Banked Curves Practice Problems

A 1000-kg van and a 2000-kg bus come closer to a circular track with a radius of 200 m. 

(i) What should be the banking angle for this circular track so that automobiles traveling at 40.0 m/s can take a turn without toppling? 

(ii) Should the bus move slower than the van?

The banked, circular highway curves are specifically designed for high-speed driving. To prevent vehicles from slipping outward, an 85 m radius highway curve is banked at a 28° inclination. Determine the maximum speed with which a high-speed car weighing 1800 kg can safely maneuver this curve without sliding off the road. The coefficient of friction between the tire and the road is 1.

A spacecraft is on a mission to explore Mars. When the spacecraft entered the Martian atmosphere, the commander made a balanced turn by tilting the spacecraft wings at an angle α to the horizontal. The commander maintained the center of gravity of the spacecraft in a horizontal plane. The speed V_s of the spacecraft was constant, and the spacecraft moved in a circle of radius R_s. Calculate the tilting angle α using V_s, Mars' gravitational acceleration g, and R_s. Note that the lift force is directed perpendicular to the wings and in the opposite direction of gravity.

What is the static friction coefficient required between the wheels and the track for a vehicle to safely navigate a level turn with a radius of 80 m at a speed of 90 km/h?

A test track for autonomous electric vehicles (EVs) features a banked curve with a certain radius, designed to test the vehicles' ability to navigate turns at a specified speed without depending on friction. This allows for testing under various environmental conditions, which can alter the coefficient of static friction between the tires and the road surface. Determine the minimum and maximum speeds, $v_{\min}$ and $v_{\max}$, at which an autonomous EV can safely handle the banked curve. These speeds should be calculated as functions of the coefficient of static friction ($\mu_{s}$), the intended test speed ($v_0$), and the curve's radius ($R$).

A 1251-kg vehicle takes a 73 m radius turn on a track inclined at 15°. If the vehicle moves at 86 km/h, will it require a frictional force? If necessary, what will be the magnitude and direction of this force?

How long does it take for a glider travelling at 511 km/h and banking at an angle of 39° to navigate a sharp turn on a level circular path?

What is the speed of a truck navigating a banked circular track of 146 m diameter at a uniform speed? Given that its motion is represented in a coordinate system with its origin at the circle's center. At a given instant, the truck's horizontal acceleration is calculated as (-16.7 î - 24.2 ĵ) m/s².

When a vehicle navigates a banked circular track 146 m in diameter at a uniform speed, what are the vehicle's coordinates (x and y) at this time? Given that its motion is represented in a coordinate system with its origin at the circle's center. At a given instant, the vehicle's horizontal acceleration is calculated as (-16.7 î - 24.2 ĵ) m/s².

Determine the frictional force exerted on a 76 kg rider when a roller coaster car, moving at a speed of 161 km/h (about 101 mi/h), takes a turn with a 571 m radius and banks at an 8.1° angle towards the curve's center.