Rollercoaster

7630 /www/nrich/html/content/id/7630/ 1 0000-00-00T00:00:00 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> Consider a circular rollercoaster track of radius $10\mathrm m$. You've carelessly forgotten to put your seat belt on. At what speed do you need to travel to not fall out of your seat when the cart is upside down? If you were $5$ kilograms lighter, how much smaller would this minimum speed be? <mdo:image alt="" src="cart.png" style="width: 400px; height: 194px;"></mdo:image> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> At the top of the loop, the forces you experience are gravity and the normal reaction force from the cart. Here, these are both pointing directly downwards. The total centripetal force experienced due to the circular motion is therefore equal to the sum of these: $$F_c = F_g + F_n \Rightarrow \frac{mv^2}{r} = mg + F_n$$ When the cart's at the smallest possible speed with you remaining in the cart, we have that at the top of the loop $F_n = 0$ i..e your seat is exerting no force on you. At this minimum speed: $$\frac{mv^2}{r} = mg \Rightarrow v^2=gr \Rightarrow v=\sqrt{gr}$$ In our example, $v=\sqrt{10g}\mathrm{ms}^{-1} \approx 10\mathrm{ms}^{-1}$. Note this answer is independent of mass. <mdo:image alt="" src="cart3.png" style="width: 481px; height: 293px;"></mdo:image> </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> The minimum speed occurs when at the top of the loop your seat exerts no force on you. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> At the top of the loop, the forces you experience are gravity and the normal reaction force from the cart. Here, these are both pointing directly downwards. The total centripetal force experienced due to the circular motion is therefore equal to the sum of these: $$F_c = F_g + F_n \Rightarrow \frac{mv^2}{r} = mg + F_n$$ When the cart's at the smallest possible speed with you remaining in the cart, we have that at the top of the loop $F_n = 0$ i..e your seat is exerting no force on you. At this minimum speed: $$\frac{mv^2}{r} = mg \Rightarrow v^2=gr \Rightarrow v=\sqrt{gr}$$ In our example, $v=\sqrt{10g}\mathrm{ms}^{-1} \approx 10\mathrm{ms}^{-1}$. Note this answer is independent of mass. <mdo:image alt="" src="cart3.png" style="width: 481px; height: 293px;"></mdo:image> </mdoxml> 2 3 0 0 0 0 1 Rollercoaster Calculate minimum speed needed to go round a circular loop on a rollercoaster.