Car A uses tires for which the coefficient of static friction is 0.357 on a particular unbanked curve. The maximum speed at which the car can negotiate this curve is 27.7 m/s. Car B uses tires for which the coefficient of static friction is 0.737 on the same curve. What is the maximum speed at which car B can negotiate the curve?
To determine the maximum speed at which car B can negotiate the curve, we can use the concept of centripetal force.
The centripetal force, Fc, is given by the equation:
Fc = (m * v^2) / r
where m is the mass of the car, v is the velocity of the car, and r is the radius of the curve.
For a car to negotiate a curve without slipping, the centripetal force must be less than or equal to the maximum static friction force available. Mathematically, this can be expressed as:
Fc ≤ μ * N
where μ is the coefficient of static friction and N is the normal force.
In this case, since the curve is unbanked, the normal force N is equal to the weight of the car, which can be calculated as:
N = m * g
where g is the acceleration due to gravity.
To solve this problem, we can set up the inequality:
(m * v^2) / r ≤ μ * m * g
Let's solve for v:
v ≤ sqrt(μ * r * g)
Now, we need to calculate the maximum speed at which car B can negotiate the curve. Given that car A has a coefficient of static friction of 0.357 on the same curve, we can use this value to calculate the maximum speed of car B. Here's the step-by-step calculation:
1. Calculate the maximum speed for car A:
v_A = sqrt(μ_A * r * g) = sqrt(0.357 * r * g)
2. Substitute the coefficient of static friction for car B (μ_B = 0.737) into the equation:
v_B = sqrt(μ_B * r * g) = sqrt(0.737 * r * g)
Therefore, the maximum speed at which car B can negotiate the curve is sqrt(0.737 * r * g).