What is the greatest safe speed, without skidding, that a 1700 kg car can have on a level curve of radius 215 meters if Mk=0.4 and Ms=.85?
The correct answer is 42.3 but idk how to get it.
μs determines when skidding will begin.
Maximum centripetal force provided by static friction
= (μs)mg
Maximum speed = v,
centrifugal force
= mv^2/r
At equilibrium,
(μs)mg = mv^2/r
Solve for v
v=sqrt(μs * rg)
=sqrt(0.85*215*9.81)
=42 m/s approximately
To find the greatest safe speed of a car without skidding on a curve, we need to consider two factors: the maximum static friction force and the force required to keep the car moving in a circular path.
First, let's calculate the maximum static friction force. The formula for static friction force is:
Fk = Mk * N
Where Fk is the maximum static friction force, Mk is the coefficient of static friction, and N is the normal force exerted on the car.
The normal force can be calculated using the formula:
N = m * g
Where m is the mass of the car and g is the acceleration due to gravity.
N = 1700 kg * 9.8 m/s^2
N = 16660 N
Now, we can calculate Fk:
Fk = 0.4 * 16660 N
Fk = 6664 N
Next, let's calculate the force required to keep the car moving in a circular path. This force is called the centripetal force and can be calculated using the formula:
Fc = m * (v^2 / r)
Where Fc is the centripetal force, m is the mass of the car, v is the velocity of the car, and r is the radius of the curve. We want to find the maximum velocity without skidding, so we'll assign the maximum static friction force as the centripetal force:
Fc = Fk
Substituting the values we know:
m * (v^2 / r) = 6664 N
Rearranging the equation, we get:
v^2 = (Fc * r) / m
v^2 = (6664 N * 215 m) / 1700 kg
v^2 = 841860 Nm / 1700 kg
v^2 ≈ 4958.59 (m/s)^2
Finally, we can find the maximum velocity by taking the square root of v^2:
v ≈ √4958.59 (m/s)
v ≈ 70.4 m/s
Therefore, the greatest safe speed the car can have on the level curve is approximately 70.4 m/s or 42.3 mph.