Cycloids and related curves questions

  • Thread starter smartblonde
  • Start date
  • Tags
    Curves
In summary, the conversation revolves around three questions related to finding the unit tangent vector and slope of a cycloid, and the coordinates and radius of a cardioid. The person is struggling to understand the concepts and is seeking guidance and clarification. Mersenne, a French mathematician, spent over 20 years studying cycloids, emphasizing the difficulty and time required to solve complex problems in mathematics.
  • #1
smartblonde
3
0
I have three questions that I'm going to roll into one. I'm going insane trying to figure these out.

1. Find the unit tangent vector T to the cycloid. Also find the speed at theta=0 and theta=Pi, if the wheel turns at dtheta/dt=1.

that dtheta/dt is the speed, right? I'm a little baffled as to how I'm supposed to find a tangent vector when I don't even know what the cycloid is. I think it has something to do with v/|v|, but that doesn't change the fact I don't know how to answer the question. I'm sorry for not having gotten further with this, I need a nudge.


2. The slope of the clycloid is infinite at theta=0: dy/dx= sin(theta)/(1-cos(theta)). Estimate the slope at theta=(1/10) and theta=(-1/10).

I know I need to use L'Hopital's rule, but the answer my key is giving me is confusing me a bit. here's my main question: how does (sin(theta))/(1 - cos(theta))=approx (theta)/(((theta)^2)/2)? after that, I think I understand.

3. For a cardioid the radius C - 1 of the fixed circle equals the radius 1 of the circle rolling outside (epicycloid with C=2). (a) The coordinates of P are x = -1+2cos(theta)- cos(2theta), y=____________. (b) the double-angle formulas yield x= 2 cos(theta)(1-cos(theta)), y=_______________. (c) x^2 + y^2 =_______________ so its square root is r=________________.

this is the one I'm really struggling with. I've been sitting here trying to make sense of it for about an hour and I just DO NOT GET IT AT ALL. I'm getting really frustrated and I really really really need someone to guide me through it. I'm sorry I don't have any work to show for my efforts, but I know nothing I've got is right. I have an answer key for this one, but it's not proving very useful.

please, I really need help!
 
Physics news on Phys.org
  • #2
Smartblonde
Read the short history of Mersenne the French mathematician and his associates.Your problem took years of work before they got an answer.Some failed completely.Mersenne studied cycloids for over 20 years.If you are impatient after a couple of hours where does that leave you?For really difficult problems ,when you explain the result nobody will understand you at all.
 

1. What is a cycloid curve?

A cycloid curve is a specific type of curve that is created by tracing a point on the edge of a rolling circle. The resulting shape resembles a flattened curve and has many interesting properties and applications in mathematics and physics.

2. How is a cycloid curve different from a parabola?

A cycloid curve and a parabola have different equations and shapes. While a parabola is formed by the graph of a quadratic equation, a cycloid curve is created by the motion of a rolling circle. Additionally, a cycloid curve has a characteristic loop at its apex, while a parabola does not.

3. What are some real-life applications of cycloid curves?

Cycloid curves have a variety of applications in real-life, including in engineering, physics, and architecture. For example, they are used in the design of gears and gearboxes, the motion of pendulums, and the shape of arches and bridges in architecture.

4. Can cycloid curves be generalized to three dimensions?

Yes, cycloid curves can be extended to three dimensions. In fact, the three-dimensional version of a cycloid curve is called a trochoid and is created by the motion of a circle rolling along the surface of a cylinder.

5. Are there any famous mathematicians associated with cycloid curves?

Yes, several famous mathematicians have studied and made contributions to the understanding of cycloid curves, including Galileo Galilei, Evangelista Torricelli, and Sir Isaac Newton. French mathematician, physicist, and philosopher René Descartes is also credited with discovering the properties of the cycloid curve.

Similar threads

  • Calculus and Beyond Homework Help
Replies
1
Views
786
  • Calculus and Beyond Homework Help
Replies
3
Views
488
  • Calculus and Beyond Homework Help
Replies
28
Views
1K
  • Calculus and Beyond Homework Help
Replies
3
Views
837
  • Calculus and Beyond Homework Help
Replies
3
Views
742
  • Calculus and Beyond Homework Help
Replies
2
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
622
  • Calculus and Beyond Homework Help
Replies
1
Views
927
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
6
Views
1K
Back
Top