# Roller Coaster and Centripetal Acceleration

• hrithikguy
In summary, the height from which the roller coaster must descend to satisfy the requirement of equal forces is 3 times the radius of the loop. This can be derived using the equations for centripetal force and conservation of energy.
hrithikguy

## Homework Statement

Part of a roller-coaster ride involves coasting down an incline and entering a loop 8.00 m in diameter. For safety considerations, the roller coaster 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 weigt. From what height above the bottom of the loop must the roller coaster descend to satisfy this requirement?

## Homework Equations

a_c = v^2/r
Conservation of Energy

## The Attempt at a Solution

m* a_c = mg

m* v^2/r = mg
v^2/r = g
v^2 = rg

mgh = 1/2 mv^2 + mg * 8
9.8 * h = 1/2 * rg + 8 * 9.8

9.8 * h = 1/2 * 4 * 9.8 + 8 * 9.8
h = 2 + 8
h = 10

Please guide me to the solution.

Thanks!

hrithikguy said:

## The Attempt at a Solution

m* a_c = mg

m* v^2/r = mg
There act two forces on the rider, one is gravity (mg) and the other is the normal force N from the seat. The problem says that the seat must exert a force on the rider equal to mg. The resultant force is equal to the centripetal force.

ehild

So does this mean that n = -mg, so

$m * a_c = 2mg ?$

$a_c = 2g$$a_c = 19.6$

$v^2/r = 19.6$
$v^2 = 19.6 * 4 = 78.4$
$v = 8.85$

$9.8 * h = 1/2 * 9.85^2 + 8 * 9.8$

$x = 12.95$

Hmm.. this is closer, but still not correct.

Last edited:
You used v = 9.85 instead of 8.85. It is much better if you do not evaluate the equations, but express the height h in terms of R.

ehild

Ah thank you! I plugged in 8.85 now and it came out to 11.996, which is close enough.
Thanks!

I show the other way: mv^2/R=2mg
hmg=mg*2R+1/2 mv^2 -->

v^2=2gR,
hg=2gR+1/2v^2--->hg = 3gR ---> h=3R.

ehild

## 1. What is a roller coaster?

A roller coaster is a type of amusement ride that is designed with tracks, steep drops, and sharp turns to create a thrilling experience for riders. It typically consists of a train-like vehicle that runs along the tracks at high speeds.

## 2. How does a roller coaster work?

A roller coaster works by converting potential energy into kinetic energy. The train is pulled up to the highest point of the track, which gives it potential energy. As it drops down the track, the potential energy is converted into kinetic energy, which propels the train forward.

## 3. What is centripetal acceleration?

Centripetal acceleration is the acceleration that an object experiences when it moves in a circular path. It is always directed towards the center of the circle and is caused by a force, such as gravity, that pulls the object towards the center.

## 4. How does centripetal acceleration affect a roller coaster ride?

Centripetal acceleration plays a crucial role in determining the speed and direction of a roller coaster ride. As the train moves along the curved track, it experiences a centripetal force that keeps it on the track and causes it to accelerate towards the center of the curve. This acceleration is what creates the thrilling sensation of being pushed against your seat or pulled towards the outside of a turn.

## 5. What factors can affect the centripetal acceleration of a roller coaster?

The factors that can affect the centripetal acceleration of a roller coaster include the speed of the train, the radius of the curved track, and the mass of the train. A faster speed, smaller radius, and heavier train will result in a higher centripetal acceleration. Additionally, the angle and banking of the track can also affect the centripetal acceleration experienced by riders.

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