Radial Distance Variation Quiz: Modeling Radial Distance Variations

  • 11th Grade
Reviewed by Editorial Team
The ProProfs editorial team is comprised of experienced subject matter experts. They've collectively created over 10,000 quizzes and lessons, serving over 100 million users. Our team includes in-house content moderators and subject matter experts, as well as a global network of rigorously trained contributors. All adhere to our comprehensive editorial guidelines, ensuring the delivery of high-quality content.
Learn about Our Editorial Process
| By Thames
T
Thames
Community Contributor
Quizzes Created: 7682 | Total Attempts: 9,547,133
| Questions: 20 | Updated: Dec 17, 2025
Please wait...
Question 1 / 20
0 %
0/100
Score 0/100
1) For r(t) = 7000 + 50 cos(0.1 t), what is the angular frequency ω (rad/s)?

Explanation

The factor multiplying t inside the cosine is ω. Here ω = 0.1 rad/s.

Submit
Please wait...
About This Quiz
Radial Distance Variation Quiz: Modeling Radial Distance Variations - Quiz

How does radial distance change when an object moves along a curved path? In this quiz, you’ll explore functions that describe how the distance from a center point varies over time or angle. You’ll practice interpreting graphs, analyzing increasing and decreasing intervals, and connecting each pattern to orbital or rotational... see morebehavior. By examining how radius changes dynamically, you’ll gain insight into real-world systems—like satellites or spinning objects—where motion depends on shifting distances.
see less

2)
You may optionally provide this to label your report, leaderboard, or certificate.
2) For r(t) = 7000 + 50 cos(0.1 t), what is the period T?

Explanation

T = 2π/|ω| = 2π/0.1 = 20π seconds. Numerically 20π ≈ 62.832.

Submit
3) For r(t) = 7000 + 50 cos(0.1 t), what is the maximum radial distance?

Explanation

Max occurs when cos(0.1 t) = 1, so r_max = R0 + A = 7000 + 50 = 7050.

Submit
4) For r(t) = 7000 + 50 cos(0.1 t), what is the minimum radial distance?

Explanation

Min occurs when cos(0.1 t) = −1, so r_min = R0 − A = 7000 − 50 = 6950.

Submit
5) For r(t) = 7000 + 50 cos(0.2 t + π), the first t ≥ 0 when r(t) is maximum is ____.

Explanation

Max when the cosine’s argument equals 2πk: 0.2 t + π = 2πk. Smallest nonnegative solution uses k = 1: 0.2 t + π = 2π ⇒ 0.2 t = π ⇒ t = π/0.2 = 5π.

Submit
6) In r(t) = R0 + A cos(ωt + φ), the amplitude A measures the wobble size around the average radius.

Explanation

The sinusoidal term oscillates between −A and +A, so the distance varies by ±A from the average R0.

Submit
7) Which changes alter the average (central) radius? Select all that apply.

Explanation

The cycle average is R0 (mean of cosine is 0). Increasing R0 directly increases the average. Adding +c to the whole function shifts the average to R0 + c. Changing A, ω, or φ does not change the average.

Submit
8) For r(t) = 9000 + 30 sin(0.05 t), what is the wobble amplitude?

Explanation

Amplitude equals the coefficient of the sinusoid: A = 30.

Submit
9) For r(t) = 9000 + 30 sin(0.05 t), what is the period?

Explanation

T = 2π/|ω| with ω = 0.05 ⇒ T = 2π/0.05 = 40π seconds (≈ 125.664).

Submit
10) For r(t) = 7000 + 50 cos(0.1 t), when is the first minimum (t > 0)?

Explanation

Min when cos(0.1 t) = −1 ⇒ 0.1 t = π ⇒ t = π/0.1 = 10π seconds.

Submit
11) For r(t) = R0 + A cos(ωt), the average of r over one full period equals ____.

Explanation

Over one period, ∫ cos(ωt) dt averages to 0, so the mean of r(t) is R0.

Submit
12) For near-circular motion with r(t) = R0 + A cos(ωt + φ) and A ≪ R0, which statements are true? Select all that apply.

Explanation

Max and min follow from cosine’s range ±A. Percent wobble is amplitude divided by central value. Period depends only on ω. Radial speed is |dr/dt|_max = |−Aω sin(…)|_max = A|ω|.

Submit
13) Replacing cos(ωt) by cos(ωt + φ) changes the average radius over time.

Explanation

Phase φ shifts the oscillation horizontally but does not change its mean value. The average remains R0.

Submit
14) For r(t) = 7500 + 40 cos(0.2 t − π/3), what is the phase φ?

Explanation

In r(t) = R0 + A cos(ωt + φ), the term added to ωt is φ. Here φ = −π/3.

Submit
15) At t = 0 for r(t) = R0 + A cos(ωt + φ), the radial distance equals:

Explanation

Substitute t = 0 into r(t): r(0) = R0 + A cos(φ).

Submit
16) For r(t) = 8000 + 20 cos(0.4 t), compute r(5).

Explanation

Angle = 0.4·5 = 2 rad. cos(2) ≈ −0.41614684. Then r = 8000 + 20(−0.41614684) ≈ 8000 − 8.3229368 = 7991.6770632 ≈ 7991.6771.

Submit
17) Which modifications leave the period unchanged? Select all that apply.

Explanation

T = 2π/|ω| depends only on ω. Changing R0, φ, A, or adding a constant does not affect period. Decreasing ω increases T, so that change does not keep period unchanged.

Submit
18) If A = 0 in r(t) = R0 + A cos(ωt), the radial distance is constant and equals R0.

Explanation

With A = 0, r(t) = R0 for all t, so the distance does not vary.

Submit
19) For r(t) = 7000 + 50 cos(0.1 t), what is the average (central) orbit radius?

Explanation

In r(t) = R0 + A cos(ωt + φ), the average over a full cycle is R0 because the mean of cos over any whole number of periods is 0. Here R0 = 7000.

Submit
20) For r(t) = 7000 + 50 cos(0.1 t), what is the wobble amplitude?

Explanation

The amplitude is the coefficient of the sinusoid, A = 50. It sets the maximum deviation from the average radius.

Submit
×
Saved
Thank you for your feedback!
View My Results
Cancel
  • All
    All (20)
  • Unanswered
    Unanswered ()
  • Answered
    Answered ()
For r(t) = 7000 + 50 cos(0.1 t), what is the angular frequency ω...
For r(t) = 7000 + 50 cos(0.1 t), what is the period T?
For r(t) = 7000 + 50 cos(0.1 t), what is the maximum radial distance?
For r(t) = 7000 + 50 cos(0.1 t), what is the minimum radial distance?
For r(t) = 7000 + 50 cos(0.2 t + π), the first t ≥ 0 when r(t) is...
In r(t) = R0 + A cos(ωt + φ), the amplitude A measures the wobble...
Which changes alter the average (central) radius? Select all that...
For r(t) = 9000 + 30 sin(0.05 t), what is the wobble amplitude?
For r(t) = 9000 + 30 sin(0.05 t), what is the period?
For r(t) = 7000 + 50 cos(0.1 t), when is the first minimum (t > 0)?
For r(t) = R0 + A cos(ωt), the average of r over one full period...
For near-circular motion with r(t) = R0 + A cos(ωt + φ) and A ≪...
Replacing cos(ωt) by cos(ωt + φ) changes the average radius over...
For r(t) = 7500 + 40 cos(0.2 t − π/3), what is the phase φ?
At t = 0 for r(t) = R0 + A cos(ωt + φ), the radial distance equals:
For r(t) = 8000 + 20 cos(0.4 t), compute r(5).
Which modifications leave the period unchanged? Select all that apply.
If A = 0 in r(t) = R0 + A cos(ωt), the radial distance is constant...
For r(t) = 7000 + 50 cos(0.1 t), what is the average (central) orbit...
For r(t) = 7000 + 50 cos(0.1 t), what is the wobble amplitude?
Alert!

Advertisement