A curve of radius 55 m is banked so that a car of mass 1.5 Mg traveling with uniform speed 57 km/hr can round the curve without relying on friction to keep it from slipping on the surface.
To determine the banking angle of the curve, we can use the concept of circular motion and the equilibrium of forces acting on the car.
Given information:
- Radius of the curve (r) = 55 m
- Mass of the car (m) = 1.5 Mg = 1.5 * 10^6 kg
- Speed of the car (v) = 57 km/hr = 57 * 1000 / 60^2 m/s (convert to m/s)
Step 1: Find the banking angle (θ) using the formula:
Tan(θ) = v^2 / (g * r)
Where g is the acceleration due to gravity (9.8 m/s^2).
Step 2: Substitute the given values into the formula and solve for θ:
Tan(θ) = (57 * 1000 / 60^2)^2 / (9.8 * 55)
Calculate the value of the right-hand side expression and then take the inverse tangent (arctan) of that value.
Step 3: Calculate θ:
θ = arctan [(57 * 1000 / 60^2)^2 / (9.8 * 55)]
Once you calculate the value of θ, you will have the banking angle of the curve that allows the car to round it without relying on friction to prevent slipping.
To determine how the curve is banked, we need to consider the forces acting on the car as it rounds the curve without relying on friction.
In this scenario, there are two primary forces at play: the gravitational force (weight) acting downward and the normal force from the road surface acting perpendicular to the car's motion.
To begin, we can calculate the weight of the car:
Weight = mass * gravity
Given that the mass of the car is 1.5 Mg (Mg = 1000 kg) and the acceleration due to gravity is approximately 9.8 m/s^2, we have:
Weight = 1.5 Mg * 9.8 m/s^2 = 14.7 kN
Next, we need to analyze the forces acting on the car during the curved motion. The normal force acts perpendicular to the road surface and can be resolved into two components: the vertical component (Fn_y) and the horizontal component (Fn_x).
Since the car is traveling without relying on friction to prevent slipping, the horizontal component of the normal force (Fn_x) provides the centripetal force required for circular motion. The centripetal force is given by:
Centripetal force = mass * acceleration (in this case, centripetal acceleration)
The centripetal acceleration is given by:
Centripetal acceleration = velocity^2 / radius
Given that the velocity of the car is 57 km/hr (which needs to be converted to m/s) and the radius of the curve is 55 m, we can calculate the centripetal force:
Centripetal acceleration = (57 km/hr * 1000 m/km) / (3600 s/hr)
= 15.83 m/s
Centripetal force = mass * centripetal acceleration
= 1.5 Mg * 15.83 m/s^2
= 23.75 kN
Since the centripetal force is provided by the horizontal component of the normal force, we can equate these two forces:
Fn_x = Centripetal force
Fn_x = 23.75 kN
Now, we can determine the vertical component of the normal force (Fn_y). Since the car is not sinking into the road or lifting off, the vertical component of the normal force should balance the weight of the car. Therefore:
Fn_y = Weight
Fn_y = 14.7 kN
Finally, we can calculate the banking angle (θ) using the following formula:
θ = arctan(Fn_y / Fn_x)
θ = arctan(14.7 kN / 23.75 kN)
≈ arctan(0.62)
Using a calculator, we find that the banking angle is approximately 31.7 degrees.
Therefore, to round the curve without relying on friction, the curve needs to be banked at an angle of approximately 31.7 degrees.