If the track is to be designed so that the passengers of the roller coaster do not experience
a normal force equal to zero or more than 4 times their weight, determine the limiting
heights ℎ𝐴 and ℎ𝑐
so that this does not occur. The roller coaster starts from rest at position
A. Neglect friction.
To determine the limiting heights hA and hc for the track so that the passengers do not experience a normal force equal to zero or more than 4 times their weight, we need to analyze the forces acting on the passengers at these points.
Let's consider the different forces acting on the passengers at positions A and C:
1. Weight (mg): This is the force acting downwards due to the gravitational pull on the passengers. Its magnitude is given by the mass of the passengers (m) multiplied by the acceleration due to gravity (g).
2. Normal force (N): The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force acts perpendicular to the track and counterbalances the weight of the passengers. It can vary depending on the curvature of the track and the orientation of the passengers.
For position A:
At the highest point of the roller coaster (position A), the passengers will experience a normal force equal to zero. This is because the track is concave upwards, and the passengers tend to move away from the track due to their inertia as the roller coaster goes up. Therefore, the normal force is momentarily zero.
This means that the limiting height hA is the height at which the normal force becomes zero. At that point, only the weight of the passengers acts on them.
For position C:
At the bottom of the roller coaster loop (position C), the passengers experience the maximum normal force. This occurs when the passengers are in contact with the bottom of the loop and are moving in the same direction as the track. In this case, the track provides an additional force, resulting in the normal force being greater than the weight of the passengers.
To calculate the maximum normal force at position C, we need to consider the net force acting on the passengers. Since no friction is mentioned, the only force acting horizontally is the normal force. Therefore, the net force will be the horizontal component of the normal force.
At position C, the net force (Fn) acting on the passengers can be written as:
Fn = N - mg
For the passengers not to experience a normal force more than four times their weight, we can write the inequality:
N - mg ≤ 4mg
Solving this inequality will give us the maximum normal force, and we can use it to determine the limiting height hc.
It is important to note that we need additional information about the shape and dimensions of the roller coaster track to determine the exact limiting heights hA and hc. Without this information, we can only provide you with the general approach to solve the problem.