How many times per minute does an object pass equilibrium? (Diff Eqs)

In summary, the conversation discusses the correct approach to solving part b of a problem involving a linear frequency and angular frequency. The correct approach involves considering the period and setting the equation equal to zero. The confusion arises from incorrectly interpreting the frequency as linear instead of angular. The expert explains that the displacement changes sinusoidally and therefore has an angular aspect to it. This is reflected in the units used for time and angular frequency.
  • #1
rugerts
153
11
Homework Statement
You place a 4 pound cauliflower on a scale. This lowers scale by 9.6 inches. Gamma = 0.5. From equilibrium, you give the scale a downward velocity of 5 fps.
What kind of damping (or not) occurs?
How many times per min does cauliflower pass equilibrium?
Solve for displacement of cauliflower from equil as fn of time.
Relevant Equations
Rcos(wt - d) = 0 (general conversion of sines and cosines to a single cosine minus some phase angle)
IMG-1980.JPG
IMG-1981.JPG
IMG-1982.JPG

Essentially, my question boils down to just part b. My initial thoughts on how to solve this are illustrated above in the second image. To me, I had just interpreted omega, the frequency, as cycles per second. From there, I simply multiplied to get it into minutes. After discussing with a prof. at office hours, he said this is incorrect and is the wrong approach. He noted that what's in the blue (final image on bottom) is the correct approach to part b. This involves considering the period. The explanation was a little rushed (as he had a meeting to attend), and it involved having to do with: the fact that omega isn't just cycles per second, and that setting this equation equal to 0 is done because this is the state of equilibrium. The setting equal to zero makes sense, but I don't see how I'm supposed to know how to interpret omega and I'm actually a little bit shakey on what's in blue so if someone could please clarify this, it would be greatly appreciated.
Thanks for your time.
 
Physics news on Phys.org
  • #2
##\omega## is the angular frequency, in rad/s. You were using it as the linear frequency ##\nu = \omega / (2\pi)##, which is expressed in cycles/s (Hz).
 
  • #3
DrClaude said:
##\omega## is the angular frequency, in rad/s. You were using it as the linear frequency ##\nu = \omega / (2\pi)##, which is expressed in cycles/s (Hz).
Thanks for the reply. How would I know to interpret this frequency as angular? In the problem, it seems as though the cauliflower is moving linearly.
 
  • #4
rugerts said:
How would I know to interpret this frequency as angular? In the problem, it seems as though the cauliflower is moving linearly.
It is moving linearly, but since it is harmonic motion, the displacement changes sinusoidally so there is an "angular" aspect to the motion.

Consider the situation in terms of units. When you have ##\cos(\omega t)##, what is in parenthesis is an angle, expressed in radians. For ##t## in seconds, ##\omega## has to be in radians per second.

Note that when writing ##\cos(2 \pi \nu t)##, the ##2 \pi## is assumed to have units of radians, so ##\nu## is in Hz.
 
  • Like
Likes rugerts

1. What is equilibrium in terms of differential equations?

Equilibrium in differential equations refers to a state in which the rate of change of a system is equal to zero. This means that the system is not changing over time and is in a stable state.

2. How is equilibrium related to the frequency of an object's passing?

The frequency of an object's passing is directly related to the rate of change of the system. When the system is in equilibrium, the rate of change is zero and therefore the object passes at a constant rate. If the system is not in equilibrium, the frequency of the object's passing will vary.

3. How can we determine the number of times an object passes equilibrium in a given time period?

This can be determined by solving the differential equation that describes the system and finding the points where the rate of change is equal to zero. These points represent the object passing equilibrium and can be used to calculate the number of times it passes in a given time period.

4. Can the frequency of an object passing equilibrium change over time?

Yes, the frequency of an object passing equilibrium can change over time if the system is not in a state of equilibrium. This can be caused by external forces or changes in the system itself.

5. How does the rate of change of a system affect the frequency of an object passing equilibrium?

The rate of change of a system directly affects the frequency of an object passing equilibrium. When the rate of change is zero, the system is in equilibrium and the object passes at a constant rate. As the rate of change increases or decreases, the frequency of the object's passing will also change accordingly.

Similar threads

  • Other Physics Topics
Replies
8
Views
1K
  • Sci-Fi Writing and World Building
3
Replies
87
Views
4K
  • Mechanical Engineering
Replies
30
Views
2K
  • Calculus and Beyond Homework Help
Replies
2
Views
4K
  • Programming and Computer Science
Replies
1
Views
1K
  • Introductory Physics Homework Help
Replies
7
Views
2K
  • Quantum Interpretations and Foundations
Replies
8
Views
432
Replies
16
Views
2K
Replies
13
Views
2K
  • Art, Music, History, and Linguistics
Replies
1
Views
1K
Back
Top