harmonics

What is oscillatory motion?

It is a repetitive back-and-forth motion about an equilibrium position.

What are the two common features of all oscillatory motions?

The motion occurs about an equilibrium position and it is periodic.

What is the Period (T) of an oscillation?

It is the time required to complete one full cycle of motion.

What is the Frequency (f) of an oscillation?

It is the number of cycles that occur per second.

What is the mathematical relationship between frequency (f) and period (T)?

f = 1/T

What defines the equilibrium position in an oscillatory system?

It is the position where there is no net force acting on the object.

In a block-spring system, the spring force always pulls the block toward the _.

equilibrium position

For an ideal spring, the restoring force is described by Hooke's Law:

F = −kx

What is Simple Harmonic Motion (SHM)?

An oscillation where the restoring force is directly proportional to the displacement from equilibrium.

In SHM, the displacement x(t) is a _ function.

sinusoidal

Is a cart on a frictionless track hitting bumpers undergoing SHM?

No — the restoring force is not proportional to displacement; force occurs only during collisions.

Is a perfectly bouncing ball undergoing SHM?

No — acceleration between bounces is constant (gravity), not proportional to displacement.

What are the three constants in the SHM equation?

Amplitude A, Angular Frequency ω, and Initial Phase φ

Term: Amplitude (A)

The maximum displacement from equilibrium.

In the SHM equation, (ωt + φ) is called the _.

phase

One full SHM cycle is completed when the phase increases by _.

2π radians

The phase constant φ determines the object's _ at t=0.

initial position

A weight oscillates between 30 cm and 50 cm. What is the amplitude?

10 cm

What is the total distance traveled in one SHM cycle with amplitude A?

4A

If x(t) = A cos(ωt + φ), what is v(t)?

v(t) = −ωA sin(ωt + φ)

What is the maximum speed v_max in SHM?

v_max = ωA

At what point in its motion is SHM speed maximum?

At equilibrium (x = 0)

What is the relationship between acceleration a and displacement x in SHM?

a = −ω² x

What is the maximum acceleration a_max in SHM?

a_max = ω² A

At what positions is acceleration maximum in SHM?

At x = +A and x = −A

For a mass-spring system, Newton’s 2nd Law gives the equation:

−kx = ma

The frequency and period of a mass-spring system depend on mass and spring constant, but not on the _.

amplitude

Given x0 and v0 at t=0, what is φ?

φ = tan⁻¹(−v0 / (ω x0))

Given x0 and v0 at t=0, what is amplitude A?

A = √( x0² + (v0/ω)² )

Why is a simple pendulum not strictly SHM?

Because its restoring force is proportional to sinθ, not θ.

Under what approximation is a pendulum SHM?

For small angles where sinθ ≈ θ.

If an SHM object is released from rest at x = +A at t=0, what is φ?

φ = 0

If an SHM object starts at x = 0 at t=0 with positive velocity, a possible φ is _.

φ = −π/2 (or −90°)

In SHM, when x = +A, the acceleration is _.

maximum negative (−a_max)

For a given mass, increasing k results in a _ frequency.

higher

For a given spring, increasing mass results in a _ period.

longer

A block floating on water experiences F = −ρAg x. What does ρAg act as?

the effective spring constant k

If x(t) = A sin(ωt), what is φ in x(t) = A cos(ωt + φ)?

φ = −π/2

When an SHM object passes equilibrium, velocity is maximum and acceleration is _.

zero

If amplitude doubles, how does period change?

The period stays the same.