Banked Roads & Aircraft | CAIE 9702 Paper 4 Notes

Paper 4

Physics9702

A car on a banked road

A slopes towards the inside of the bend.
At the , no friction is needed.
The vertical part of the normal contact force balances the weight. Its horizontal part provides the inward resultant.

At the design speed, only the normal contact force and weight act. The dashed arrows are parts of the normal force, not extra forces.

If the road angle $\theta$ is measured from the horizontal, the vertical part of the contact force balances the weight.
The horizontal part of the contact force is the inward resultant.
Divide the inward equation by the vertical equation to remove the normal contact force and the mass.

N\cos\theta=mg

N\sin\theta=\dfrac{mv^2}{r}

\tan\theta=\dfrac{v^2}{rg}

Worked example

The design speed of a road

A road is banked at 15°. The bend radius is 50 m. Find the design speed.

Rearrange the result for speed.
$v=\sqrt{rg\tan\theta}$
Substitute the radius and angle.
$v^2=50\times9.81\times\tan15^\circ$
$v=11.5\text{ m s}^{-1}$

Answer

v=11.5\text{ m s}^{-1}

An aircraft in a banked turn

is the force from the air on the aircraft wings.
When the aircraft banks, the lift force tilts with it.
The vertical part of lift balances the weight. The horizontal part turns the aircraft.
The pilot increases the total lift so that its vertical part still balances the weight.

Lift is perpendicular to the wings. Its vertical part balances weight, and its inward part turns the aircraft.

If $\theta$ is the angle between the wings and the horizontal, the vertical part of lift balances the weight.
The horizontal part of lift is the inward resultant.

L\cos\theta=mg

L\sin\theta=\dfrac{mv^2}{r}

Your turnQuick check

An aircraft turns at 50 m s⁻¹ in a horizontal circle of radius 500 m. Find its bank angle from the horizontal.

Show worked answer

Divide the force equations, then substitute the values.

\tan\theta=\dfrac{v^2}{rg}

\tan\theta=\dfrac{50^2}{500\times9.81}

\tan\theta=0.510

\theta=27.0^\circ

Friction away from the design speed

Below the design speed, the car would move down the slope without friction, so friction acts up the slope.
Above the design speed, the car would move up the slope without friction, so friction acts down the slope.
Decide the possible motion first. Friction acts in the opposite direction.

Common mistake

Friction does not always point towards the centre on a banked road. Its direction depends on whether the car is below or above the design speed.