Answered

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An engineer wishes to design a roller coaster so that
the cars will not fall when they are at the top of their
circular path.
A 500 kg roller coaster car is designed to travel at 20
m/s at the top of the circle. The engineer wants to
make the circle as large as possible while still being
at least one quarter the maximum radius for safety
reasons. How large can the radius of the circle be?

Sagot :

To prevent cars from falling, the radius at the top of the circle should be

small such cars inverted at the top remain attached during motion.

Correct response;

The radius of the coaster can be C) 20 m

Method by which the above option is selected

Mass of roller coaster car, m = 500 kg

Speed at the top of the circle, v = 20 m/s

Required:

The maximum radius of the circular path the roller coaster car.

Solution:

[tex]\displaystyle Centrifugal \ force, \, F_{c} = \mathbf{\frac{m \cdot v^2}{r}}[/tex]

Where;

r = The radius of the circular path.

Weight of the roller coaster car = m·g = The centripetal force

Where;

g = Acceleration due to gravity = 9.81 m/s²

At equilibrium, we have;

Centrifugal force = Centripetal force

[tex]\displaystyle \frac{m \cdot v^2}{r} = \mathbf{ m \cdot g}[/tex]

Therefore;

[tex]\displaystyle r = \mathbf{ \frac{v^2}{g}}[/tex]

Which gives;

[tex]\displaystyle r = \frac{20^2}{9.81} \approx 40.77[/tex]

The maximum radius for safety of a roller coaster, r ≈ 40.77 meters

[tex]\displaystyle Range \ of \ radius \ of \ the \ circle = \frac{40.77}{4} \leq Radius \ of \ circle \leq 40.77[/tex]

Which gives;

Range of the radius of the circle = 10.2 ≤ Radius of circle ≤ 40.77

The correct option for safety considerations is therefore;

  • C) 20 m

The possible question options are;

A) 5 m  B) 10 m  C) 20 m  D) 40 m  E) 80 m

Learn more about centripetal force here:

https://brainly.com/question/12674230

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