8. Centripetal Forces & Gravitation

A bobsled turn banked at 78° is taken at 24 m/s. Assume it is ideally banked and there is no friction between the ice and the bobsled. Calculate the centripetal acceleration of the bobsled.

A $1125$-kg car and a $2250$-kg pickup truck approach a curve on a highway that has a radius of $225$ m. At what angle should the highway engineer bank this curve so that vehicles traveling at $65.0$ mi/h can safely round it regardless of the condition of their tires? Should the heavy truck go slower than the lighter car?

A car drives at a constant speed around a banked circular track with a diameter of 145 m. The motion of the car can be described in a coordinate system with its origin at the center of the circle. At a particular instant the car’s acceleration in the horizontal plane is given by $\vec{a}=(-15.7\hat{i}-23.2\hat{j}){\text{m/s}}^2$. What is the car’s speed?

(II) How large must the coefficient of static friction be between the tires and the road if a car is to round a level curve of radius 85 m at a speed of 95 km/h?

An airplane feels a lift force $\overrightarrow{L}$ perpendicular to its wings. In level flight, the lift force points straight up and is equal in magnitude to the gravitational force on the plane. When an airplane turns, it banks by tilting its wings, as seen from behind, by an angle from horizontal. This causes the lift to have a radial component, similar to a car on a banked curve. If the lift had constant magnitude, the vertical component of $\overrightarrow{L}$ would now be smaller than the gravitational force, and the plane would lose altitude while turning. However, you can assume that the pilot uses small adjustments to the plane's control surfaces so that the vertical component of $\overrightarrow{L}$ continues to balance the gravitational force throughout the turn. Find an expression for the banking angle $\theta$ needed to turn in a circle of radius $r$ while flying at constant speed $v$.

A car is traveling around a curve of radius $100$ m that is banked at an angle of $20.0°$ with respect to the horizontal. What is the ideal speed at which the car can take the curve without relying on friction? (Assume $g=9.8$ m/s$2^{-2}$)

When a curved road is banked at an angle $\theta$ for a car of mass $m$ traveling at speed $v$ around a curve of radius $r$, what is the primary purpose of banking the road?

When driving on a banked curve designed for a specific speed, how should you leave the curve to maintain maximum safety and control?