250+ TOP MCQs on Harmonic Motion and Mass – Spring System and Answers

Ordinary Differential Equations MCQs focuses on “Harmonic Motion and Mass – Spring System”.

1. A particle moves along the x-axis according to the law (frac{d^2 x}{dt^2} + 6 frac{dx}{dt} + 25x = 0.) If the particle is started at x=0 with an initial velocity of 12ft/sec to the left, determine x in terms of t.
a) x=-3e^-3t sin⁡4t
b) x=-e^-3t (sin⁡4t+cos 4t)
c) x=-3e^-3t (sin⁡4t+cos⁡4t)
d) x=-3e^-3t cos⁡4t
Answer: a
Explanation: x” (t) + 6x’ (t) + 25x(t) = 0 and at t=0, (frac{dx}{dt}) = -12….since it is moving to the left.
m²+6m+25=0…..A.E
(m=frac{-6±sqrt{6^2-4*25}}{2} = frac{-6±8i}{2} = -3±4i)
complementary solution x_c = e^-3t (c₁ cos⁡4t + c₂ sin⁡4t)
at t=0, x=0 therefore c₁ = 0 –> x_c = e^-3t c₂ sin⁡4t
differentiating (frac{dx_c}{dt}) = -3e^-3t c₂ sin⁡4t + 4e^-3t c₂ cos⁡4t
at t=0 (frac{dx_c}{dt}) = -12… given R.H.S = 4c₂ –> c₂ = -3
x_c = x = -3e^-3t sin⁡4t……as there is no particular solution.

2. Solve the problem of resonance damped vibration of a spring .If the governing D.E is given by (m frac{d^2 y}{dt^2} + c frac{dy}{dt} + ky=0;) c>0 with initial conditions as y(0)=y₀ and y’(0)=y₁ and assume c/m=2λ, k/m=μ² and (v = sqrt{μ^2-λ^2}).
a) y = e^-λt (y₀ cos⁡vt + y₁ sin⁡vt)
b) y = e^(- λt) (y₀ cos⁡vt+(left(frac{y_1+ λy_0}{v}right)) sin⁡vt)
c) y = e^(- λt) ((left(frac{y_1+ λy_0}{v}right)) cos⁡vt + y₀ sin⁡vt)
d) y = e^-λt (y₀ cos⁡vt+λy₁ sin⁡vt)
Answer: b
Explanation: The governing D.E is given by (m frac{d^2 y}{dt^2} + c frac{dy}{dt} + ky=0;) c>0
–> ( frac{d^2 y}{dt^2} + frac{c}{m} frac{dy}{dt} + frac{k}{m}y = 0) given c/m=2λ, k/m=μ² thus D.E takes the form
( frac{d^2 y}{dt^2} + 2λ dfrac{dy}{dt} + μ^2 y = 0) it’s A.E is m² + 2λm+ μ² = 0
(m=frac{-2λ±sqrt{4λ^2-4μ^2}}{2} = – λ±vi)…. given (v = sqrt{μ^2-λ^2})
y_c = y = e^{– λt} (c₁ cos⁡vt + c₂ sin⁡vt ) to find constants we use initial condition
at t=0 y = y₀ –> c₁ = y₀ therefore y = e^-λt (y₀ cos⁡vt + c₂ sin⁡vt)
now differentiate and substituting y’(0)=y₁
y’ = – λe^{– λt} (y₀ cos⁡vt + c₂ sin⁡vt) + e^{– λt} (-y₀ vsin⁡vt + vc₂ cos⁡vt)
at t=0, y₁ = – λy₀ + vc₂ –> c₂ = (left(frac{y_1+ λy_0}{v}right)) thus y=e^{– λt} (y₀ cos⁡vt+(left(frac{y_1+ λy_0}{v}right)) sin⁡vt).

3. Solve the problem of un-damped forced vibrations of a spring in the case where the forcing function is f(t)=A sin ωt. D.E associated with the problem is (m frac{d^2 y}{dt^2} + ky = f(t)), with initial conditions as y(0)=y₀ and y’(0)=y₁ and assume λ² = k/m, μ=A/m.
a) y = y₀ cos⁡λt + y₁ sin⁡λt + (frac{λ cos ωt}{-ω^2+λ^2})
b) y = y₀ cos⁡λt + (y₁/μ)sin⁡λt + (frac{cos ωt}{ω^2+λ^2})
c) y = y₀ cos⁡λt + (y₁ λ)sin⁡λt +(frac{sin ωt}{ω^2+λ^2})
d) y = y₀ cos⁡λt + (y₁/λ)sin⁡λt + (frac{μ sin ωt}{-ω^2+λ^2})
Answer: d
Explanation: (m frac{d^2 y}{dt^2} + ky = A ,sin ,ωt, ,or, frac{d^2 y}{dt^2} + frac{k}{m} y = frac{A}{m} sin ωt rightarrow frac{d^2 y}{dt^2} + λ^2 y = μ sin ωt)
A.E is m² + λ² = 0 –> m = ±λi –> y = c₁ cos⁡λt + c₂ sin⁡λt at t=0 y=y₀
–> c₁ = y₀ therefore y_c = y₀ cos⁡λt + c₂ sin⁡λt
now differentiate and substituting y’(0)=y₁
y’ = -y₀ λsin⁡λt + λc₂ cos⁡λt at t = 0 y₁/λ = c₂ and y_c = y₀ cos⁡λt + (y₁/λ)sin⁡λt
to find particular solution (y_p = frac{μ sin ωt}{D^2+λ^2}) assuming μ as constant (y_p = frac{μ sin ωt}{-ω^2+λ^2})
∴y=y₀ cos⁡λt + (y₁/λ)sin⁡λt + (frac{μ sin ωt}{-ω^2+λ^2}).

4. A particle undergoes forced vibrations according to the law x”(t) + 25x(t) = 21 sin⁡t. If the particle starts from rest at t=0, find the displacement at any time t>0.
a) (frac{21 cos t}{25})
b) (frac{7 sin t}{8})
c) (frac{3 cos t}{8})
d) (frac{7 sin t}{9})
Answer: b
Explanation: x” (t) + 25x(t) = 21 cos⁡t comparing it with ( frac{d^2 y}{dt^2} + frac{k}{m} y= frac{A}{m} sin ωt)
( frac{d^2 y}{dt^2} + λ^2 y = μ sin ,ωt) where λ²=k/m, μ=A/m and its solution is
y=y₀ cos⁡λt + (y₁/λ)sin⁡λt + ( frac{μ sin ωt}{-ω^2+λ^2}) where y(0)=y₀ and y’(0)=y₁.
we got λ² = 25 –> λ=5 and μ=21, ω=1. Its solution is
x = x₀ cos⁡5t + (x₁/5)sin⁡5t + (frac{21 sin t}{-1^2+25}) particle is at rest at t = 0 –> x(0) = 0 = x₀
and x’(0) = 0 = x₁ therefore (x = frac{21 sin t}{24} = frac{7 sin t}{8}.)

Global Education & Learning Series – Ordinary Differential Equations.

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