Deriving Max. Velocity for Banked Curve with Friction (Centripetal Acceleration)

• Novii
In summary, the conversation discusses a circular curve on a highway designed for safe travel at a certain speed on glare ice. It explains the consequences of traveling too slowly or too quickly and how the coefficient of static friction affects a car's ability to stay on the road. The task is to derive a formula for the maximum safe speed, vmax, in terms of the coefficient of static friction, µ, initial speed, v0, and radius, R. The attempt at a solution involves using equations for centripetal acceleration, force of static friction, and the angle of elevation, but further simplification is needed to solve for vmax.
Novii

Homework Statement

A circular 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, then it will slip away from the center of the circle. If the coefficient of static friction increases, a car can stay on the road while traveling at any speed within a range from vmin to vmax.

Derive formula for vmax, as a function of µ (coefficient of static friction), v0, and R.

Homework Equations

F=ma
centripetal acceleration = v2/R
Force of static friction = µs * N

The Attempt at a Solution

I tried expressing the angle of the bank with v0 and R first, then substituting it in for equations found for vmax:

θ=arctan((v0^2)/(9.8R))
x: sinθN + cosθFfr= ma= m (vmax^2)/R
y: mg + sinθFfr= cosθN = 9.8m + sinθ * µ* N

Where θ is the angle of elevation, N is normal force, and Ffr is friction force.

But the x and y equations turned out to be pretty complicated, and I'm not sure how to proceed now. Are my equations right?

Your equations look fine. Rearranging them slightly, you get

\begin{align*} N\sin\theta + \mu N \cos \theta &= mv_{max}^2/R \\ N\cos\theta - \mu N \sin \theta &= mg \end{align*}

Try dividing the first equation by the second and go from there.

1. What is the formula for deriving maximum velocity for a banked curve with friction?

The formula for deriving maximum velocity for a banked curve with friction is v = √(rgtanθ - μrgcosθ), where v is the maximum velocity, r is the radius of the curve, g is the acceleration due to gravity, θ is the angle of the banked curve, and μ is the coefficient of friction.

2. Why is it important to consider friction when deriving maximum velocity for a banked curve?

Friction plays a significant role in determining the maximum velocity for a banked curve. Without considering friction, the calculated velocity may be too high and can result in the vehicle or object sliding off the curve. Therefore, it is crucial to include friction in the calculation to ensure the safety and stability of the object on the curve.

3. How does the angle of the banked curve affect the maximum velocity?

The angle of the banked curve, θ, is a crucial factor in determining the maximum velocity. As the angle increases, the maximum velocity also increases. This is because a steeper angle provides a greater centripetal force, allowing the object to travel at a higher speed without sliding off the curve.

4. Can the maximum velocity for a banked curve with friction ever be negative?

No, the maximum velocity for a banked curve with friction cannot be negative. The velocity is always a positive value, as it represents the speed at which the object can safely traverse the curve without sliding off. If the calculated velocity is negative, it means that the object would not be able to maintain the centripetal force required to travel on the curve and would slide off.

5. What other factors can affect the maximum velocity for a banked curve with friction?

Aside from the angle of the banked curve and coefficient of friction, other factors that can affect the maximum velocity include the mass of the object, the radius of the curve, and the acceleration due to gravity. A heavier object would require a higher velocity to maintain the necessary centripetal force, while a larger radius or higher acceleration due to gravity would allow for a lower maximum velocity.

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