What is the path that traces the circle in the plane

What is the path that traces the circle in the plane

Trigonometric Functions
asked 2021-01-08
What is the path that traces the circle in the plane =1 with radius =5 and center (−7,1,1) with constant speed 15?

Answers (1)

Technically the path is independent of the speed, what you're talking about is a parametrization of the path. And even then, your description does not determine a single parametrization because you did not specify where on the path the parametrization should start, which way around the circle we want to go, and how many times we should loop around.
Nonetheless, we can find a parametrization that fits this bill. Firstly, let's look at the "standard" parametrization of the circle of radius 5 centered at the origin going around at a speed of \omega exactly once:
\(r(t)=5\cos(\omega t) i+5\sin(\omega t) j,t\in [0,\omega 2\times \pi)\)
To tilt this on its side so that it's in the y=0 plane instead of the z=0 plane, we can just replace j with k:
\(r(t)=5\cos(\omega t) i+5\sin(\omega t) k,t\in [0,2\pi \omega )\)
To have it go at a speed of 15 radians per unit of time, we just insert 15 for \omega :
\(r(t)=5\cos(15t) i+5\sin(15t) k,t\in [0,2\pi 15)\)
\(r(t)=5\cos(15t) i+5\sin(15t) k,t\in [0,152\pi ​)\)
And finally to move it so that it's center is at (−7,1,1) (which incidentally will also move it from the y=0 plane to the y=1 plane), we just add the constant vector −7 i+j+k to it:
\(r(t)=[−7+5\cos(15t)] i+j+[1+5\sin(15t)] k,t\in [0,2\pi 15)\)
\(r(t)=[−7+5\cos(15t)] i+j+[1+5\sin(15t)] k,t\in [0,152\pi ​)\)
And we're done!

Relevant Questions

asked 2021-01-10
What is the path that traces the circle in the plane =1 with radius =5 and center (−7,1,1) with constant speed 15?
asked 2021-02-19
A 10 kg objectexperiences a horizontal force which causes it to accelerate at 5 \(\displaystyle\frac{{m}}{{s}^{{2}}}\), moving it a distance of 20 m, horizontally.How much work is done by the force?
A ball is connected to a rope and swung around in uniform circular motion.The tension in the rope is measured at 10 N and the radius of thecircle is 1 m. How much work is done in one revolution around the circle?
A 10 kg weight issuspended in the air by a strong cable. How much work is done, perunit time, in suspending the weight?
A 5 kg block is moved up a 30 degree incline by a force of 50 N, parallel to the incline. The coefficient of kinetic friction between the block and the incline is .25. How much work is done by the 50 N force in moving the block a distance of 10 meters? What is the total workdone on the block over the same distance?
What is the kinetic energy of a 2 kg ball that travels a distance of 50 metersin 5 seconds?
A ball is thrown vertically with a velocity of 25 m/s. How high does it go? What is its velocity when it reaches a height of 25 m?
A ball with enough speed can complete a vertical loop. With what speed must the ballenter the loop to complete a 2 m loop? (Keep in mind that the velocity of the ball is not constant throughout the loop).
asked 2021-03-03
Figure shows a nonconducting rod with a uniformly distributed charge +Q. The rod forms a 10/22 of circle with radius R and produces an electric field of magnitude Earc at its center of curvature P. If the arc is collapsed to a point at distance R from P, by what factor is the magnitude of the electric field at P multiplied?
asked 2021-04-06
A race car enters a flat 200 m radius curve at a speed of 20 m/swhile increasing its speed at a constant 2 m/s2. If the coefficient of static friction is .700, what will the speed of thecar be when the car beings to slide?
A) 24.3 m/s
B) 31.5 m/s
C) 37.1 m/s
D) 36.2 m/s
E) 28.7 m/s
asked 2021-04-15
A car initially traveling eastward turns north by traveling in a circular path at uniform speed as in the figure below. The length of the arc ABC is 235 m, and the car completes the turn in 33.0 s. (Enter only the answers in the input boxes separately given.)
(a) What is the acceleration when the car is at B located at an angle of 35.0°? Express your answer in terms of the unit vectors \(\displaystyle\hat{{{i}}}\) and \(\displaystyle\hat{{{j}}}\).
1. (Enter in box 1) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+{\left({E}{n}{t}{e}{r}\in{b}\otimes{2}\right)}{P}{S}{K}\frac{{m}}{{s}^{{2}}}\hat{{{j}}}\)
(b) Determine the car's average speed.
3. ( Enter in box 3) m/s
(c) Determine its average acceleration during the 33.0-s interval.
4. ( Enter in box 4) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{i}}}+\)
5. ( Enter in box 5) \(\displaystyle\frac{{m}}{{s}^{{2}}}\hat{{{j}}}\)
asked 2021-04-25
A stunt man whose mass is 70 kg swings from the end ofa 4.0 m long rope along thearc of a vertical circle. Assuming that he starts from rest whenthe rope is horizontal, find the tensions in the rope that are required to make him follow his circular path at each of the following points.
(a) at the beginning of his motion N
(b) at a height of 1.5 m above the bottom of the circular arc N
(c) at the bottom of the arc N
asked 2021-03-22
A box is sliding with a speed of 4.50 m/s on a horizontal surface when, at point P, it encounters a rough section. On the rough section, the coefficient of friction is not constant but starts at .100 at P and increases linerly with distance past P, reaching a value of .600 at 12.5 m past point P. (a) Use the work energy theorem to find how far this box slides before stopping. (b) What is the coefficient of friction at the stopping point? (c) How far would the box have slid iff the friciton coefficient didn't increase, but instead had the constant value of .1?
asked 2021-03-24
A 2.4-kg object is attached to a horizontal spring of forceconstant k=4.5 kN/m. The spring is stretched 10 cm fromequilibrium and released. Find (a) the frequency of themotion, (b) the period, (c) the amplitude, (d) the maximum speed,and (e) the maximum acceleration. (f) When does the objectfirst reach its equilibrium position? What is itsacceleration at this time?
Two identical blocks placed one on top of the other rest on africtionless horizontal air track. The lower block isattached to a spring of spring constant k= 600 N/m. Whendisplaced slightly from its equilibrium position, the systemoscillates with a frequency of 1.8 Hz. When the amplitude ofoscillation exceeds 5 cm, the upper block starts to slide relativeto the lower one. (a) What are the masses of the twoblocks? (b) What is the coefficient of static frictionbetween the two blocks?
asked 2021-05-13
A movie stuntman (mass 80.0kg) stands on a window ledge 5.0 mabove the floor. Grabbing a rope attached to a chandelier, heswings down to grapple with the movie's villian (mass 70.0 kg), whois standing directly under the chandelier.(assume that thestuntman's center of mass moves downward 5.0 m. He releasesthe rope just as he reaches the villian).
a) with what speed do the entwined foes start to slide acrossthe floor?
b) if the coefficient of kinetic friction of their bodies withthe floor is 0.250, how far do they slide?
asked 2021-02-22
A truck engine transmits 28 kW (27.5 hp) to the driving wheels when the truck is traveling at a constant velocity of magnitude 60 km/h (37.3 m/h) on a level road. (a) What is the resisting force acting on the truck? (b) Assume that 65% of the resisting force is due to rolling friction and that the remainder is due to air resistance. If the force of rolling friction is independent of speed and the force of air resistance is proportional to the square of speed, what power will drive the truck at 30 km/h? at 120 km/h? give your answers in kilowatts and in horsepower.