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.