MATH 261, Spring 2001 Due Date: Name(s):
Extra Project 14.4: Motion Around a Circular Path
Objective
In this project we discuss motion around a circular path. This project is not a Maple project: it only
involves computations you can make by hand.
Narrative
If an object moves around a circular path of radius R then its position can be described by the parametric
equations in time t:
x = x(t) = R cos (t), y = y(t) = R sin (t)
where (t) is the angular displacement of the object at time t, or by the vector equation
r = r(t) = R cos (t), R sin (t) = R cos (t), sin (t) . (1)
Differentiating (1) with respect to t, we find that
r = R - sin ,  cos  = R - sin , cos  . (2)
Thus:
1. r is perpendicular to r since
r  r = (R - sin , cos  )  (R cos , sin  ) = R2
 (- sin  cos  + sin  cos ) = 0,
and the unit tangent vector
T = T(t) =
r
||r ||
= - sin , cos 
is perpendicular to r(t) for all t.
2. the linear velocity v = v(t) of the object
v = ||r || = ||R - sin , cos  || = R| |.
So if we denote the (absolute value of the) angular velocity  of the object by , then
v = R. (3)
3. The derivative
T = - cos , - sin  =  - cos , - sin 
of T with respect to t is perpendicular to T since
T  T = ( - cos , - sin  )  - sin , cos  =  (sin  cos  - sin  cos ) = 0.
Indeed, the unit normal vector
N =
T
||T ||
= - cos , - sin  = -
r
R
for each t.
Differentiating (2) with respect to t, we find that
r = R( )2
- cos , - sin  + R - sin , cos  = R T + R( )2
N (4)
(or r = R T + R2
N). Using the facts that (3) implies that the linear acceleration
a = v = R = R
and that
R( )2
= R
v
R
2
=
v2
R
,
we may write (4) as
a = aT +
v2
R
N. (5)
Thus:
1. the magnitude of the tangential component of the acceleration vector a is a,
2. the magnitude of the normal component of the acceleration vector a -- the centripetal acceleration --
is v2
/R, and
(a) as R decreases, the centripetal acceleration increases, and
(b) as the linear velocity v increases, the centripetal acceleration increases (by the square of v).
Task
Assuming R = 2 and  = (t) = t3
, compute:
1. r, 2. r , 3. T, 4. v, 5. ,
6. T , 7. N, 8. a, 9. a, 10. v2
/R.
Finally,
11. sketch the path of the projectile, and
12. sketch and label the vectors T and N when t = 3
3/4.