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19 Nov 2019

The position of a particle moving on the XY plane is given by R=2.0cos(3.0rad s/t)i+2.0sin(3.0rad s/t)j, where r is in meters and t is in t seconds. A) Show that this represents circular motion of radius 2.0 m centered at the origin. B) Determine the speed? C) determine the magnitude of the acceleration? D) Show that a=v^2/r? E) Show that the acceleration vector always points toward the center of the circle? I cant seem to figure this one out. Please help?

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Deanna Hettinger
Deanna HettingerLv2
1 Nov 2019

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Related textbook solutions

Physics: Principles with Applications
7th Edition, 2013
Giancoli
ISBN: 9780321625922

Related questions

A particle P moves with constant angular speed Ļ‰ around a circle whose center is at the origin and whose radius is R.The particle is said to be in a uniform circular motion. Assume that the motion is counterclockwise and that the particle is at the point (R, 0) when t = 0.The position vector at time t ā‰„ 0 is r(t) = R cos Ļ‰t i + R sin Ļ‰t j.

(a) Find the velocity vector v and show that v Ā· r = 0. Conclude that v is tangential to the circle and points in the direction of the motion.

(b) Show that the speed of the particle is constant Ļ‰R. The period T of the particle is the time required for one complete revolution. Conclude T = 2Ļ€/Ļ‰.

Ā 

A particleĀ PĀ moves with constant angular speedĀ Ļ‰Ā around a circle whose center is at the origin and whose radius isĀ R. The particle is said to be inĀ uniform circular motion. Assume that the motion is counterclockwise and that the particle is at the pointĀ (R,0)Ā whenĀ t=0. The position vector at timeĀ tā‰„0Ā isĀ r(t)=RcosĻ‰tĀ iĀ +Ā RsinĻ‰tĀ j.

(a) Find the velocity vectorĀ vĀ and show thatĀ vā‹…r=0. Conclude thatĀ vĀ is tangent to the circle and points in the direction of the motion.

(b) Show that the speedĀ |v|Ā of the particle is the constantĀ Ļ‰R. TheĀ period TĀ of the particle is the time required for one complete revolution. Conclude that

T=2Ļ€R/|v|=2Ļ€/Ļ‰

Ā 

Chapter 13.P, Problem 1P, A particle P moves with constant angular speed around a circle whose center is at the origin and

(A) Show that the position of a particle on a circle of radius Rwith center at the origin (x=0,y=0) is given in unit vectors byr=cosƎĀøi+sinƎĀøj, where ƎĀø is the angle of the position vector with thex axis.

B) If the particle moves with constant speed v and period T,starting on the x axis at t=0, find an expression for ƎĀø in terms ofthe time and T.

C) Differentiate the position vector twice with respect to timeto find the acceleration and show that it is the centripetalacceleration, whose magnitude is given by equation (a=(v^2)/r) andwhose direction is toward the center of the circle. (Not that theunit vectors i and j are constants, independent of time.)

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