Question

In: Advanced Math

Cycloid is the curve traced by a point P on the circumference of a circle of radius a rolling along a straight line in a plane. See the figure. The parametric equation of the cycloid is

Cycloid is the curve traced by a point P on the circumference of a circle of radius a rolling along a straight line in a plane. See the figure. The parametric equation of the cycloid is

                       

(a) Find the tangent to the cycloid at the point where \theta=\pi / 3

(b) At what points is the tangent horizontal? When is it vertical?

(c) Find the length and area under one arch of the cycloid.

Solutions

Expert Solution

(a) Find the tangent to the cycloid at the point where \theta=\pi / 3

To find the tangent of the cyloid, consider equation of tangent as:

where m=dy/dx

Therefore, we can find the slop m=\frac{d y}{d x} at point of \theta=\frac{\pi}{3} as following

               

After that, we can find the tangent of the cyloid as following:

(b) At what points is the tangent horizontal? When is it vertical?

(c) Find the length and area under one arch of the cycloid.

Therefore, the length of one arch of the cyloid is L=8a (unit of length)

Therefore, the area of one arch of the cyloid is A=3 a^{2} \pi (unit of square length)

 

SUN BUNRA answered 2 years ago
Next > < Previous

Related Solutions

Write parametric equations for each of the following curves. (a) The straight line segment traced from...
Write parametric equations for each of the following curves. (a) The straight line segment traced from (3,2) to (5,8) as t goes from 0 to 1. (b) A circle centered at (3,2), of radius 4, traced out two times, counterclockwise, as t goes from 0 to 2pi.
Find the equation of the tangent plane and the parametric equations for the normal line to...
Find the equation of the tangent plane and the parametric equations for the normal line to the surface x2 + y2 - z = 0 at the point P(4,-1, 6). Show all steps
Three point charges lie along a straight line as shown in the figure below, where q1...
Three point charges lie along a straight line as shown in the figure below, where q1 = 5.58 µC, q2 = 1.54 µC, and q3 = -1.88 µC. The separation distances are d1 = 3.00 cm and d2 = 2.00 cm. Calculate the magnitude and direction of the net electric force on each of the charges. Three charges lie along a horizontal line. Positive charge q1 is on the left. Positive charge q2 is a distance d1 to the right...
Three point charges lie along a straight line as shown in the figure below, where q1...
Three point charges lie along a straight line as shown in the figure below, where q1 = 6.12 µC, q2 = 1.57 µC, and q3 = -2.08 µC. The separation distances are d1 = 3.00 cm and d2 = 2.00 cm. Calculate the magnitude and direction of the net electric force on each of the charges. (a) q1? (b) q2? (c) q3?
Three point charges lie along a straight line as shown in the figure below, where q1...
Three point charges lie along a straight line as shown in the figure below, where q1 = 6.48 µC, q2 = 1.42 µC, and q3 = -2.08 µC. The separation distances are d1 = 3.00 cm and d2 = 2.00 cm. Calculate the magnitude and direction of the net electric force on each of the charges. Three charges lie along a horizontal line. Positive charge q1 is on the left. Positive charge q2 is a distance d1 to the right...
1.) Build the parametric equations of a circle centered at the point (2,-3) with a radius...
1.) Build the parametric equations of a circle centered at the point (2,-3) with a radius of 5 that goes counterclockwise and t=0 gives the location (7,-3) 2.) Build the parametric equations for an ellipse centered at the point (2, -3) where the major axis is parallel to the x-axis and vertices at (7, -3) and (-3, -3), endpoints of the minor axis are (2, 0) and (2, -6). The rotation is counterclockwise 3.) Build the parametric equations for a...
Figure 1. A ring of radius R and charge +Q lies in the x,y-plane. Point P...
Figure 1. A ring of radius R and charge +Q lies in the x,y-plane. Point P is distance a from the centre of the ring. Consider a thin ring (width dr) with radius r that contains a total charge Q uniformly distributed along the ring. A point P is a distance a above the centre of the ring as shown in this figure. In this question, use ϵ0 rather than Coulomb's constant in your expressions. The potential an infinite distance...
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.
3. (5 points) (a): Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.$$ x=e^{-t} \cos t, \quad y=e^{-t} \sin t, \quad z=e^{-t} ; \quad(1,0,1) $$(b): Find the unit tangent vector \(\mathbf{T}\), the principal unit normal \(\mathbf{N}\), and the curvature \(\kappa\) for the space curve,$$ \mathbf{r}(t)=<3 3="" 4="" sin="" cos="" t="">$$
1. Find an equation for the line in the xy−plane that is tangent to the curve...
1. Find an equation for the line in the xy−plane that is tangent to the curve at the point corresponding to the given value of t. Also, find the value of d^2y/dx^2 at this point. x=sec t, y=tan t, t=π/6 2. Find the length of the parametric curve: x=cos t, y=t+sin t, 0 ≤ t ≤ π. Hint:To integrate , use the identity,  and complete the integral.
Give a vector parametric equation for the line that passes through the point (2,0,5), parallel to...
Give a vector parametric equation for the line that passes through the point (2,0,5), parallel to the line parametrized by 〈t−3,t−3,−3−3t〉: Find the simplest vector parametric expression r⃗ (t) for the line that passes through the points P=(0,0,−1) at time t = 1 and Q=(0,−4,−1) at time t = 5 Need help with Calculus III
ADVERTISEMENT
Subjects
ADVERTISEMENT
Latest Questions
ADVERTISEMENT