A curve with a 120-m radius on a level road is banked at the correct angle for a speed of 20 m/s

A curve with a 120-m radius on a level road is banked at the...

Key Concepts

Static Friction

Static friction is the force that resists the relative motion between two surfaces in contact. In the context of vehicles on a curved, banked road, static friction acts to supplement the centripetal force when the speed differs from the design speed, helping to prevent skidding by opposing the tendency of the vehicle to slide outward.

Force Decomposition

Force decomposition involves breaking down forces into their components, typically parallel and perpendicular to the road surface. This is critical in analyzing banked curves because the normal force has components that contribute to both the vertical force balance (against gravity) and the horizontal force (providing centripetal acceleration).

Centripetal Force

Centripetal force is the inward force required to keep an object moving in a circular path. It is calculated as the mass times the square of the speed divided by the radius of the curve. In vehicle dynamics, this force must be provided by the combination of the lateral components of the normal force and friction.

Banked Curve Dynamics

Banked curves involve tilting the roadway so that part of the normal force provides the necessary centripetal force for circular motion. When designed for a specific speed, the bank angle reduces reliance on friction, allowing vehicles to negotiate the curve safely without skidding under ideal conditions.

Transcript

00:01 In this problem, we're given that the curve is banked for 20 meters per second motion.

00:06 So we know if you're moving 20 meters per second, no friction is necessary for you to go around this curve.

00:12 Now, if you revert to example 5 .22, they derive an equation that is the banking angle is equal to the arc tan of v squared over g times r.

00:32 And this is true whenever friction is not necessary to round a curve.

00:38 And so we can plug in what velocity that is, in our case, it's 20 meters per second, and a radius of our curve.

00:44 And we get that the angle must be 18 .79 degrees.

00:52 That's not the final answer.

00:53 That's just going to help us in the end.

00:55 Now we need to draw a new force diagram.

00:58 In this case, we have our car here.

01:00 We have our normal force going this way.

01:03 We have an angle beta here and we have the vertical component of the normal force which is in cosine beta the horizontal component is here and it's in sine beta we also have gravity pulling us down m g and the car is going to want to go up the ramp this way which means friction is pointing this way so this is force of friction this angle here is also beta which means that we can break this up into its components too.

01:47 And so this is force of friction times sine beta.

01:55 And this part down here is, oh, my pen's glitching out.

01:59 One second.

02:02 This down here is force of friction cosine beta.

02:07 So this is kind of messy.

02:09 So let's just start writing down newton's second law.

02:13 Oh, by the way, there's an acceleration, a radial acceleration, towards the center of the ramp this way.

02:17 So now if i define this to be my x direction and this to be my y direction, then newton's second law in the x direction gives me n sine beta plus f, which i'm calling capital f, capital f, so a little f times cosine beta is equal to mass times acceleration in the radial direction.

02:47 And then if i do the same in the y direction, i get n cosine beta, minus force of friction, sine beta.

02:57 Is equal to mg...

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