A roller coaster at an amusement park has a dip that bottoms out in a vertical circle of radius (r). A passenger feels the seat of the car pushing upward on her with a force equal to two and a half times her weight as she goes through the dip. If r = 24.0 m, how fast is the roller coaster traveling at the bottom of the dip?
To find the speed of the roller coaster at the bottom of the dip, we can use the concept of centripetal force and equate it to the net force acting on the passenger.
First, let's analyze the forces acting on the passenger. The passenger experiences two forces: the gravitational force (weight) pulling her downward and the normal force exerted by the seat of the car pushing her upward. At the bottom of the dip, these forces combine to provide the net force.
Let's denote the passenger's weight as W. The normal force exerted by the seat will be 2.5 times her weight, which we can write as 2.5W.
At the bottom of the dip, the net force acting on the passenger is given by:
Net Force = Normal Force - Weight
So, Net Force = 2.5W - W = 1.5W
Now, we can relate this net force to the centripetal force acting on the passenger as she moves in a vertical circle. At the bottom of the dip, the net force is the centripetal force, given by:
Centripetal Force = Net Force = 1.5W
The centripetal force required to keep an object moving in a circle is given by the formula:
Centripetal Force = (mass × velocity^2) / radius
In this case, the mass of the passenger cancels out, so we get:
Velocity^2 / radius = 1.5W
Now, we can rearrange the equation to solve for velocity:
Velocity = √(1.5W × radius)
Substituting the value of the radius (r = 24.0 m), we can find the speed at the bottom of the dip by calculating:
Velocity = √(1.5W × 24.0)
However, we still need the value of the passenger's weight (W). Let's assume an average weight of 68 kg for an adult.
Plugging in the values, we get:
Velocity = √(1.5 × 68 × 9.8 × 24.0)
Evaluating this expression will give us the speed of the roller coaster at the bottom of the dip.