogous to mass)

ƒ = (1 / 2π) • √(κ / I)

• ƒ = frequency

• κ = torsion constant (analogous to spring constant)

• I = moment of inertia (analogous to mass)

The angular displacement θ as a function of time is given by __________.

θ = ϴcos(ωt + φ)

• θ = angular displacement

• ϴ = angular amplitude

• ω = angular frequency

• φ = phase constant

The angular frequency of a simple pendulum with small amplitude is _________.

ω = √(g / L)

• ω = angular frequency

• g = acceleration due to gravity

• L = pendulum length

What is a physical pendulum?

A real pendulum that has an irregular shape in contrast to a simple pendulum with all its mass concentrated at a point. In equilibrium, the CG is directly below the pivot and the distance between them is d. The axis of rotation is through the pivot with a moment of inertia (I). When the object is displaced, the weight (mg) causes a restoring torque.

The equation for the restoring torque in a physical pendulum is __________. From this, we can derive the equations for ω and T as ___________ and ___________.

τ_z = -(mgd)θ (can only use θ and not sinθ since the motion is is approx. SHM)

• τ_z = restoring torque

• m = mass of the pendulum

• g = acceleration due to gravity

• d = distance between pivot and center of gravity

• θ = angular displacement

ω = √(mgd / I)

T = 2π√(I / mgd)

• I = moment of inertia

We can use a ____________ to keep oscillations going. If we apply this periodically with angular frequency ω_d to a damped harmonic oscillator, the motion that results is called a ____________. There is an amplitude peak and driving frequencies close to the natural frequency of the system called ____________.

driving force , forced oscillation , resonance