Applications of UCM

Vehicle Along a Horizontal Circular Track:

Figure shows vertical section of a car on a horizontal circular track of radius r. Plane of figure is a vertical plane, perpendicular to the track but includes only centre C of the track. Forces acting on the car are

(i) weight mg, vertically downwards,

(ii) normal reaction N, vertically upwards that balances the weight mg and

(iii) force of static friction fs between road and the tyres.

This is static friction because it prevents the vehicle from outward slipping or skidding. This is the resultant force which is centripetal. 

Well (or Wall) of Death:

This is a vertical cylindrical wall of radius r inside which a vehicle is driven in horizontal circles. As shown in the figure, the forces acting on the vehicle (assumed to be a point) are

(i) Normal reaction N acting horizontally and towards the centre,

(ii) Weight mg acting vertically downwards, and

(iii) Force of static friction fs acting vertically upwards between vertical wall and the tyres. Magnitude of static friction is equal to mg, as this is the only upward force. Normal reaction N is thus the resultant centripetal force. Thus, in magnitude, 

Vehicle on a Banked Road:

While taking a turn on a horizontal road, the force of static friction between the tyres of the vehicle and the road provides the necessary centripetal force. However, the frictional force has an upper limit with a non-constant value. Thus, in real life, we should not depend upon it, as far as possible.

For this purpose, the surfaces of curved roads are tilted with the horizontal with some angle θ. This is called banking of a road or the road is said to be banked. Figure shows the vertical section of a vehicle on a curved road of radius r banked at an angle  with the horizontal.

Considering the vehicle to be a point and ignoring friction and other non-conservative forces like air resistance, there are two forces acting on the vehicle,

(i) weight mg, vertically downwards and

(ii) normal reaction N, perpendicular to the surface of the road.

As the motion of the vehicle is along a horizontal circle, the resultant force must be horizontal and directed towards the centre of the track. It means, the vertical force mg must be balanced. Thus, we have to resolve the normal reaction N along the vertical and along the horizontal. Its vertical component Ncosθ balances weight. Horizontal component Nsinθ being the resultant force, must be the necessary centripetal force. Thus, in magnitude,  

are (i) weight mg acting vertically downwards and

(ii) normal reaction N acting perpendicular to the road.

Only at this speed, the resultant of these two forces (which is Nsinθ) is the necessary centripetal force. In practice, vehicles never travel exactly with this speed. For speeds other than this, the component of force of static friction between road and the tyres helps us, up to a certain limit. 

Conical Pendulum:

A tiny mass (assumed to be a point object and called a bob) connected to a long, flexible, massless, inextensible string, and suspended to a rigid support is called a pendulum. When the string is revolved in such a way that the string moves along the surface of a right circular cone of vertical axis and the point object performs a (practically) uniform horizontal circular motion, it is called a conical pendulum.

Figure shows the vertical section of a conical pendulum having bob of mass m and string of length L. In a given position B, the forces acting on the bob are

(i) its weight mg directed vertically downwards and

(ii) the force T0 due to the tension in the string, directed along the string, towards the support A.

As the motion of the bob is a horizontal circular motion, the resultant force must be horizontal and directed towards the centre C of the circular motion. For this, all the vertical forces must cancel. Hence, we