Name: Group member: TAM 212 Workheet 3 Solution The workheet i concerned with the deign of the loop-the-loop for a roller coater ytem. Old loop deign: The firt generation of loop wa circular, a hown below. R New loop deign: The modern loop ha evolved into the tear-drop track hape pictured below. 1
1. Briefly predict what iue might arie due to a circular loop, and how a tear-drop loop might reolve thoe problem. After a train enter thi circular loop, the paenger who will feel thi a a udden jerk come from the centripetal acceleration. Rarely, thi jerk even caue a neck injury. Circular Track The firt generation of loop were circular, a illutrated on Page 1. Although not trictly accurate, we ll aume for thi ection that the roller coater train maintain a contant peed a it travel along the track. 2. For the circular loop, plot the curvature κ of the track a a function of, the total ditance covered. Label the important point on the vertical axi in term of the loop radiu R. Note that 1 and 2 denote the point where the train enter and leave the loop, repectively. κ 1/R 1 2 3. Plot the tangential a t and normal a n component of the train acceleration a = a t ê t + a n ê n a a function of, the total ditance covered. Label the important point on the vertical axi in term of the train peed v and the loop radiu R. a t a n v 2 /R 1 2 1 2 2
4. The circular loop deign, popular in the earliet inverion roller coater, wa in fact reponible for many broken bone and neck injurie. Why do you think thi may have occurred? The normal component of the acceleration will uddenly jump from zero on the traight-line egment to v 2 /R on the loop. Thi will not be afe for paenger, who will be jerked upward a a reult of the udden change in acceleration. 5. Now plot the a z (upward, ˆk) component of the train acceleration a a function of, the total ditance covered. Label all ignificant point on the vertical axi. a z v 2 /R 1 2 -v 2 /R Tear-Drop Track The modern loop ha evolved into the teardrop-like hape a exhibited by the roller coater on Page 1. Alo, we ll aume for thi ection that the roller coater train maintain a contant peed. 6. To reduce rik of injury, the teardrop hape for the loop hown below i now commonly employed. How do you think thi hape help to give paenger a moother ride? The curvature of the loop appear to vary more continuouly, tarting from zero and increaing continuouly a we approach the top, and then decreaing continuouly back to zero a we exit the curve. The continuou variation in curvature mean there will not be a jump in the acceleration. 3
7. A curve for which the curvature varie linearly with the ditance covered i known a a clothoid or Euler piral. The teardrop hape above cloely reemble two ymmetric clothoid, in which the curvature increae linearly with after the train enter the curve, reache a maximum value of 1/R at the top of the curve, and then linearly decreae back to zero when the train exit the loop. For uch a loop, plot the curvature κ a a function of. Note that A and E denote the point where the train enter and leave the loop (a illutrated on the next page). κ 1/R A E 8. For the teardrop loop, ketch the normal and tangential acceleration (labeling each) at the five point demarked below. C B D A E Normal acceleration hown; there i no tangential acceleration ince the train peed i aumed contant. 9. For the teardrop loop, plot the tangential a t and normal a n component of the train acceleration a = a t ê t + a n ê n a a function of, the total ditance covered. Label the important point on the vertical axi in term of the train peed v and the radiu or curvature at the top of the loop R. a t a n v 2 /R A B C D E A B C D E 4
10. Now plot the a z (upward, ˆk) component of the train acceleration on the teardrop loop a a function of, the total ditance covered. Where i the acceleration in the ˆk direction felt by the paenger larget? How large i thi? a z A B C D E -v 2 /R The acceleration experienced by the paenger i larget at the top of the loop (magnitude v 2 /R, directed downward). 5
Challenge: Energy Analyi 11. A you have likely experienced on roller coater, the peed doe not tay contant a you travere the loop. Intead, the peed decreae a you travel up the curve and and increae a you move back down the track. Ue conervation of energy to calculate the expected peed at the top of a 25 m tall loop when the initial peed i 10 m/, where kinetic energy i KE = 1 2 mv2 and potential energy i P E = mgh. 12. Baed on your energy analyi, i contant peed a reaonable aumption? How would your analyi in the firt two ection change if the velocity become dependent on track poition? 13. What other phyic would you include to make your analyi even more accurate? 6