Chapter 1 - Simple Harmonic Motion

Question 1

How would you derive the equation of simple harmonic motion, and give the equation.

Answer 1

1) Start with F = -sx
2) mx^.. = -sx
3) mx^.. + sx = 0
4) x^.. + s/mx = 0

let ω² = 2πv = s/m since s/m = v²

(v = period)

The Wave equation is: x^.. + ω²x = 0.

Question 2

What is the general solution of the equation for SHM:

x^.. + ω²x = 0

Answer 2

x = A cosωt + B sinωt

with

x^.. = -ω²(A cosωt + B sinωt) = -ω²x

Where A and B are determined by the initial conditions, and

A = a sinϕ and B = a cosϕ

Gives:

x = a sin (ωt + ϕ)

Question 3

Describe each term in the equation:

x = a sin (ωt + ϕ)

Answer 3

1. a is the amplitude of displacement, determined by the maximum energy.
2. (ωt + ϕ) is called the phase, and varies with time.
3. ω is the angular frequency. Measured in radians per second.
4. ϕ is the phase constant. Measured in radians, defines the position in the cycle of oscillation at time t = 0. The phase constant allows the motion to be defined from any starting point in the cycle.

Question 4

What is the values of the velocity and acceleration in SHM for:

x = a sin (ωt + ϕ)

Answer 4

Velocity

dx/dt = aω cos (ωt + ϕ)

aω = velocity amplitude.

Acceleration

d²x/dt² = -aω² sin (ωt + ϕ)

aω² = acceleration amplitude.

Remeber the - ! !

Question 5

Describe the relationship between the position, velocity and acceleration of SHM.

Answer 5

The velocity leads the displacement by a phase angle of π/2 rad and its maxima and minima are always a quarter of a cycle ahead of those of the displacement.

The velocity is maximum when the displacement is zero and is zero at maximum displacement.

The acceleration is ‘anti-phase’ (π rad) with respect to displacement.

Question 6

Define the concepts of ‘in phase’ and ‘anti-phase’

Answer 6

• If the one system is at x = +a whilst the other is at x = -a, the systems are ‘anti-phase’ and the total phase difference is:

ϕ1 – ϕ2 = nπ rad

n is an odd integer.

• Identical system are ‘in phase’ and have:

ϕ1 – ϕ2 = 2nπ rad

n is any integer.

Question 7

Give the equation of the total energy of a SHM and the total energy at any instant of time.

Answer 7

The total energy is given by:

E = ½ mẋ ² + ½ sx²

Remember the total energy is kinetic and potential energy and

x = a sin (ωt + ϕ) and s = ω²m

The total energy at any instant is

E = ½sa²

Question 8

Give the equivalent Oscillating electrical system term of each SHM term below:

1. Stiffness (s)
2. Force (F) equation (mx^.. + sx = 0)
3. Kinetic and Potential energy

Answer 8

1. Stiffness (s) = The compliance (Where s = 1/C)
2. The voltage equation (v) (Lq^.. + q/C = 0)
3. Magnetic field energy (Inductance) and Electric field energy (Capacitance)

Question 9

What is the equivelant equation in an oscillating electrical system of the equation of SHM,

x ̇ ̇ + ω²x = 0?

Answer 9

Lq^.. + q/c = 0

or

q^.. + ω² q = 0

where

ω² = 1/LC

Question 10

Give the equations for the:

1. Magnetic field inertia (Inductance)
2. Electric field energy (Capacitance)
3. Total Energy

Answer 10

Inductance

E_L = ½LI² = ½Lq^{. 2}

Capacitance

Ec = ½CV² = ½q²/c

Total energy

E = ½Lq^.2 + ½q²/c

Question 11

1. How would you find the resulting motion of a system moving in the x direction under the simultanious effect of two simple harmonic oscillations of equal angular frequencies but different amplitudes and phases.
2. Give the equation of the displacement of the resulting SHM formed by:

x₁ = a₁ cos (ωt + ϕ₁)

and x₂ = a₂ cos (ωt + ϕ₂)

Answer 11

1. Represent each simple harmonic motion by its appropriate vector and carry out a vector addition.
2. x = R cos (ωt + θ)

where

R = a₁² + a₂² + 2a₁a₂ cos δ

where δ = ϕ₁ - ϕ₂

and

Tanθ = a₁Sinϕ₁ + a₂Sinϕ₂

 a<sub>2</sub>Cosϕ<sub>1</sub> + a<sub>2</sub>Cosϕ<sub>2</sub>

Question 12

1. Suppose two vibrations of equal amplitudes but different frequencies are superposed, where

x₁ = a sin ω₁t and x₂ = a sin ω₂t.

(ω₂ > ω₁)

Give in words the resulting displacement.

 2. Give the equation of the displacement.

Answer 12

1. It is a sinusoidal oscillations at the average frequency ½ (ω₁ + ω₂) having a displacement amplitude of 2a which modulates between 2a and zero under the influence of the cosine term of a much slower frequency equal to halve the difference ½ (ω₂ - ω₁) between the original frequencies.
2. x = x₁ + x₂ = 2a Sin ½ (ω₁ + ω₂)t Cos ½ (ω₂ – ω₁)t

Question 13

Give the equation of motion for two perpendicular waves.

and

Describe the motion for different phase constants.

Answer 13

x² + y² – 2xy cos(ϕ₂ – ϕ₁) = sin²(ϕ₂ – ϕ₁)

             a<sub>1</sub><sup>2</sup>    a<sub>2</sub><sup>2</sup>  a<sub>1</sub>a<sub>2</sub>

1. If ϕ₂ – ϕ₁ = π/2, an ellipse with semi-axes a₁ and a₂
2. If ϕ₂ – ϕ₁ = 0, 2π, 4π etc, a straight line with slope a₂\a_1.
3. If ϕ₂ – ϕ₁ = π, 3π etc, a straight line with slope -a₂\a_1.