A Proof
of
The relationships between the force, the
mass, the speed, and the radius of an
object moving in Uniform Circular Motion
Consider an object moving in UCM.
A
r

vA
AB
r
B
vB

v

vB

 vA
Both triangles are isosceles.
Since all three corresponding
angles are congruent, the
triangles are similar.
Hence, corresponding sides
are proportional.


Fnet  ma

 v
a
t
 
 vB  v A
a
 t 
 v B   v A 
a
t
v AB

v
r
To make a  a, let t  0 (In other words,
move the points closer together)
Results in AB  Arc AB
v Arc AB

v
r
But since the object is moving at a
constant speed
d
v
t
d  vt
Arc AB  vt
v vt

v
r
2
vt
v 
r
v v 2

t
r
v2
a
r
By Newton' s Second Law
Fnet  ma
mv 2
F
r
No Error!