A motorcycle moves around a vertical circle with a constant speed under the influence of the force of gravity , the force of friction between the wheels and the track , and the nor- mal force between the wheels and the track (a) Which of the following quantities has a constant magnitude?..(b) Which of the following quantities, when nonzero, is always directed toward the center of the circle?

The correct option is:

(a)
To moves around the vertical circle with a constant speed, constant force must act on the motorcyclist, and must be directed towards the center of the center. The centripetal acceleration necessary for the same is provided by the frictional force between the tires and the surface.
The magnitude of normal force depends on the position of the motorcyclist on the circular platform because it depends upon the force exerted by the motorcyclist on the ground, and is not always equal to its weight.
Therefore, the quantity that has constant magnitude either has frictional force or the action of gravity or both. Hence the correct option is (C) corresponding to .
(b)
It has been stated above that to move in a circle, a force must act on the motorcyclist to change its direction of motion. This force appears in the form of frictional force between the tires of motorcycle and the surface.
It is important to note that the effect of gravitational force always act downwards, but it does not always passes through the center and can be ruled out.
Therefore, (A) is the correct option, while the rest are ruled out.