A roller coaster track is designed so that the car travels upside down on a certain portion of the track, as shown in Figure P5.20. What is the minimum speed the roller coaster can have without falling from the track? Assume the track has a radius of curvature of 32.1 m.
To determine the minimum speed the roller coaster can have without falling from the track, we can use the concept of centripetal force.
Centripetal force is the force that keeps an object moving in a circular path. It is given by the formula:
F_centripetal = (m * v^2) / r
Where:
F_centripetal is the centripetal force,
m is the mass of the roller coaster,
v is the velocity of the roller coaster, and
r is the radius of curvature of the track.
In this case, we want to find the minimum speed, so we consider the roller coaster at the point where it is just about to fall off the track. At this point, the centripetal force is equal to the force of gravity.
Let's assume the mass of the roller coaster is m.
The force of gravity is given by the formula:
F_gravity = m * g
Where:
g is the acceleration due to gravity.
Since the centripetal force and the force of gravity are equal at the point of almost falling off the track, we can set the two equations equal to each other:
(m * v^2) / r = m * g
Simplifying the equation, we have:
v^2 = r * g
Now we can solve for the minimum speed (v) by taking the square root of both sides of the equation:
v = sqrt(r * g)
Given that the radius of curvature of the track is 32.1 m and the acceleration due to gravity is approximately 9.8 m/s^2, we can plug in these values into the equation to find the minimum speed:
v = sqrt(32.1 * 9.8)
Calculating the value, we find:
v ≈ 17.7 m/s
Therefore, the minimum speed the roller coaster can have without falling from the track is approximately 17.7 m/s.
To determine the minimum speed the roller coaster can have without falling from the track, we can use the concept of centripetal force and gravitational force.
The minimum speed occurs when the centripetal force provided by the track just balances the gravitational force acting on the car. At this speed, the car is on the verge of falling off the track but doesn't actually fall.
Let's break down the forces involved:
1. Centripetal force (Fc): This is the force that keeps an object moving in a curved path and is directed toward the center of the circle. In this case, it is provided by the normal force from the track pushing inward on the car.
2. Gravitational force (Fg): This is the force due to gravity pulling the car downward, acting vertically downward.
For the car to remain on the track, the centripetal force (Fc) must be equal to or greater than the gravitational force (Fg).
The centripetal force (Fc) can be calculated using the equation:
Fc = mv^2 / r
Where:
- m is the mass of the car
- v is the velocity of the car
- r is the radius of curvature of the track
The gravitational force (Fg) can be calculated using the equation:
Fg = mg
Where:
- m is the mass of the car
- g is the acceleration due to gravity (approximately 9.8 m/s^2)
Since the mass (m) of the car cancels out in both equations, we can ignore it for simplicity.
Setting Fc equal to Fg and rearranging the equation, we get:
mv^2 / r = mg
Canceling out 'm' from both sides of the equation, we get:
v^2 / r = g
Solving for v, we have:
v = sqrt(r * g)
Now we can substitute in the given values:
r = 32.1 m
g = 9.8 m/s^2
Calculating the minimum speed (v), we get:
v = sqrt(32.1 * 9.8) = 17.96 m/s
Therefore, the minimum speed the roller coaster can have without falling from the track is approximately 17.96 m/s.