In the vertical loop within a hallow cylindrical surface, the cyclist tends to move straight in accordance with its natural tendency. The curvature of cylinder, however, forces the cyclist to move along circular path (by changing direction). As such, the body has the tendency to press the surface of the cylindrical surface. In turn, cylindrical surface presses the body towards the center of the circular path.
The free body diagram of the cyclist at an angle “θ” is shown in the figure. We see that the resultant of normal force and component of weight in the radial direction meets the requirement of centripetal force in radial direction,
N

m
g
cos
θ
=
m
v
2
r
N

m
g
cos
θ
=
m
v
2
r
The distinguishing aspects of circular motion in vertical plane are listed here :
 Motion in a vertical loop is a circular motion – not uniform circular motion. It is so because there are both radial force (N – mg cosθ) and tangential force (mg sin θ). Radial force meets the requirement of centripetal force, whereas tangential force accelerates the particle in the tangential direction. As a result, the speed of the cyclist decreases while traveling up and increases while traveling down.
 Centripetal force is not constant, but changing in magnitude as the speed of the cyclist is changing and is dependent on the angle “θ”.
The cyclist is required to maintain a minimum speed to avoid free fall. The possibility of free fall is most stringent at the highest point of the loop. We, therefore, analyze the motion at the highest point with the help of the free body diagram as shown in the figure.
N
+
m
g
=
m
v
2
r
N
+
m
g
=
m
v
2
r
We can also achieve the result as above by putting the value θ=180° in the equation obtained earlier.
Th minimum speed of the cyclist corresponds to the situation when normal force is zero. For this condition,
m
g
=
m
v
2
r
v
=
√
(
r
g
)
m
g
=
m
v
2
r
v
=
√
(
r
g
)
This motion is same as discussed above. Only difference is that tension of the string replaces normal force in this case. The force at the highest point is given as :
T
+
m
g
=
m
v
2
r
T
+
m
g
=
m
v
2
r
Also, the minimum speed for the string not to slack at the highest point (T = 0),
m
g
=
m
v
2
r
v
=
√
(
r
g
)
m
g
=
m
v
2
r
v
=
√
(
r
g
)
The complete analysis of circular motion in vertical plane involves considering forces on the body at different positions. However, external forces depend on the position of the body in the circular trajectory. The forces are not constant forces as in the case of circular motion in horizontal plane.
We shall learn subsequently that situation involving variable force is best analyzed in terms of energy concept. As such, we will revisit vertical circular motion again after studying different forms of mechanical energy.