GRE Physics Companion: Damped Harmonic Motion

GRE Physics Companion: Damped Harmonic Motion

The standard linear damped oscillator is

(1)

with

(2)

The characteristic roots are

(3)

1 High-value GRE facts

1. γ = b∕(2m) for the convention ẍ + 2γẋ + ω₀²x = 0.
2. ω₀ = is the undamped natural angular frequency.
3. Underdamped: γ < ω₀.
4. Critical: γ = ω₀.
5. Overdamped: γ > ω₀.
6. ω_d = in the underdamped regime.
7. The underdamped envelope decays as e^−γt.
8. critical damping requires b_c = 2.
9. mechanical energy satisfies dE∕dt = −bv².
10. logarithmic decrement is Λ = 2πγ∕ω_d.

Part I: Original GRE-style problems

Problem 1: damping rate

For

(4)

the damping rate in

(5)

is

1. b∕m
2. b∕(2m)
3. 2b∕m
4. k∕b
5. 

Problem 2: regime classification

A system has

(6)

It is

1. undamped
2. underdamped
3. critically damped
4. overdamped
5. unstable

Problem 3: damped frequency

For the system in Problem 2,

(7)

is

1. 2 rad/s
2. 3 rad/s
3. 4 rad/s
4. 5 rad/s
5. rad/s

Problem 4: amplitude decay

For an underdamped oscillator with

(8)

the amplitude envelope after

(9)

is multiplied by

1. e⁻²
2. e⁻¹
3. e^−1∕2
4. 2
5. 1

Problem 5: critical coefficient

For

(10)

critical damping requires

1. 5 kg/s
2. 10 kg/s
3. 20 kg/s
4. 25 kg/s
5. 100 kg/s

Problem 6: critical solution

The general critically damped displacement has the form

1. A cos ω₀t
2. Ae^−γt cos ω _dt
3. (C + Dt)e^−ω₀t
4. Ce^ω₀t + De^−ω₀t
5. C + Dt

Problem 7: energy loss rate

For viscous damping,

(11)

equals

1. +bv²
2. −bv²
3. −kx²
4. −mgv
5. zero

Problem 8: phase plane

The underdamped phase-plane trajectory is approximately

1. a closed ellipse
2. an outward spiral
3. an inward spiral
4. a straight line
5. a hyperbola

Problem 9: logarithmic decrement

Successive same-sign peaks are

(12)

The logarithmic decrement is

1. 1∕2
2. ln 2
3. 2
4. e²
5. π

Problem 10: overdamped roots

For an overdamped oscillator, the characteristic roots are

1. purely imaginary
2. complex conjugates with negative real part
3. two distinct negative real numbers
4. one positive and one negative number
5. both zero

Problem 11: weak-damping quality factor

For weak viscous damping,

(13)

is approximately

1. 2γ∕ω₀
2. ω₀∕(2γ)
3. γω₀
4. ω₀²∕γ
5. 1

Problem 12: damping and period

For an underdamped oscillator,

1. ω_d > ω₀
2. ω_d = ω₀ for all damping
3. ω_d < ω₀
4. ω_d = 0 always
5. ω_d is imaginary

Part II: Complete worked solutions

Solution 1

Dividing by mass gives

(14)

Therefore

(15)

so

(16)

Answer: (B).

Solution 2

Because

(17)

(18)

The system is underdamped. Answer: (B).

Solution 3

ωd	=	(19)
	=	(20)
	= 4 rad∕s.	(21)

Answer: (C).

Solution 4

The envelope factor is

(22)

Answer: (B).

Solution 5

bc	= 2	(23)
	= 2	(24)
	= 20 kg∕s.	(25)

Answer: (C).

Solution 6

A repeated root produces

(26)

Answer: (C).

Solution 7

The damping force power is

(27)

Thus

(28)

Answer: (B).

Solution 8

Damping removes energy, so the closed undamped phase-plane orbit contracts toward the origin. In the underdamped case it spirals inward. Answer: (C).

Solution 9

(29)

Answer: (B).

Solution 10

For

(30)

(31)

is real and smaller than

(32)

Thus both roots are real, distinct, and negative. Answer: (C).

Solution 11

For weak damping,

(33)

Answer: (B).

Solution 12

(34)

Answer: (C).

2 GRE checklist

1. Compute ω₀ and γ before selecting a solution form.
2. Compare γ directly with ω₀.
3. Keep the convention 2γ = b∕m consistent.
4. For underdamping, use ω_d = .
5. For critical damping, remember the extra factor of t in the second independent solution.
6. For overdamping, expect two decaying exponentials, not a sinusoid.
7. Use dE∕dt = −bv² as a sign and physics check.
8. Use logarithmic decrement only with corresponding same-sign peaks.

References

References