Split Harmonic Motion into Two Harmonic Motions

Answer

Solution

The given harmonic motion is:

x = 10 sin(ωt + π/6)

Since

π/6 = 30°

Therefore,

x = 10 sin(ωt + 30°)

Step 1: Write the two harmonic motions

The first motion has a phase angle of 0°:

x₁ = A sin(ωt)

The second motion has a phase angle of 46°:

x₂ = B sin(ωt + 46°)

Since the original motion is the sum of these two motions:

x = x₁ + x₂

Therefore,

10 sin(ωt + 30°) = A sin(ωt) + B sin(ωt + 46°)

Step 2: Expand the left-hand side

Using the identity:

sin(α + β) = sinα cosβ + cosα sinβ

We get:

10 sin(ωt + 30°) = 10[sin(ωt) cos30° + cos(ωt) sin30°]

Since:

cos30° = √3/2

sin30° = 1/2

Therefore:

10 sin(ωt + 30°) = 5√3 sin(ωt) + 5 cos(ωt)

Step 3: Expand the second motion

We have:

B sin(ωt + 46°)

Using the same identity:

B sin(ωt + 46°) = B[sin(ωt) cos46° + cos(ωt) sin46°]

Therefore:

x₁ + x₂ = A sin(ωt) + B cos46° sin(ωt) + B sin46° cos(ωt)

Grouping the terms:

x₁ + x₂ = (A + B cos46°) sin(ωt) + B sin46° cos(ωt)

Step 4: Compare the coefficients

We have:

5√3 sin(ωt) + 5 cos(ωt)

Comparing the coefficients of sin(ωt):

A + B cos46° = 5√3

Comparing the coefficients of cos(ωt):

B sin46° = 5

Step 5: Find B

From:

B sin46° = 5

Therefore:

B = 5 / sin46°

Using:

sin46° ≈ 0.71934

We get:

B ≈ 6.951

Step 6: Find A

From:

A + B cos46° = 5√3

Therefore:

A = 5√3 - B cos46°

Substituting the value of B:

A = 8.6603 - (6.951)(0.6940)

Therefore:

A ≈ 3.832

Final Answer

The first harmonic motion having phase angle 0° is:

x₁ ≈ 3.832 sin(ωt)

The second harmonic motion having phase angle 46° is:

x₂ ≈ 6.951 sin(ωt + 46°)

Hence:

x ≈ 3.832 sin(ωt) + 6.951 sin(ωt + 46°)

which is equivalent to:

x = 10 sin(ωt + 30°)