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2 dimensional motion problem

(a) What are the period and speed of a person on a carousel if the person has an acceleration with a magnitude of 0.86 m/s2 when she is standing 3.6 m from the axis?

(b) What are her acceleration magnitude and speed if she then moves to a distance of 1.9 m from the carousel center and the carousel keeps rotating with the same period?

Comments

Robert, I disagree with your answer.  You are using the angular acceleration as angular velocity.  Also part A has 2 parts to it.  You only attempted to answer 1 part.  Part A: angular acceleration=angular velocity squared/radius.  Rearranging this equation gives you angular velocity = angular acceleration*radius =(.86)(3.6) = 3.096 m/s.  The period is the time it takes to complete one revolution- in this case at 3.096 m/s.  Period = 2*pi*radius/angular velocity =(2)(3.14)(3.6)/(3.096) =7.3 s.

Part B:find a and w given r=1.9m and T=7.3s

w=2*pi*r/T=(2)(3.14)(1.9)/(7.3) =1.63 m/s (doing a quick logic test tells you that this answer makes sense.  Think about doing a spin.  When your arms are outstretched, they are farther away from your body, the axis, and therefore must travel faster to keep up than your shoulders which are closer to the center of rotation).  Now that we have w, we can find the new a.  a=wsquared/r =(1.63)squared/(1.9) =1.40 m/s2.  Let me know if you have any other questions!!

You are right. I misread acceleration as velocity.

Marsha, I think you have your terms mixed up - angular acceleration and angular velocity - this question is only dealing with centripetal acceleration and tangential velocity. You also forgot to take the square root in part a, when finding the velocity.

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2 Answers

(a)

from  a = v²/r  we get  0.86 = v²/3.6

which leads to v = 1.76 m/s

period is the time it takes to complete a full cirle (your distance)

from v = d/t we get 1.76 = 2π(3.6) / T

which leads to T = 12.9s

 

 

(b)

again from v = d/t we get v = 2π(1.9)/12.9

which leads to v = 0.925 m/s

and from a = v²/r we get a = (.925)²/1.9

which leads to a = 0.45 m/s²

 

a)

a = v^2/r

8.6 = v^2/3.6

So, v = sqrt(.86*3.6) = 1.76 m/s

Period = 2pi r/v = 2pi*3.6/1.76 = 12.9 sec

b)

a = w^2 r = (2pi/T)^2 r = (2pi/12.9)^2 * 1.9 = .454 m/sec^2

v = w*r = (2pi/12.9) * 1.9 = .925 m/sec

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{Edited}

Comments

Robert, it looks like you used acceleration in place of velocity in part a. Velocity should be 1.76m/s2, period should be 12.9 seconds.

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