loop_the_loop

loop_the_loop - Analysis of the forces acting on a pilot...

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Unformatted text preview: Analysis of the forces acting on a pilot during a loop. 1. At the top (inside the loop) Since At the top the two forces r Nt there are two possibilities to consider: acting on the pilot are (1) r the normal force ( N t ) due r mg Top to the seat, and (2) the gravitational force due to the Earth. In executing the mv 2 − mg , Nt = R • If mv 2 ≥ mg , Nt > 0, i.e., the seat applies a force R on the pilot, but it is less than his true weight (mg). By maneuver, the net downward Newton’s 3rd Law this is the same as the force exerted force produces the centripetal R by the pilot on the seat. Therefore, the pilot experiences force, i.e., Ftop = Nt + mg = mv 2 . R So, the magnitude of the force of the seat on the pilot, i.e., the nomal force, is: Nt = an apparent weight (Nt) that is less than mg. • If mg > mv 2 , Nt < 0, the pilot will “fall” from the R seat unless restrained by a seat-belt! (The seat can only 2 mv − mg . R provide a positive force.) 1 Analysis of the forces acting on the riders on a roller 2. At the bottom (inside the loop): coaster ride doing a loop: At the bottom, the free 3. At the top (inside the loop) body diagram tells us the centripetal force is: R mv 2 = Nb − mg . R Bottom v So, the magnitude of the r Nb oo N mg force exerted by the seat on the pilot is r mg Nb = mg + mv 2 r which is, clearly, greater than mg, the true weight of the pilot. But, by the 3rd Law, Nb is also the magnitude of the force the pilot on the seat, i.e., the apparent weight of the pilot. Thus, his apparent weight is greater than his true weight. R The analysis is the same as case #1, i.e., the centripetal force experienced by the rider is mv 2 = N + mg , R and so the force applied by the seat on the rider is mv 2 − mg. N= R 2 4. At the top (outside the loop) N mg v R 2 So, if mg > mv , N < 0, and the riders will “fall” from R their seat (as a seat can only provide a positive force!) The free body diagram gives: mv 2 = mg − N, R Consequently, there is a minimum speed ( v min ) for a roller coaster to carry out this maneuver safely, i.e., v min = Rg . so the force applied by the seat on the rider is With a speed less than this value, the riders will fall N = mg − from their seats. (Note, it does not depend on the rider’s mass.) mv 2 . R Providing v < Rg , then N > 0 , so the rider remains in their seat. However, if v > Rg , then N < 0 the rider “flies” from their seat (unless restrained)! Thus, there is a maximum speed for this maneuver to be safely executed. 3 ...
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