Loop the Loop a “Fifth Gear” stunt special
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Transcript of Loop the Loop a “Fifth Gear” stunt special
Loop the Loopa “Fifth Gear” stunt special
Hugh Hunt
Cambridge University Engineering Department
2R
h = 5R/2
Vo = 0
V1 = 5gR
A
B
C
V1 = 5gR V1 = 5gR
V2 = gR
From A to B, conserve KE+PEmgh = ½mV1
2 so h = V1
2/2g = 5R/2
G-forces are V2/R : At B : 5g + g = 6g At C : g – g = 0
From B to C, conserve KE+PE ½mV1
2 = ½mV22 + 2mgR
so V12 = V2
2 + 4gR .or V1 = 5gR .
At C (top of the loop) centrifugal force exactly balances weight, so
mg = mV22 / R
or V2 = gR
Simple physics
12m
A
B
C
V1 = 39mph
V2 = 17mph
V1 = 39mph V1 = 39mph
15m
Fifth-gear Stunt Special:
R = 6m V1 = 17.2 ms-1 (39 mph) V2 = 7.7 ms-1 (17 mph) and G-force at B = 6g
12m
B
C V2 = 17mph
V1 = 39mph V1 = 39mph
g g
6g
zero g
bad assumptions:1. particle2. no friction3. rigid suspension4. perfect tyres5. short wheelbase6. G-force not important7. ignore transients8. engine not powering9. planar – no steering10. rigid structure
Optimum shape for the loop(not a circle!)
It all depends on the criteria. For these rollercoasters, height is not an issue. Fifth-gear could have constructed such a shape, but it would somehow have been
less impressive…CONSTRAINT: entry and exit horizontal and coincident
Steering
The line drawn in here is correct – no steering
would be required
The line actually painted has a kink in it – steering is needed !
Transient
What happens to the suspension and tyres when the weight of the car goes from 1000kg to 6000kg ?
Here is a notional graph of the motion of a simple vehicle subject suddenly to a change in g to 6g . The tyre and
suspension behave non-linearly.
1/6 ?
The suspension bottoms out, the vehicle grinds out, the tyre blows out – all pretty undesirable. solutions: stiffen suspension increase tyre pressure from 30psi to …. ???
Engine power and friction
Toyota Aygo: 67bhp, say 40kW
PE increase = 110 kJ
If time to complete loop = 4.5 seconds then engine provides 90kJ on the way up
Friction: = 1 , mean “G-force” = 3g, energy lost to friction = 1000 kJ
Conclude: Most important to avoid “grinding out” – friction (even as low as = 0.1) may cause the stunt to fail.