Solve Equations of Curves - Get Help Now!

  • Context: Graduate 
  • Thread starter Thread starter wilhelmsir
  • Start date Start date
  • Tags Tags
    Curves
Click For Summary
SUMMARY

The discussion focuses on deriving the equation of a curve described by a fixed point on a rolling circle of diameter A, which rolls without slipping around a stationary circle of the same diameter. The center of the rolling circle is located at a distance of 2A from the origin, with coordinates defined as x = 2A cos(θ) and y = 2A sin(θ). The coordinates of the fixed point on the rolling circle are expressed as x = A cos(φ) + 2A sin(θ) and y = A sin(φ) + 2A sin(θ). The relationship between the angles θ and φ is crucial for simplifying the equations.

PREREQUISITES
  • Understanding of polar coordinates
  • Knowledge of trigonometric functions and their relationships
  • Familiarity with the concept of rolling motion without slipping
  • Basic skills in drawing geometric figures and right triangles
NEXT STEPS
  • Explore the derivation of parametric equations for rolling circles
  • Study the application of polar coordinates in curve sketching
  • Learn about the relationship between angles in rolling motion
  • Investigate the use of right triangles in solving geometric problems
USEFUL FOR

Mathematics students, geometry enthusiasts, and anyone interested in the application of polar coordinates to solve problems involving curves and rolling motion.

wilhelmsir
Messages
2
Reaction score
0
[SOLVED] equations of curves

a circle of diameter A rolls without slipping along the outer circumference of a stationary circle of the same diameter. use polar coordinates to derive the equation of a curve described by some fixed point on the rolling circle. "can anyone help me out in this?"thanks 4 yo kindness
 
Physics news on Phys.org
Assume the stationary circle has center at (0, 0). Let [itex]\theta[/itex] be the angle the center of the rolling circle makes with the x-axis. Then the center of the rolling circle is distance 2A from the origin and has [itex]x= 2A cos(\theta)[/itex], [itex]y= 2A sin(\theta)[/itex] Now let [itex]\phi[/itex] be the angle the line from the center of the rolling circle to the given point. Then the given point has coordinates [itex]x= A cos(\phi)+ 2A sin(\theta)[/itex] and [itex]y= A sin(\phi)+ 2A sin(\theta)[/itex].

Of course, [itex]\theta[/itex] and [itex]\phi[/itex] are not independent. Draw a few pictures and right triangles to see how they are related. Then you can replace one by the other to get a single parameter.
 
thanks a lot HallsofIvy
 

Similar threads

Undergrad Question about Integrals to Determine Volume vs Surface Area
  • · Replies 5 ·
Replies
5
Views
4K
MHB Equation of Normal to Curve at (1,5): Solved?
  • · Replies 4 ·
Replies
4
Views
2K
High School Calculating shaded area: I'm getting discrepancies between methods
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Algorithm to determine the roulette curve
  • · Replies 1 ·
Replies
1
Views
4K
Graduate Deducing the existence of a disconnected solution
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Nonsensical (lack of) relation between area and arc-length of polar curves
  • · Replies 7 ·
Replies
7
Views
4K
Graduate Finding Minimal Mean Distance Curves on the Unit Sphere
  • · Replies 2 ·
Replies
2
Views
2K
Calculate this Integral around the Circular Path using Green's Theorem
  • · Replies 3 ·
Replies
3
Views
2K
MHB Implicit Differentiation to find equation of a tangent line
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Does Time Drive Black Hole Travelers to a Central Singularity?
  • · Replies 13 ·
Replies
13
Views
2K