Jamie B.

asked • 12/06/16

centripetal forces

Mark and Sue are spinning together on a merry-go-round.

Mark yells across to Sue “Why are you on the outside far from the centre? According to the equation there is an inverse relation between r and the acceleration. Therefore, if r gets smaller the acceleration will be bigger and the ride will be more exciting”

Sue does not seem impressed with Mark’s abilities in Physics and yells back “According to the equation there is a direct relation between r and ac. If I remain on the outside my acceleration will be bigger and the ride will be more exciting.”

Both equations are valid for centripetal acceleration. Who is correct? What was the flaw in logic that caused the confusion?

1 Expert Answer

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Arturo O. answered • 12/06/16

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ac = v2/r
 
v = tangential speed
 
ω = angular rate of rotation of the merry-go-round, which is the same everywhere on the merry-go-round
 
v = ωr
 
Then 
 
ac = (ωr)2/r = ω2r 
 
The person farther from the center experiences a higher centripetal acceleration.  Sue is correct.  Mark was right about the inverse relation with r, but neglected the direct relation with v2.
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Steven W.

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To add a more explicit statement to Arturo's solution, which is already complete:  though ac should increase with decreasing r according to ac = v2/r, decreasing the radius also decreases the linear velocity v.  This is because someone at smaller r travels around a circle of smaller circumference, in the same time he or she used to travel around a circle of larger circumference.  So that person goes slower in v as the radius goes down.  
 
Since ac depends on v2, but only on the first power of r, decreasing v decreases ac faster than decreasing r increases ac.  So going toward the center (which decreases r and v at the same time) causes a net decrease in ac.
 
However, as Arturo mentioned, ω is the same at any radius, so it does not change as one moves out or in along a radial line.  So it provides a better comparison for how ac changes with r, making Sue correct.
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12/06/16

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