7.46.
Riding a Loop-the-Loop. A car in an amusement park ride rolls without friction around the
track shown in the figure. It starts from rest at point A at a height h above the bottom of the loop.
Treat the car as a particle. (a) What is the minimum value of h (in terms of R) such that the care
moves around the loop without falling off at the top (point B)? (b) If h = 3.50R and R = 20.0 m,
compute the speed, radial acceleration, and tangential acceleration of the passengers when the car
is at point C, which is at the end of a horizontal diameter. Show these acceleration components in
a diagram, approximately to scale.


F  ma
Identify: Apply K1 + U1 + WNC = K2 + U2 to relate h and vB . Apply 
the minimum speed required at B for the car not to fall off the track.
at point B to find
2
2
At B, a  vB / R , downward. The minimum speed is when n  0 and mg  mvB / R . The
Set up:
minimum speed required is vB  gR . K1  0 and Wother  0 .
Execute:
2
1
(a) KA + UA = KB + UB applied to points A and B gives U A  U B  2 mvB . The speed at the
top must be at least
gR .
1
5
mg (h  2R)  mgR, or h  R.
2
2
Thus,
(b) Apply KA + UA = KC + UC to points A and C. U A  UC  (2.50) Rmg  K C , so
vC  (5.00) gR  (5.00)(9.80 m/s2 )(20.0 m)  31.3 m/s.
arad 
vC2
 49.0 m/s 2 .
R
The tangential direction is down, the normal
The radial acceleration is
force at point C is horizontal, there is no friction, so the only downward force is
2
gravity, and atan  g  9.80 m/s .
Evaluate: If h  2 R , then the downward acceleration at B due to the circular motion is greater
than g and the track must exert a downward normal force n. n increases as h increases and
5
hence vB increases.