A problem on centripetal forces and friction

[Chiara]

hey this is important, I am having this discussion with my father on a Physics problem and i don't think his answer is right. How would you solve this problem?
For a car traveling with speed v around a curve of radiur r, determine a formula for the angle at which the road should be banked so that no friction is required.

 

[FZ+]

Draw a diagram for the system, and mark on the forces involved. Gravity downwards. Reaction force at right angles to the surface of the curve.

Now, you know the component of the reaction force in the direction towards the centre of the curve (and so at right angles to the ground) as equal to m * V^2/r (from pretty basic circular motion). You also know it's component in the downwards direction (as mg). You can then divide one from the other to get the tan(angle), using geometry to get the angle of the plane.

Much easier to understand with good diagrams.

 

[M98Ranger]

To solve this problem, we first need to understand the forces at play. The car is experiencing two forces: the centripetal force, which is directed towards the center of the curve, and the force of gravity, which is directed downwards. In order for the car to stay on the curve without slipping, these two forces must be balanced. This means that the component of the force of gravity perpendicular to the surface of the road must equal the centripetal force.

Using trigonometry, we can determine that the angle of the road, θ, is given by the equation θ = arctan(v^2 / rg), where v is the speed of the car, r is the radius of the curve, and g is the acceleration due to gravity.

To verify this formula, we can plug in some values. Let's say the car is traveling at 20 m/s and the curve has a radius of 50 m. Plugging these values into the formula, we get θ = arctan((20 m/s)^2 / (50 m)(9.8 m/s^2)) = 21.8 degrees.

We can also think about this problem from a different perspective. If the road is banked at the correct angle, the normal force (perpendicular to the road) will be equal to the force of gravity. This means that the frictional force will be zero, since no force is needed to counteract the component of the force of gravity parallel to the surface of the road. This is why no friction is required for the car to stay on the curve.

In conclusion, the correct formula for the angle at which the road should be banked is θ = arctan(v^2 / rg). I hope this explanation helps you in your discussion with your father. Remember to always double check your calculations and make sure to use the correct units. Good luck!