M06-01 Simple Harmonic Motion

Simple Harmonic Motion

Simple harmonic motion is the motion generated by a linear restoring acceleration:

(1)

Equivalently,

(2)

The displacement

(3)

is measured from equilibrium.

The constant

(4)

is the angular frequency.

The defining physical feature is that acceleration is:

• proportional to displacement,
• opposite in direction to displacement.

The solution is sinusoidal, but the deeper structure is broader than a sine wave.

Because the equation of motion is second order, the complete instantaneous state requires both position and velocity:

(5)

Once

(6)

and

(7)

are specified, the equation of motion determines the complete future and past motion.

1 Spring-mass derivation

For an ideal spring,

(8)

where

(9)

is the spring constant.

Newton’s second law gives

(10)

Therefore

(11)

Define

(12)

Then

(13)

The angular frequency has dimensions

(14)

2 Why the force is called restoring

If

(15)

then

(16)

If

(17)

then

(18)

In either case, the force points toward

(19)

The equilibrium is therefore stable.

3 Solving the differential equation

Consider

(20)

A general real solution is

(21)

Differentiate:

(22)

Differentiate again:

(23)

Thus

(24)

as required.

4 Position, velocity, and acceleration phase relations

For the special form

(25)

the velocity is

(26)

and the acceleration is

(27)

Therefore:

• velocity is shifted by one-quarter cycle relative to displacement,
• acceleration is opposite displacement,
• velocity is largest in magnitude at equilibrium,
• acceleration is largest in magnitude at the turning points.

5 Period and frequency

The motion repeats when the phase increases by

(28)

Thus

(29)

Therefore

(30)

The ordinary frequency is

(31)

so

(32)

For the spring-mass oscillator,

(33)

The ideal linear oscillator period is independent of amplitude.

6 Initial conditions determine the motion

Let

(34)

Start with

(35)

At

(36)

(37)

From

(38)

we obtain

(39)

Therefore

(40)

The initial-value solution is

(41)

The corresponding velocity is

(42)

The factor

(43)

is required both by the initial condition and by dimensional consistency.

7 Amplitude from initial position and velocity

Write the solution in the form

(44)

Use the angle-addition identity:

x(t)

= A sin ωt cos ϕ + A cos ωt sin ϕ.

(45)

Compare with

(46)

Therefore

(47)

and

(48)

Square and add:

(49)

Hence

(50)

Thus the amplitude is determined by the complete initial state, not by initial position alone.

8 Dimensional check

The two terms under the square root must have the same dimensions.

Because

(51)

and

(52)

(53)

Therefore

(54)

has dimensions

(55)

as required for

(56)

Dimensional analysis provides a fast check on initial-condition formulas.

9 Phase from the initial state

From

(57)

and

(58)

the phase can be obtained with the quadrant-aware function

(59)

Using a one-argument inverse tangent can lose quadrant information.

Different sine or cosine phase conventions produce equivalent formulas with correspondingly shifted phase constants.

10 Example 1: amplitude and phase from an initial state

Let

(60)

with

(61)

Then

(62)

The amplitude is

A	=	(63)
	=	(64)
	≈ 0.361 m.	(65)

For the sine convention,

(66)

Thus

(67)

11 The state of the oscillator

The equation

(68)

is second order.

Define

(69)

Then it becomes two first-order equations:

ẋ = v,		(70)
v = −ω2x.		(71)

The pair

(72)

is therefore a complete state description.

At a given position, velocity distinguishes whether the oscillator is:

• moving toward increasing x,
• moving toward decreasing x,
• instantaneously at rest.

Position alone is not enough to specify the state.

12 State-space matrix form

Define the state vector

(73)

Then

(74)

The exact state evolution is

(75)

Thus the initial state uniquely determines the state at every later time.

13 The phase plane

Eliminate time using the amplitude relation.

For any time,

(76)

and

(77)

Therefore

(78)

Equivalently,

(79)

The trajectory in the

(80)

phase plane is an ellipse.

14 Scaled phase plane

Define a scaled velocity coordinate

(81)

Then

(82)

The phase-space ellipse becomes a circle.

The amplitude is simply the radius:

(83)

In particular, at

(84)

(85)

15 Direction of motion in phase space

At

(86)

the acceleration is

(87)

Therefore the state begins moving toward negative velocity.

In an

(88)

horizontal,

(89)

vertical phase plane, the standard SHO trajectory is traversed clockwise.

The direction matters: the same position coordinate can occur twice per cycle with opposite velocities.

16 Mechanical energy

For a spring-mass oscillator,

(90)

and

(91)

The total mechanical energy is

(92)

Use

(93)

Then

E	= m	(94)
	= mω2.	(95)

Since

(96)

(97)

Since

(98)

(99)

Therefore

(100)

Amplitude is therefore a direct measure of the conserved energy.

17 Energy from initial conditions

At

(101)

(102)

Set this equal to

(103)

Then

(104)

Since

(105)

(106)

This gives an independent energy derivation of the amplitude formula.

18 Exchange of kinetic and potential energy

Using

(107)

(108)

and

(109)

Therefore

(110)

Energy continually transfers between kinetic and potential forms while the total remains constant.

19 Maximum speed and acceleration

At equilibrium,

(111)

so all the energy is kinetic:

(112)

Thus

(113)

At a turning point,

(114)

so

(115)

Therefore

(116)

20 Example 2: speed from displacement

A spring-mass oscillator has

(117)

Find the speed when

(118)

From the phase-plane equation,

(119)

Thus

v2	= ω2(A2 − x2)	(120)
	= 25	(121)
	= 0.64.	(122)

Therefore

(123)

The sign of

(124)

depends on which half of the phase-plane ellipse the oscillator occupies.

21 Superposition and linearity

The SHO equation

(125)

is linear.

If

(126)

and

(127)

are solutions, then

(128)

is also a solution for constants

(129)

That is why

(130)

and

(131)

can serve as a basis for all real SHO solutions.

Linearity is also the foundation for normal modes, Fourier methods, and wave mechanics.

22 Complex exponential representation

Euler’s identity gives

(132)

Therefore a convenient complex trial solution is

(133)

Then

(134)

so

(135)

The physical displacement can be taken as the real part:

(136)

Complex notation becomes especially valuable for driven oscillators and waves.

23 Simple harmonic motion near a stable equilibrium

The harmonic oscillator is much more general than an ideal spring.

Suppose a one-dimensional system has potential energy

(137)

with a stable equilibrium at

(138)

At equilibrium,

(139)

For stability,

(140)

Let

(141)

Expand the potential near equilibrium:

(142)

Then

(143)

For mass

(144)

(145)

Therefore small motion near any smooth stable equilibrium is approximately simple harmonic, with

(146)

24 Small-angle pendulum

A pendulum of length

(147)

obeys

(148)

For small angles measured in radians,

(149)

Then

(150)

Therefore

(151)

and

(152)

The pendulum is therefore approximately a simple harmonic oscillator only in the small-angle regime.

25 Vertical spring and shifted equilibrium

For a vertical spring, gravity changes the equilibrium position but not the angular frequency of small ideal oscillations about that equilibrium.

Let downward displacement from the spring’s natural length be

(153)

Newton’s law is

(154)

The equilibrium extension satisfies

(155)

Define displacement from equilibrium:

(156)

Then

(157)

so

(158)

Gravity shifts the equilibrium but does not change the ideal linear frequency.

26 Canonical phase space

The state

(159)

is a useful mechanical state-space description.

In Hamiltonian mechanics the canonical momentum is

(160)

The canonical phase-space state is

(161)

The Hamiltonian is

(162)

Hamilton’s equations are

ẋ	= = ,	(163)
ṗ	= − = −kx.	(164)

Combining them gives

(165)

Thus the elementary SHO is also one of the simplest examples of Hamiltonian phase-space dynamics.

27 What amplitude and phase really represent

The constants

(166)

and

(167)

are another way to encode the same two pieces of state information carried by

(168)

and

(169)

In the scaled phase plane:

• amplitude is the radius of the orbit,
• phase is the angular position of the state on that orbit.

For the ideal conservative oscillator:

• amplitude remains constant,
• phase advances uniformly at rate ω.

This geometric interpretation becomes useful when damping, forcing, coupled oscillators, and nonlinear oscillations are introduced.

28 Common mistakes

1. Treating any periodic motion as simple harmonic motion.
2. Forgetting that SHM requires acceleration proportional to negative displacement.
3. Measuring displacement from the wrong equilibrium point.
4. Confusing angular frequency ω with ordinary frequency f.
5. Writing T = 1∕ω instead of T = 2π∕ω.
6. Forgetting the factor 1∕ω multiplying initial velocity in the initial-value solution.
7. Adding x₀² directly to v ₀² when computing amplitude.
8. Using a one-argument inverse tangent and losing the correct phase quadrant.
9. Assuming position alone completely specifies the oscillator state.
10. Forgetting the sign of velocity when using the phase-plane ellipse.
11. Assuming the phase-plane orbit is a circle in unscaled (x,v) coordinates.
12. Forgetting that kinetic and potential energies vary while total energy remains constant.
13. Applying the small-angle pendulum formula at large amplitude.
14. Assuming gravity changes the ideal vertical spring frequency instead of shifting equilibrium.
15. Confusing the mechanical state plane (x,v) with canonical phase space (x,p) without noting that p = mv.

29 Practice exercises

1. Starting from Hooke’s law and Newton’s second law, derive the SHO equation and identify ω.
2. Verify directly that x = C cos ωt + D sin ωt solves the SHO equation.
3. Starting from x(0) = x₀ and v(0) = v₀, derive the initial-value solution.
4. Derive A = by comparing sine and cosine coefficients.
5. Derive the same amplitude formula using conservation of energy.
6. Explain why x₀² + v ₀² is not a valid expression for A² if v ₀ is a velocity.
7. Derive the state-transition matrix for (x,v).
8. Eliminate time between x(t) and v(t) to derive the phase-plane ellipse.
9. Show that the scaled variables (x,v∕ω) move on a circle of radius A.
10. Derive v_max = ωA and a_max = ω²A.
11. For A = 0.20 m, ω = 5 rad/s, and x = 0.12 m, reproduce the speed in Example 2.
12. Show that K and U oscillate at twice the displacement angular frequency while their sum is constant.
13. Derive the SHO approximation for motion near a smooth stable equilibrium U(q).
14. Derive the small-angle pendulum angular frequency.
15. Derive Hamilton’s equations for the harmonic oscillator and recover the Newtonian equation of motion.

30 Summary

Simple harmonic motion is defined by

(170)

For a spring-mass system,

(171)

The initial state

(172)

determines the entire motion:

(173)

The amplitude is

(174)

The phase-plane orbit is

(175)

The total mechanical energy is

(176)

The same mathematics appears approximately near any smooth stable equilibrium.

This makes the harmonic oscillator one of the central local models of classical physics and the starting point for the broader study of oscillations.

References

References