Explanation Given: The position of an object in circular motion is modeled by the following parametric equations, x = 2 sin t , y = 2 cos t … ( 1 ) Approach: The relation between polar coordinates and rectangular coordinates are r 2 = x 2 + y 2 … ( 2 ) and tan θ = y x , ( x ≠ 0 ) … ( 3 ) Calculation: Substitute x = 2 sin t and y = 2 cos t in equation ( 2 ) as follows, r 2 = ( 2 sin t ) 2 + ( 2 cos t ) 2 = 4 ( cos 2 t + sin 2 t ) = 4 ( 1 ) = 4 r = 4 = 2 Thus, the radius of the circle is 2 unit . Substitute t = 0 in equation ( 1 ) as follows, x = 2 sin ( 0 ) = 2 ( 0 ) = 0 And y = 2 cos ( 0 ) = 2 ( 1 ) = 2 Thus, the position of the object at t = 0 is ( 0 , 2 ) . Substitute t = π 2 in x = 2 sin t as follows, x = 2 sin ( π 2 ) = 2 ( 1 ) = 2 Substitute t = π in x = 2 sin t as follows,

Algebra and Trigonometry (MindTap Course List)

4th Edition

ISBN: 9781305071742

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

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Chapter 8.4, Problem 28E

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The characteristics of an object in circular motions, by stating the radius of circle, the position of the point at t=0, the orientation of the motion and time t to complete one revolution around the circle.

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Chapter 8.4, Problem 27E

Chapter 8.4, Problem 29E

Chapter 8 Solutions

Algebra and Trigonometry (MindTap Course List)

Ch. 8.1 - CONCEPTS We can describe the location of a point...Ch. 8.1 - Prob. 2E Ch. 8.1 - Prob. 3E Ch. 8.1 - Prob. 4E Ch. 8.1 - Prob. 5E Ch. 8.1 - Prob. 6E Ch. 8.1 - SKILLS 5-10 Plotting Points in Polar Coordinates...Ch. 8.1 - Prob. 8E Ch. 8.1 - Prob. 9E Ch. 8.1 - Prob. 10E

Ch. 8.1 - Prob. 11E Ch. 8.1 - Prob. 12E Ch. 8.1 - SKILLS 11-16 Different Polar Coordinates for the...Ch. 8.1 - Prob. 14E Ch. 8.1 - Prob. 15E Ch. 8.1 - Prob. 16E Ch. 8.1 - Prob. 17E Ch. 8.1 - Prob. 18E Ch. 8.1 - SKILLS 17-24 Points in Polar Coordinates...Ch. 8.1 - Prob. 20E Ch. 8.1 - Prob. 21E Ch. 8.1 - Prob. 22E Ch. 8.1 - Prob. 23E Ch. 8.1 - Prob. 24E Ch. 8.1 - SKILLS 25-26 Rectangular Coordinates to Polar...Ch. 8.1 - Prob. 26E Ch. 8.1 - Prob. 27E Ch. 8.1 - Prob. 28E Ch. 8.1 - Prob. 29E Ch. 8.1 - Prob. 30E Ch. 8.1 - 29-36 Polar Coordinates to Rectangular Coordinates...Ch. 8.1 - Prob. 32E Ch. 8.1 - Prob. 33E Ch. 8.1 - Prob. 34E Ch. 8.1 - Prob. 35E Ch. 8.1 - Prob. 36E Ch. 8.1 - SKILLS 37-44 Rectangular Coordinates to Polar...Ch. 8.1 - Prob. 38E Ch. 8.1 - Prob. 39E Ch. 8.1 - Prob. 40E Ch. 8.1 - Prob. 41E Ch. 8.1 - Prob. 42E Ch. 8.1 - 37-44 Rectangular Coordinates to Polar Coordinates...Ch. 8.1 - Prob. 44E Ch. 8.1 - Prob. 45E Ch. 8.1 - Prob. 46E Ch. 8.1 - Prob. 47E Ch. 8.1 - Prob. 48E Ch. 8.1 - SKILLS 45-50 Rectangular equations to polar...Ch. 8.1 - Prob. 50E Ch. 8.1 - Prob. 51E Ch. 8.1 - Prob. 52E Ch. 8.1 - Prob. 53E Ch. 8.1 - Prob. 54E Ch. 8.1 - SKILLS 51-70 Polar Equations to Rectangular...Ch. 8.1 - Prob. 56E Ch. 8.1 - Prob. 57E Ch. 8.1 - Prob. 58E Ch. 8.1 - Prob. 59E Ch. 8.1 - Prob. 60E Ch. 8.1 - SKILLS 51-70 Polar Equations to Rectangular...Ch. 8.1 - Prob. 62E Ch. 8.1 - Prob. 63E Ch. 8.1 - Prob. 64E Ch. 8.1 - Prob. 65E Ch. 8.1 - Prob. 66E Ch. 8.1 - SKILLS 51-70 Polar Equations to Rectangular...Ch. 8.1 - Prob. 68E Ch. 8.1 - Prob. 69E Ch. 8.1 - Prob. 70E Ch. 8.1 - Prob. 71E Ch. 8.1 - Prob. 72E Ch. 8.2 - Prob. 1E Ch. 8.2 - Prob. 2E Ch. 8.2 - Prob. 3E Ch. 8.2 - Prob. 4E Ch. 8.2 - Prob. 5E Ch. 8.2 - Prob. 6E Ch. 8.2 - Prob. 7E Ch. 8.2 - Prob. 8E Ch. 8.2 - Prob. 9E Ch. 8.2 - Prob. 10E Ch. 8.2 - Prob. 11E Ch. 8.2 - Prob. 12E Ch. 8.2 - Prob. 13E Ch. 8.2 - Prob. 14E Ch. 8.2 - Prob. 15E Ch. 8.2 - Prob. 16E Ch. 8.2 - Prob. 17E Ch. 8.2 - Prob. 18E Ch. 8.2 - 17-22 Polar to Rectangular Sketch a graph of the...Ch. 8.2 - Prob. 20E Ch. 8.2 - Prob. 21E Ch. 8.2 - Prob. 22E Ch. 8.2 - Prob. 23E Ch. 8.2 - Prob. 24E Ch. 8.2 - Prob. 25E Ch. 8.2 - Prob. 26E Ch. 8.2 - Prob. 27E Ch. 8.2 - Prob. 28E Ch. 8.2 - Prob. 29E Ch. 8.2 - Prob. 30E Ch. 8.2 - Prob. 31E Ch. 8.2 - Prob. 32E Ch. 8.2 - Prob. 33E Ch. 8.2 - Prob. 34E Ch. 8.2 - Prob. 35E Ch. 8.2 - Prob. 36E Ch. 8.2 - Prob. 37E Ch. 8.2 - Prob. 38E Ch. 8.2 - Prob. 39E Ch. 8.2 - Prob. 40E Ch. 8.2 - Prob. 41E Ch. 8.2 - Prob. 42E Ch. 8.2 - 2346 Graphing Polar EquationsSketch a graph of the...Ch. 8.2 - Prob. 44E Ch. 8.2 - Prob. 45E Ch. 8.2 - Prob. 46E Ch. 8.2 - Prob. 47E Ch. 8.2 - Prob. 48E Ch. 8.2 - Prob. 49E Ch. 8.2 - Prob. 50E Ch. 8.2 - Prob. 51E Ch. 8.2 - Prob. 52E Ch. 8.2 - Prob. 53E Ch. 8.2 - Prob. 54E Ch. 8.2 - Prob. 55E Ch. 8.2 - Prob. 56E Ch. 8.2 - Prob. 57E Ch. 8.2 - Prob. 58E Ch. 8.2 - Prob. 59E Ch. 8.2 - Prob. 60E Ch. 8.2 - Prob. 61E Ch. 8.2 - Prob. 62E Ch. 8.2 - Prob. 63E Ch. 8.2 - Prob. 64E Ch. 8.2 - Prob. 65E Ch. 8.2 - Prob. 66E Ch. 8.2 - DISCUSSDISCOVERPROVEWRITE DISCUSS: Choosing a...Ch. 8.3 - CONCEPTS A complex number z=a+bi has two parts: a...Ch. 8.3 - Prob. 2E Ch. 8.3 - Prob. 3E Ch. 8.3 - Prob. 4E Ch. 8.3 - Prob. 5E Ch. 8.3 - Prob. 6E Ch. 8.3 - SKILLS 514 A Complex Number and Its Modulus Graph...Ch. 8.3 - Prob. 8E Ch. 8.3 - Prob. 9E Ch. 8.3 - Prob. 10E Ch. 8.3 - Prob. 11E Ch. 8.3 - Prob. 12E Ch. 8.3 - SKILLS 514A Complex Number and Its Modulus Graph...Ch. 8.3 - Prob. 14E Ch. 8.3 - SKILLS 15-16Graphing Complex Numbers. Sketch the...Ch. 8.3 - Prob. 16E Ch. 8.3 - Prob. 17E Ch. 8.3 - Prob. 18E Ch. 8.3 - SKILLS 19-20Graphing Complex Numbers. Sketch the...Ch. 8.3 - Prob. 20E Ch. 8.3 - Prob. 21E Ch. 8.3 - Prob. 22E Ch. 8.3 - Prob. 23E Ch. 8.3 - 21-28 Graphing Sets of Complex Numbers Sketch the...Ch. 8.3 - 21-28 Graphing Sets of Complex Numbers Sketch the...Ch. 8.3 - Prob. 26E Ch. 8.3 - Prob. 27E Ch. 8.3 - Prob. 28E Ch. 8.3 - Prob. 29E Ch. 8.3 - Prob. 30E Ch. 8.3 - 2948 Polar Form of Complex Numbers Write the...Ch. 8.3 - Prob. 32E Ch. 8.3 - Prob. 33E Ch. 8.3 - Prob. 34E Ch. 8.3 - Prob. 35E Ch. 8.3 - Prob. 36E Ch. 8.3 - 2948 Polar Form of Complex Numbers Write the...Ch. 8.3 - Prob. 38E Ch. 8.3 - Prob. 39E Ch. 8.3 - Prob. 40E Ch. 8.3 - Prob. 41E Ch. 8.3 - Prob. 42E Ch. 8.3 - 2948 Polar Form of Complex Numbers Write the...Ch. 8.3 - Prob. 44E Ch. 8.3 - Prob. 45E Ch. 8.3 - Prob. 46E Ch. 8.3 - Prob. 47E Ch. 8.3 - Prob. 48E Ch. 8.3 - SKILLS 49-56Product and Quotients of Complex...Ch. 8.3 - Prob. 50E Ch. 8.3 - Prob. 51E Ch. 8.3 - Prob. 52E Ch. 8.3 - Prob. 53E Ch. 8.3 - Prob. 54E Ch. 8.3 - 49-56 Product and Quotients of Complex numbersFind...Ch. 8.3 - Prob. 56E Ch. 8.3 - Prob. 57E Ch. 8.3 - Prob. 58E Ch. 8.3 - Prob. 59E Ch. 8.3 - Prob. 60E Ch. 8.3 - 57-64 Product and Quotients of Complex...Ch. 8.3 - Prob. 62E Ch. 8.3 - Prob. 63E Ch. 8.3 - Prob. 64E Ch. 8.3 - Prob. 65E Ch. 8.3 - Prob. 66E Ch. 8.3 - SKILLS 65-76Powers Using De Moivres TheoremFind...Ch. 8.3 - SKILLS 65-76Powers Using De Moivres TheoremFind...Ch. 8.3 - Prob. 69E Ch. 8.3 - Prob. 70E Ch. 8.3 - Prob. 71E Ch. 8.3 - Prob. 72E Ch. 8.3 - SKILLS 65-76Powers Using De Moivres TheoremFind...Ch. 8.3 - Prob. 74E Ch. 8.3 - Prob. 75E Ch. 8.3 - Prob. 76E Ch. 8.3 - Prob. 77E Ch. 8.3 - Prob. 78E Ch. 8.3 - SKILLS 77-86Roots of Complex NumbersFind the...Ch. 8.3 - Prob. 80E Ch. 8.3 - Prob. 81E Ch. 8.3 - 77-86Roots of Complex NumbersFind the indicated...Ch. 8.3 - Prob. 83E Ch. 8.3 - Prob. 84E Ch. 8.3 - 77-86 Roots of Complex NumbersFind the indicated...Ch. 8.3 - Prob. 86E Ch. 8.3 - Prob. 87E Ch. 8.3 - Prob. 88E Ch. 8.3 - Prob. 89E Ch. 8.3 - Prob. 90E Ch. 8.3 - Prob. 91E Ch. 8.3 - Prob. 92E Ch. 8.3 - Prob. 93E Ch. 8.3 - Prob. 94E Ch. 8.3 - Prob. 95E Ch. 8.3 - Prob. 96E Ch. 8.3 - Prob. 97E Ch. 8.3 - Prob. 98E Ch. 8.3 - Prob. 99E Ch. 8.3 - Prob. 100E Ch. 8.3 - Prob. 101E Ch. 8.4 - Prob. 1E Ch. 8.4 - Prob. 2E Ch. 8.4 - Prob. 3E Ch. 8.4 - Prob. 4E Ch. 8.4 - Prob. 5E Ch. 8.4 - Prob. 6E Ch. 8.4 - Prob. 7E Ch. 8.4 - Prob. 8E Ch. 8.4 - Prob. 9E Ch. 8.4 - Prob. 10E Ch. 8.4 - Prob. 11E Ch. 8.4 - Prob. 12E Ch. 8.4 - Prob. 13E Ch. 8.4 - Prob. 14E Ch. 8.4 - Prob. 15E Ch. 8.4 - Prob. 16E Ch. 8.4 - Prob. 17E Ch. 8.4 - Prob. 18E Ch. 8.4 - Prob. 19E Ch. 8.4 - Prob. 20E Ch. 8.4 - Prob. 21E Ch. 8.4 - Prob. 22E Ch. 8.4 - Prob. 23E Ch. 8.4 - Prob. 24E Ch. 8.4 - Prob. 25E Ch. 8.4 - Prob. 26E Ch. 8.4 - Prob. 27E Ch. 8.4 - Prob. 28E Ch. 8.4 - Prob. 29E Ch. 8.4 - Prob. 30E Ch. 8.4 - Prob. 31E Ch. 8.4 - Prob. 32E Ch. 8.4 - Prob. 33E Ch. 8.4 - Prob. 34E Ch. 8.4 - Prob. 35E Ch. 8.4 - Prob. 36E Ch. 8.4 - Prob. 37E Ch. 8.4 - Prob. 38E Ch. 8.4 - Prob. 39E Ch. 8.4 - Prob. 40E Ch. 8.4 - Prob. 41E Ch. 8.4 - Prob. 42E Ch. 8.4 - Prob. 43E Ch. 8.4 - Prob. 44E Ch. 8.4 - Prob. 45E Ch. 8.4 - Prob. 46E Ch. 8.4 - Prob. 47E Ch. 8.4 - Prob. 48E Ch. 8.4 - Prob. 49E Ch. 8.4 - Prob. 50E Ch. 8.4 - Prob. 51E Ch. 8.4 - Prob. 52E Ch. 8.4 - Prob. 53E Ch. 8.4 - Finding Parametric Equations for a Curve Two...Ch. 8.4 - Prob. 55E Ch. 8.4 - Prob. 56E Ch. 8.4 - Prob. 57E Ch. 8.4 - Prob. 58E Ch. 8.4 - Prob. 59E Ch. 8.4 - Prob. 60E Ch. 8.4 - Prob. 61E Ch. 8.4 - Prob. 62E Ch. 8.4 - Prob. 63E Ch. 8.4 - Epicycloid If the circle C of Exercise 63 rolls on...Ch. 8.4 - Longbow CurveIn the following figure, the circle...Ch. 8.4 - Prob. 66E Ch. 8.4 - Prob. 67E Ch. 8.4 - Prob. 68E Ch. 8.4 - Prob. 69E Ch. 8.4 - Prob. 70E Ch. 8.4 - Prob. 71E Ch. 8.CR - Prob. 1CC Ch. 8.CR - Prob. 2CC Ch. 8.CR - Prob. 3CC Ch. 8.CR - Prob. 4CC Ch. 8.CR - a How do we express the complex number z in polar...Ch. 8.CR - Prob. 6CC Ch. 8.CR - Prob. 7CC Ch. 8.CR - Prob. 8CC Ch. 8.CR - Prob. 9CC Ch. 8.CR - Prob. 1E Ch. 8.CR - Prob. 2E Ch. 8.CR - Prob. 3E Ch. 8.CR - Prob. 4E Ch. 8.CR - Prob. 5E Ch. 8.CR - Prob. 6E Ch. 8.CR - Prob. 7E Ch. 8.CR - Prob. 8E Ch. 8.CR - Prob. 9E Ch. 8.CR - Prob. 10E Ch. 8.CR - Prob. 11E Ch. 8.CR - Prob. 12E Ch. 8.CR - Prob. 13E Ch. 8.CR - Prob. 14E Ch. 8.CR - Prob. 15E Ch. 8.CR - Prob. 16E Ch. 8.CR - Prob. 17E Ch. 8.CR - Prob. 18E Ch. 8.CR - Prob. 19E Ch. 8.CR - Prob. 20E Ch. 8.CR - Prob. 21E Ch. 8.CR - Prob. 22E Ch. 8.CR - Prob. 23E Ch. 8.CR - Prob. 24E Ch. 8.CR - Prob. 25E Ch. 8.CR - Prob. 26E Ch. 8.CR - Prob. 27E Ch. 8.CR - Prob. 28E Ch. 8.CR - Prob. 29E Ch. 8.CR - Prob. 30E Ch. 8.CR - Prob. 31E Ch. 8.CR - Prob. 32E Ch. 8.CR - Prob. 33E Ch. 8.CR - Prob. 34E Ch. 8.CR - Prob. 35E Ch. 8.CR - Prob. 36E Ch. 8.CR - Prob. 37E Ch. 8.CR - Prob. 38E Ch. 8.CR - Prob. 39E Ch. 8.CR - Prob. 40E Ch. 8.CR - Prob. 41E Ch. 8.CR - Prob. 42E Ch. 8.CR - Prob. 43E Ch. 8.CR - Prob. 44E Ch. 8.CR - Prob. 45E Ch. 8.CR - Prob. 46E Ch. 8.CR - Prob. 47E Ch. 8.CR - Prob. 48E Ch. 8.CR - Prob. 49E Ch. 8.CT - Prob. 1CT Ch. 8.CT - Prob. 2CT Ch. 8.CT - Prob. 3CT Ch. 8.CT - Prob. 4CT Ch. 8.CT - Prob. 5CT Ch. 8.CT - Find the cube roots of 27i, and sketch these roots...Ch. 8.CT - Prob. 7CT Ch. 8.CT - Prob. 8CT Ch. 8.CT - Prob. 9CT Ch. 8.FOM - Trajectories Are Parabolas From the graphs in...Ch. 8.FOM - Path of a Baseball Suppose a baseball is thrown at...Ch. 8.FOM - Path of a Rocket Suppose that a rocket is fired at...Ch. 8.FOM - Firing a Missile The initial speed of a missile is...Ch. 8.FOM - Prob. 5P Ch. 8.FOM - Shooting into the Wind Suppose that a projectile...Ch. 8.FOM - Shooting into the Wind Using the parametric...Ch. 8.FOM - Prob. 8P

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Incline Railway An incline railway is approximately 170 feet long with a 36% uphill grade (see figure). (a) Find the inclination of the railway. (b) Find the change in elevation from the base to the top of the railway. (c) Using the origin of a rectangular coordinate system as the base of the inclined plane, find an equation of the line that models the railway track. (d) Sketch a graph of the equation you found in part (c).
Gear Trains Figure 8 shows a single-stage gear train. Gear trains are used in many products, such as clocks and automotive transmissions, to reduce or increase the angular velocity of a component. The size of each gear is measured by the number of teeth rather than the radius. Suppose the first gear has n1 and the second gear has n2 teeth. Because the spacing of the teeth is the same for both gears, the ratio of their radii will be equivalent to the corresponding ratio of the number of teeth. When two gears are meshed together, they share the same linear velocity. If 1 and 2 are the angular velocities of the first and second gears, respectively, then v2=v1r22=r112=r1r212=n1n21 The first gear in a single-stage gear train has 42 teeth and an angular velocity of 2 revolutions per second. The second gear has 7 teeth. Find the angular velocity of the second gear.
Gear Trains Figure 8 shows a single-stage gear train. Gear trains are used in many products, such as clocks and automotive transmissions, to reduce or increase the angular velocity of a component. The size of each gear is measured by the number of teeth rather than the radius. Suppose the first gear has n1 and the second gear has n2 teeth. Because the spacing of the teeth is the same for both gears, the ratio of their radii will be equivalent to the corresponding ratio of the number of teeth. When two gears are meshed together, they share the same linear velocity. If 1 and 2 are the angular velocities of the first and second gears, respectively, then v2=v1r22=r112=r1r212=n1n21 The second gear in a single-stage gear train has 6 teeth and an angular velocity of 90 revolutions per minute. The first gear has 54 teeth. Find the angular velocity of the first gear.

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The characteristics of an object in circular motions, by stating the radius of circle , the position of the point at t = 0 , the orientation of the motion and time t to complete one revolution around the circle.

Solution Summary: The author explains how the position of an object in circular motion is modeled by the following parametric equations.

Videos

To find:

The characteristics of an object in circular motions, by stating the radius of circle, the position of the point at t=0, the orientation of the motion and time t to complete one revolution around the circle.

Want to see the full answer?

Chapter 8 Solutions

The characteristics of an object in circular motions, by stating the radius of circle , the position of the point at t = 0 , the orientation of the motion and time t to complete one revolution around the circle.

Solution Summary: The author explains how the position of an object in circular motion is modeled by the following parametric equations.

Videos

To find: The characteristics of an object in circular motions, by stating the radius of circle, the position of the point at t=0, the orientation of the motion and time t to complete one revolution around the circle.

Want to see the full answer?

Chapter 8 Solutions

To find:

The characteristics of an object in circular motions, by stating the radius of circle, the position of the point at t=0, the orientation of the motion and time t to complete one revolution around the circle.