Physics of Roller-Coaster Design

Introduction

• Discussion on designing a roller-coaster
• Focus on determining minimum speed at various points: top, bottom, and horizontal level
• Use of a specific numerical example involving a vertical loop with a radius of 15 meters

Key Problems

• Calculate minimum speed at the bottom to complete a vertical loop
• Determine speed at the top and horizontal level given minimum speed

Physics Concepts

Centripetal Force

• Required for circular motion
• Formula: ( F_c = \frac{mv^2}{r} )
• Example: Earth's orbit due to gravitational force

Free Body Diagram (FBD)

1. At the Top of the Loop
   • Forces:
     • Weight (( mg )) acting downwards
     • Normal force (( N_t )) acting downwards
   • Centripetal force provided by normal force and weight
   • Minimum speed condition: Normal force = 0
     • Formula: ( v = \sqrt{gr} )
     • Calculation: ( r = 15 \text{ m}, g = 9.8 \text{ m/s}^2 )
     • Minimum speed at the top: ( 12.12 \text{ m/s} )

Conservation of Energy

• Total energy = Kinetic energy + Potential energy
• Energy is conserved at any point in the loop

2. At the Bottom of the Loop

   • Forces:
     • Weight (( mg )) acting downwards
     • Normal force (( N_b )) acting upwards
   • Centripetal force formula: ( N_b - mg = \frac{mv_b^2}{r} )

3. Energy Considerations

   • At the Top:
     • Kinetic energy: ( \frac{1}{2}mv_t^2 )
     • Potential energy: ( mg \times 2r )
   • At the Bottom:
     • Only kinetic energy due to reference level
   • Minimum speed at bottom: ( v_b = \sqrt{5gr} )
     • Calculation: ( 27.1 \text{ m/s} )

4. Horizontal Level

   • Compare energy at this point and another
   • Height = ( r )
   • Kinetic energy + potential energy at horizontal level
   • Calculated horizontal speed: ( 21 \text{ m/s} )

Conclusion

• Minimum speed at the top, bottom, horizontal level are different
• For a complete loop, start speed must be ( 27.1 \text{ m/s} )
• Ideal calculations ignore friction and air resistance

Final Thoughts

• Encouragement to ask questions, like, share, and subscribe