1. Non-Central Coordinate System
In addition to the 1-D, 2-D, and 3-D coordinate systems discussed in previous lectures, there is another useful coordinate system known as normal/tangential coordinates. The main difference in this coordinate system is that origin is no longer fixed. Instead, this coordinate system references particle motion by the tangential velocity along its path of motion
where v is the velocity magnitude and is a unit vector denoting the tangential direction of the path. As long as the motion is primarily confined to two dimensions, this coordinate system proves to be very useful in solving problems.

2. For example, given the path trajectory s, the associated directions of the tangential unit vector, which is parallel to the instantaneous velocity, and normal unit vector are shown below.
However, the normal unit vector could be drawn any number of ways that is perpendicular to the tangential direction. Why is it drawn this way? What significance is this normal direction, given the radius of curvature?

3. Unit Vector Time Derivatives
The time derivative of a unit vector is proportional to how quickly it changes direction. For example, suppose you have a vector rotating about an axis with angular velocity vector
where is the frequency of rotation, denoted in revolutions (or radians) per second, and is the axis around which the vector is rotating. The direction change is shown mathematically above.

4. Let’s provide a model system equivalent with cylindrical coordinates to make the derivation easier. For example, in the system of a particle in a stable circular orbit around a point, the vectors can be written as follows:

5. Thus, the normal-tangential coordinate system is fundamentally different than all the previously studied coordinate systems in that the origin is not at a fixed location. To see why, consider the path of a particle shown in the diagram below…
Why would we ever want to use this coordinate system, whose origin is constantly shifting?

6. Indianapolis 500 speedtrack
1/8 mile
straight-
away
1/4 mile
turn
5/8 mile straightaway
Given the average speed of a car for 1 lap is around 220 mph and the driver is moving at a constant speed, what is the acceleration around the turn?

7. In ¼ mile, the car goes through a 90 degree turn
In ¼ mile, the car goes through a 90 degree turn. Thus, the circumference of the circle is 1 mile, and its radius can be computed from:
The acceleration of the car can be computed from:
However, assuming the car has constant velocity:
The acceleration becomes:

8. With advances in automotive technology, how fast can a car safely drive in the Indy 500?
The catch here, is how we define “safely”. According to my definition, the word “safely” means the speed at which the car does not slip on the road.

9. In general, we know that the acceleration of the car is given by:
We also have the relation for force:
From this, we can arrive at two relations to solve for the two unknowns, T and N

10. Given the 2 equations and 2 unknowns:
The solution is given as follows:
The condition for slipping becomes:

11. Case 1: Assume no angle on the bank
Qualitatively, one can see that any acceleration (braking or accelerating) reduces the maximum speed at which you can take the turn without slipping.

12. Case 2: Assume no tangential acceleration
Thus, the velocity can be expressed as:
Curiously, a banked curve can support potentially “infinite” velocity if the denominator equals zero: (i.e.)
Given typical values for the friction coefficient, this would require that the angle of the bank exceeds 45 degrees, which would be too scary to drive on!

13. With advances in automotive technology, how fast can a car safely drive in the Indy 500?
So, we have arrived at 2 qualitative conclusions… The first is that when taking a steep curve, you should maintain constant velocity to prevent slipping. The second is that the maximum speed you can take the curve depends on the friction coefficient and the bank angle. If you were on the NASCAR team, you would be well-advised to know these parameters in designing the car. In the INDY 500, the bank angle around the turns is 9o 12’. Assuming a friction coefficient of 0.3, what is the maximum speed you can take around the turn?

14. At the Indy500, the maximum speed around a turn assuming μ=0.3 is:

15. Problem 1 The driver of an automobile decreases her speed at a
constant rate from 45 to 30 mi/h over a distance of 750 ft
along a curve of 1500-ft radius. Determine the magnitude
of the total acceleration of the automobile after the
automobile has traveled 500 ft along the curve.

16. Problem 1
The driver of an automobile decreases her speed at a
constant rate from 45 to 30 mi/h over a distance of 750 ft
along a curve of 1500-ft radius. Determine the magnitude
of the total acceleration of the automobile after the
automobile has traveled 500 ft along the curve.
1. Use tangential and normal components: These components
are used when the particle travels along a circular path. The
unit vector et is tangent to the path (and thus aligned with the
velocity) while the unit vector en is directed along the normal to
the path and always points toward its center of curvature.
2. Determine the tangential acceleration: For a constant
tangential acceleration, the acceleration can be determined from
v dv = at dx vo v xo x

17. Problem 1
The driver of an automobile decreases her speed at a
constant rate from 45 to 30 mi/h over a distance of 750 ft
along a curve of 1500-ft radius. Determine the magnitude
of the total acceleration of the automobile after the
automobile has traveled 500 ft along the curve.
3. Determine the normal acceleration: For known velocity and
radius of curvature, the tangential acceleration is determine by
an =
4. Determine the magnitude of the total acceleration: The
magnitude of the total acceleration is given by
a = at2 + an2 v2 r

18. 45 mi/h = 44 ft/s 45 mi/h = 66 ft/s v dv = at dx v dv = at dx 1
Problem 1 Solution
45 mi/h
45 mi/h = 44 ft/s
45 mi/h = 66 ft/s at
750 ft
Determine the tangential acceleration.
30 mi/h vo v xo x
v dv = at dx
1500 ft 66 44
750
v dv = at dx 2 1
( ) = at ( )
at = ft/s2

19. v dv = at dx v dv = -1.613 dx 1 (v12 - 662 ) = -1.613 ( 500 - 0 ) 2
Problem 1 Solution
at = ft/s2
1500 ft
45 mi/h v1 an
500 ft
The speed after the automobile
traveled 500 ft. vo v xo x
v dv = at dx 66 v1
500
v dv = dx 2 1
(v ) = ( )
v1 = 52.4 ft/s

20. v2 an = r ( 52.4 ft/s )2 an = = 1.828 ft/s2 1500 ft a = at2 + an2
Problem 1 Solution
45 mi/h
at = ft/s2
Determine the normal acceleration.
500 ft v2 r
52.4 ft/s an
an =
( 52.4 ft/s )2
an = = ft/s2
1500 ft
1500 ft
Determine the magnitude of the total acceleration.
a = at2 + an2
a = ( ft/s2 )2 + ( ft/s2 )2
a = 2.44 ft/s2

21. Problem 2 A a v0 B Knowing that the conveyor belt
15 ft A
25 ft a v0 B
Knowing that the conveyor belt
moves at the constant speed
v0 = 24 ft/s, determine the angle a
for which the sand is deposited on
the stockpile at B.

22. Knowing that the conveyor belt moves at the constant speed
Problem 2
15 ft A
25 ft a v0 B
Knowing that the conveyor belt
moves at the constant speed
v0 = 24 ft/s, determine the angle a
for which the sand is deposited on
the stockpile at B.
1. Analyzing the motion of a projectile: Consider the the vertical
and the horizontal motion separately.
2. Consider the horizontal motion: When the resistance of the air
can be neglected, the horizontal component of the velocity
remains constant (uniform motion). The distance, velocity, and
time are related by
x = x0 + (vx)0 t

23. Knowing that the conveyor belt moves at the constant speed
Problem 2
15 ft A
25 ft a v0 B
Knowing that the conveyor belt
moves at the constant speed
v0 = 24 ft/s, determine the angle a
for which the sand is deposited on
the stockpile at B.
3. Consider the vertical motion: When the resistance of the air
can be neglected, the vertical component of the acceleration is
constant (uniformly accelerated motion). The distance, velocity,
acceleration and time are related by
y = y0 + (vy)0 t g t2 1 2

24. (-15 ft) = 0 + (24 ft/s) sin a t - (32.2 ft/s2) t2 1 2
Problem 2 Solution
15 ft A
25 ft a v0 B x y
Consider the horizontal motion.
x = x0 + (vx)0 t
x = -25 ft
x0 = 0
(vx)0 = v0 cos a
(vx)0 = (24 ft/s) cos a
Consider the vertical motion
(25 ft) = 0 + (24 ft/s) cos a t 1 2
y = y0 + (vy)0 t g t2
y = -15 ft
y0 = 0
(vy)0 = v0 sin a
(vy)0 = (24 ft/s) sin a
Two equations with
two unknowns, a and t.
(-15 ft) = 0 + (24 ft/s) sin a t (32.2 ft/s2) t2 1 2

25. Problem 2 Solution
15 ft A
25 ft a v0 B x y
25 = cos a t
-15 = sin a t t2 1 2
Solve the equations for a and t. 25
24 t
eliminate a: sina = cos2a = ( )2 25
24 t
-15 = 24 t ( ) t2
t t = 0
two solutions:
t = s , a = 6.09o
t = s , a = 52.9o
a = 6.09o or 52.9o