What is the direction of the force exerted by the car on the passenger as the car goes over the top of the amusement ride under the following circumstances: (a) the car goes over the top at such a speed that the gravitational force is the only force acting? (b) The car goes over the top faster than this speed? (c) The car goes over the top slower than this speed?

The centripetal force causes a body to follow a curved path. It always moves in the opposite direction of the body, towards the fixed point of the path's instantaneous centre of curvature.

When a car goes in a vertical loop the two forces acting on the car in the case are its weight $\left(\mathrm{W}\right)$and the normal force of the car $\left(\mathrm{F}\right)$. When the car is on the top of a vertical loop these two forces must add up to give an external centripetal force that pushes you forward and counters at the centripetal force acting towards the centre and does the car does not fall off the rails . Also an inertial force act on the passenger that presses the body of the passenger to the outside of the loop as a ka spin around.

The car goes over the top at such a speed that the gravitational force is the only force acting: When the car goes over the top in such a way that the gravitational force is the only force acting, then the net centripetal force is due to the weight of the passenger and not the normal force on the car. Thus, the force acting on passengers by the car is in the downwards direction.

When the speed of the car is faster than this speed then the centripetal force increases which tends to push the passengers towards the centre of the vertical loop of the ride.

When the speed of the car is lower than its speed, then the centripetal force is less than the gravitational force and so the car pushes the passengers away from the centre.