What is the minimum height of a roller coaster in terms of "r" such that you can easily complete the loop- de- loop safely?
To determine the minimum height of a roller coaster in terms of "r" (the radius of the loop-de-loop) to safely complete the loop, we need to consider the forces acting on the rider. The key is to ensure that the net force at the top of the loop is sufficient to keep the rider in contact with the track.
At the top of the loop-de-loop, there are two forces acting on the rider:
1. The gravitational force (mg), which acts downward.
2. The normal force (N), which acts perpendicular to the track.
For the rider to stay in contact with the track at the top of the loop, the net force must be pointing toward the center of the loop. This means that the net force is the difference between the gravitational force and the normal force.
Net force (F_net) = mg - N
At the topmost point of the loop, the net force should be greater than or equal to zero to ensure the rider doesn't lose contact with the track. Therefore:
F_net ≥ 0
Substituting the values:
mg - N ≥ 0
Since N represents the force required to provide the necessary centripetal acceleration to keep the rider moving in a circle, it can be calculated as follows:
N = m * v^2 / r
Where:
m = mass of the rider
v = velocity of the rider at the top of the loop (which depends on the total energy of the system)
Combining these equations, we have:
mg - m * v^2 / r ≥ 0
To find the minimum height (h) in terms of the radius (r), we need to first express v in terms of r.
Using conservation of energy, at the top of the loop, the sum of potential energy (m * g * h) and kinetic energy (1/2 * m * v^2) should be equal to the initial potential energy (m * g * h0) of the coaster at the highest point before the loop.
m * g * h + 1/2 * m * v^2 = m * g * h0
Simplifying the equation:
v^2 = 2 * g * (h0 - h)
Substituting this expression for v^2 in the earlier equation:
mg - m * (2 * g * (h0 - h)) / r ≥ 0
Simplifying further:
mrg - 2mgh0 + 2mgh ≥ 0
Factoring out the common term "mg":
mg (r - 2h0 + 2h) ≥ 0
For the inequality to hold true, the term in parentheses must be greater than or equal to zero:
r - 2h0 + 2h ≥ 0
Simplifying this equation:
2h ≥ 2h0 - r
Dividing by 2:
h ≥ h0 - r/2
Therefore, the minimum height (h) of the roller coaster in terms of "r" such that you can easily complete the loop-de-loop safely is:
h ≥ h0 - r/2
Where "h0" is the initial height of the coaster before the loop.