8. Centripetal Forces & Gravitation

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

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?

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$2^{-2}$)

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.

Consider a train that rounds a curve with a radius of 530 m at a speed of 160 km/h (approximately 100 mi/h ). Calculate the friction force on the passenger if the train tilts at an angle of 8.0° toward the center of the curve.

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 concrete highway curve of radius 70 m is banked at a 15° angle. What is the maximum speed with which a 1500 kg rubber-tired car can take this curve without sliding?

A banked curve of radius R in a new highway is designed so that a car traveling at speed v₀ can negotiate the turn safely on glare ice (zero friction). If a car travels too slowly, then it will slip toward the center of the circle. If it travels too fast, it will slip away from the center of the circle. If the coefficient of static friction increases, it becomes possible for a car to stay on the road while traveling at a speed within a range from v_min to v_max. Derive formulas for v_min and v_max as functions of μ_s, v₀ and R.

(II) A 1250-kg car rounds a curve of radius 68 m banked at an angle of 14°. If the car is traveling at 85 km/h, will a friction force be required? If so, how much and in what direction?

An airplane traveling at 510 km/h needs to reverse its course. The pilot decides to accomplish this by banking the wings at an angle of 38° while moving in a level circular path. Find the time needed to reverse course.

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 $\vec{a}=(-15.7\hat{i}-23.2\hat{j}){\text{m/s}}^2$. Where (x and y) is the car at this instant?