
Concept explainers
PROBLEM PLUS
FIGURE FOR PROBLEM 1
1. A particle P moves with constant angular speed ω around a circle whose center is at the origin and whose radius is R. The particle is said to be in uniform circular motion. Assume that the motion is counterclockwise and that the panicle is at the point (R, 0) when t = 0, The position
(a) Find the velocity vector v and show that v · r = 0, Conclude that v is tangent to the circle and points in the direction of the motion
(b) Show that the speed |v| of the particle is the constant ωR. The period T of the particle is the time required for one complete resolution. Conclude that
(c) Find the acceleration vector a. Show that it is proportional to r and that it points toward the origin. An acceleration with this property is called a centripetal acceleration. Show that the magnitude of the acceleration vector is |a| = Rω2.
(d) Suppose that the particle has mass m. Show that the magnitude of the force F that is required to produce this, motion, called a centripetal force. is

Want to see the full answer?
Check out a sample textbook solution
Chapter 13 Solutions
Student Solutions Manual, Chapters 10-17 for Stewart's Multivariable Calculus, 8th (James Stewart Calculus)
- Find the exact area inside r=2sin(2\theta ) and outside r=\sqrt(3)arrow_forwardA 20 foot ladder rests on level ground; its head (top) is against a vertical wall. The bottom of the ladder begins by being 12 feet from the wall but begins moving away at the rate of 0.1 feet per second. At what rate is the top of the ladder slipping down the wall? You may use a calculator.arrow_forwardExplain the focus and reasons for establishment of 12.4.1(root test) and 12.4.2(ratio test)arrow_forward
- Use 12.4.2 to determine whether the infinite series on the right side of equation 12.6.5, 12.6.6 and 12.6.7 converges for every real number x.arrow_forwarduse Cauchy Mean-Value Theorem to derive Corollary 12.6.2, and then derive 12.6.3arrow_forwardExplain the focus and reasons for establishment of 12.5.4arrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781305652224Author:Charles P. McKeague, Mark D. TurnerPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage Learning


