The figure shows part of a curve traced by a point on the circumference of a circle of radius 4 that rotates, without slipping, around a fixed circle of radius 2. The rotating circle starts with angle t = 0 radians and the point P (x, y) at (10, 0). In this problem you will find parametric equations of the resulting epicycloid.
In the figure, t = 0.9 radian. Find the coordinates (rounded to the nearest hundredth) of the center of the large moving circle.
Because the big circle rotates wtihout slipping, arc a on the big circle equals arc a on the small circle. Find a when t = 0.9 radian, as in the figure. Use the answer to find the measure of angle A that subtends arc a on the big circle.
a = units
angle A = radians
Angle θ at the center of the big circle has measure equal to t + A. Find θ when t = 0.9 radian. θ = radians.
Use the answers above to find the coordinates of point P when t = 0.9. (Round to the nearest hundredth).
( , )
In general, what does θ equal as a function of t?
θ = t
By repeating the process you used to arrive at the coordinates of point P when t = 0.9, write parametric equations for x and y as functions of t.
How many revolutions of t are needed to generate the entire graphs?
Diagram can be found blondebeliever.tumblr.[com]/precalc (on my blog) under question 3!
Let r be the radius of the small inside circle
Let R be the radius of the large outside circle
Let C be the center of the large circle
Cx = (r+R)cos(t)
Cy = (r+R)sin(t)
a = rt
A = a/R
θ = t+A
Px = Cx + Rcosθ
Py = Cy + Rsinθ
r = 2
R = 4
when t = 0.9
Cx = 6cos.9 = 3.73
Cy = 6sin.9 = 4.70
a = 2t = 1.80
A = 1.8/4 = 0.45
θ = t+A = 1.35
Px = 3.73 + 4cos1.35 = 4.61
Py = 4.70 + 4sin1.35 = 8.60
Px = 6cost + 4cos3t/2
Py = 6sint + 4sin3t/2
After t has gone once around, the outer circle has only made a half turn. So, after 2 turns of t, we have 3 turns of A.
fooplot has a good app for x-y plots, parametric plots, and polar plots
To find the coordinates of the center of the large moving circle when t = 0.9 radians, we need to consider that the big circle rotates without slipping. Since the radius of the big circle is 4 and the radius of the small circle is 2, the ratio of their circumferences is 2:1. This means that the arc length on the big circle is twice the arc length on the small circle.
When t = 0.9 radians, the arc length on the small circle is 0.9 units (since the radius is 2, so the circumference is 2π = 2π * 0.9 = 1.8 units).
Since the arc length on the big circle is twice the arc length on the small circle, the arc length on the big circle is also 1.8 units. The center of the big circle is located at this point on its circumference.
To find the angle A that subtends arc a on the big circle, we can use the formula: arc length = radius * angle (in radians).
Since the arc length on the big circle is 1.8 units and the radius is 4, plugging in these values into the formula: 1.8 = 4 * A. Solving for A, we get A = 0.45 radians.
The angle θ at the center of the big circle is equal to the sum of t and A. When t = 0.9 radians and A = 0.45 radians, θ = 0.9 + 0.45 = 1.35 radians.
To find the coordinates of point P when t = 0.9 radians, we can use the formula for the parametric equations of the resulting epicycloid.
The x-coordinate of P can be found using the formula: x = (R + r) * cos(t) - r * cos((R + r) / r * t), where R is the radius of the big circle (4), r is the radius of the small circle (2), and t is the angle.
Plugging in these values, we get: x = (4 + 2) * cos(0.9) - 2 * cos((4 + 2) / 2 * 0.9). When simplified and rounded to the nearest hundredth, x ≈ 1.71.
The y-coordinate of P can be found using the formula: y = (R + r) * sin(t) - r * sin((R + r) / r * t), where R is the radius of the big circle (4), r is the radius of the small circle (2), and t is the angle.
Plugging in these values, we get: y = (4 + 2) * sin(0.9) - 2 * sin((4 + 2) / 2 * 0.9). When simplified and rounded to the nearest hundredth, y ≈ 4.86.
Therefore, when t = 0.9 radians, the coordinates of point P are approximately (1.71, 4.86).
In general, θ is equal to t plus A. So, θ = t + A.
To write the parametric equations for x and y as functions of t, we can use the formulas mentioned earlier.
The parametric equation for x is: x = (4 + 2) * cos(t) - 2 * cos((4 + 2) / 2 * t).
The parametric equation for y is: y = (4 + 2) * sin(t) - 2 * sin((4 + 2) / 2 * t).
To determine how many revolutions of t are needed to generate the entire graph, we need to find the period of the parametric equations.
In this case, the period is the smallest positive value of t for which both x and y repeat their values. This occurs when t completes one full revolution around the unit circle, which is 2π.
Therefore, it takes 2π revolutions of t to generate the entire graph.