Quick questions on Uniform circular motion, centripetal force and banked curves (QCE Physics Unit 3)

On a curve banked at angle $\theta$ to the horizontal, the normal force is perpendicular to the road surface. Resolving:

Expect a banked-curve diagram with the bank angle and a speed, asked to determine whether the vehicle stays on the curve, to find the design speed, or to extract the friction coefficient required at a non-design speed. Alternatively, a conical pendulum stimulus with multiple period measurements at different angles.

Common designs: a conical pendulum with mass, period and angle measured to test $T = 2 \pi \sqrt{L \cos\theta / g}$; a rotating-mass setup on a horizontal turntable with friction varied; a ball-in-a-cone or marble-in-a-bowl rolling at different heights. Strong reports linearise the relationship before fitting (e.g. $T^2$ vs $\cos\theta$ slopes give $4\pi^2 L / g$) and propagate uncertainty cleanly.

State the centripetal acceleration and force formulas. [2 marks]

A $1200 \text{ kg}$ car rounds a flat horizontal curve of radius $50 \text{ m}$ at $15 \text{ m s}^{-1}$. Calculate the required centripetal force and identify the source. [3 marks]

A Bremer River bridge banked approach: $r = 80 \text{ m}$, $\theta = 6^\circ$. (a) Calculate the frictionless design speed. (b) Determine the friction force on a $1500 \text{ kg}$ car at $20 \text{ m s}^{-1}$ ($72 \text{ km h}^{-1}$).