Part of a roller-coaster ride involves coasting down an incline and entering a loop 27.0 m in diameter. For safety considerations, the roller coasters speed at the top of the loop must be such that the force of the seat on a rider is equal in magnitude to the rider’s weight. From what height above the bottom of the loop must the roller coaster descend to satisfy this requirement?
To determine the height from which the roller coaster must descend to satisfy the requirement, we need to consider the forces acting on the rider at the top of the loop.
At the top of the loop, the rider experiences two forces: the gravitational force (weight) and the normal force exerted by the seat. In order for these two forces to be equal and the rider to feel a normal force equal to their weight, there must be no net force acting on the rider in the vertical direction.
To find the height, we can use the concept of centripetal force. At the top of the loop, the centripetal force is provided by the normal force from the seat. This centripetal force is equal to the product of mass (m) and the centripetal acceleration (ac).
The centripetal acceleration is given by ac = v^2 / r, where v is the velocity of the roller coaster at the top of the loop, and r is the radius of the loop (half the diameter).
The weight of the rider (mg) is acting downward, and the normal force (N) is acting upward. At the top of the loop, the net force in the vertical direction is given by N - mg, which must be equal to zero.
So, we have N - mg = 0, which implies N = mg.
Now, equating the centripetal force and the normal force, we have:
N = m * (v^2 / r).
Since N = mg, we can rewrite the equation as:
mg = m * (v^2 / r).
Simplifying, we get:
g = v^2 / r.
Rearranging the equation to solve for v, we have:
v^2 = g * r.
Taking the square root of both sides:
v = sqrt(g * r).
Now, we can consider the conservation of mechanical energy. At the top of the loop, the roller coaster has both potential energy (PE) and kinetic energy (KE).
PE = m * g * h, where h is the height above the bottom of the loop.
KE = (1/2) * m * v^2.
The sum of PE and KE at the top of the loop will be equal to the potential energy at the bottom of the loop (when the height is zero).
m * g * h + (1/2) * m * v^2 = m * g * 0.
Simplifying, we get:
h + (1/2) * v^2 / g = 0.
Substituting the value of v^2 from the earlier equation:
h + (1/2) * (g * r) / g = 0.
Simplifying further:
h + (1/2) * r = 0.
Solving for h:
h = - (1/2) * r.
Since height cannot be negative, we discard the negative sign:
h = (1/2) * r.
Finally, substituting the given radius of 27.0 m:
h = (1/2) * 27.0.
Therefore, the roller coaster must descend from a height of 13.5 meters above the bottom of the loop to satisfy the requirement.