Radial Distance Variation Quiz: Modeling Radial Distance Variations

T = 2π/ω = 2π/0.05 = 40π seconds ≈ 125.66 seconds. Option A gives π/0.05 = 20π which incorrectly uses π instead of 2π in the numerator. Option B gives 20π corresponding to ω = 0.1. Option C gives 10π corresponding to ω = 0.2. Only 40π correctly applies T = 2π/0.05.

The answer is True. With A = 0 the sinusoidal term vanishes and r(t) = R0 for all t. The radial distance no longer varies with time and the orbit becomes a perfect circle of constant radius R0. This represents the special case where there is no wobble and the satellite maintains a fixed distance from the central body at all times.

T = 2π/ω depends only on ω. Increasing R0 shifts the central radius without touching ω, so T is unchanged, confirming A. Changing phase φ only shifts the oscillation horizontally in time without affecting ω or T, confirming C. Doubling A increases wobble size but leaves ω and therefore T completely unchanged, confirming D. Option B directly reduces ω which increases T = 2π/ω, so the period is not preserved.

Angle = 0.4×5 = 2 radians. cos(2) ≈ -0.41615. r = 8000 + 20×(-0.41615) = 8000 - 8.323 ≈ 7991.677. Option B gives 7980 = 8000 - 20, requiring cos(2) = -1. Option C gives 8020 = 8000 + 20, requiring cos(2) = 1. Option D gives 8008.32, requiring cos(2) ≈ +0.416, the wrong sign.

Substituting t = 0 gives r(0) = R0 + A×cos(φ). Option B replaces cos(φ) with ω, mixing angular frequency into the amplitude term incorrectly. Option C replaces A with ω, using the wrong coefficient. Option D negates the cosine term without justification. Only option A correctly evaluates the function at t = 0.

In r(t) = R0 + A×cos(ωt + φ), the phase φ is the constant added to ωt inside the cosine. Here the argument is 0.2t - π/3, so φ = -π/3. Option B gives +π/3, the wrong sign. Option C gives ω = 0.2, the angular frequency. Option D gives A = 40, the amplitude.

The answer is False. Adding a phase φ shifts the oscillation horizontally in time but does not change its mean value. Cosine still oscillates symmetrically between -1 and +1 regardless of phase, so its average over a full period remains 0. The average of r(t) therefore stays at R0 whether or not a phase term is present.

Maximum occurs when cosine = 1 giving R0 + A, confirming A. Minimum occurs when cosine = -1 giving R0 - A, confirming B. T = 2π/ω depends only on ω and is completely independent of R0 and A, confirming C. Option D is false — percent wobble = 100×A/R0, meaning amplitude divided by central radius. Option D inverts this ratio giving 100×R0/A which produces a very large number, not a small percentage.

Over one full period the integral of cos(ωt) averages to 0. Therefore the average of r(t) = R0 + A×cos(ωt) equals R0 + A×0 = R0. Option A gives just the amplitude. Option B gives the maximum value, not the average. Option C gives 0, ignoring R0 entirely. Only R0 correctly represents the time-averaged central radius.

Minimum occurs when cos(0.1t) = -1, meaning 0.1t = π, giving t = π/0.1 = 10π seconds. Option B gives 5π, where 0.1×5π = π/2, giving cos(π/2) = 0, not the minimum. Option C gives π, where 0.1×π = π/10, also not the minimum. Option D gives 20π, the full period, which is the second minimum occurrence.

In r(t) = R0 + A×cos(ωt + φ), the average over a full cycle equals R0 because the mean of cosine over any whole number of periods is 0. Here R0 = 7000. Option B gives the amplitude A = 50. Option C gives the angular frequency ω = 0.1. Option D gives half the central radius, which has no geometric meaning here.

The amplitude equals the coefficient of the sinusoidal term regardless of whether sine or cosine is used. Here A = 30. Option B gives R0 = 9000, the central radius. Option C gives ω = 0.05, the angular frequency. Option D gives R0 + A = 9030, the maximum distance, not the amplitude.

The cycle average of r(t) equals R0 because the mean of cosine over a full period is 0. Directly increasing R0 raises the average, confirming A. Adding a constant +c shifts the average to R0 + c, confirming D. Changing A only affects wobble size. Changing ω only affects period. Changing φ only shifts the oscillation horizontally. None of B or C affect the average.

The answer is True. The sinusoidal term A×cos(ωt + φ) oscillates between -A and +A. Adding R0 shifts this range to between R0 - A and R0 + A. The radial distance therefore varies by exactly ±A from the central value R0. A larger amplitude means greater wobble while A = 0 produces a perfectly circular orbit with constant radius R0.

Maximum occurs when cos(0.2t + π) = 1, meaning 0.2t + π = 2πk. The smallest nonneg solution uses k = 1: 0.2t = π, so t = 5π. Option B gives 10π corresponding to k = 2. At option C t = π the argument = 0.2π + π = 1.2π and cos(1.2π) ≠ 1. Option D gives 2.5π which does not satisfy the equation.

Minimum occurs when cos(0.1t) = -1, giving r_min = R0 - A = 7000 - 50 = 6950. Option B gives R0 = 7000, the average radius. Option C gives R0 + A = 7050, the maximum distance. Option D gives R0 - 2A = 6900, subtracting the amplitude twice incorrectly.

Maximum occurs when cos(0.1t) = 1, giving r_max = R0 + A = 7000 + 50 = 7050. Option A gives R0, the average radius. Option B gives R0 - A = 6950, the minimum distance. Option D gives R0 + 2A = 7100, adding the amplitude twice incorrectly.

T = 2π/ω = 2π/0.1 = 20π seconds ≈ 62.83 seconds. Option B gives 2π which requires ω = 1, not 0.1. Option C gives 10π which requires ω = 0.2. Option D gives π which requires ω = 2. Only 20π correctly applies the period formula with ω = 0.1.

The angular frequency ω is the coefficient of t inside the cosine. Here the argument is 0.1t, so ω = 0.1 rad/s. Option B inverts the value giving 1/ω = 10. Option C gives 2π, which is the factor used in the period formula but is not ω here. Option D gives ω squared = 0.01.

The amplitude is the coefficient of the sinusoidal term, A = 50. It sets the maximum deviation from the average radius R0. Option A gives R0 = 7000, the central radius. Option B gives ω = 0.1, the angular frequency. Option D gives R0 + A = 7050, the maximum radial distance, not the amplitude itself.