A car at the Indianapolis-500 accelerates uniformly from the pit area, going from rest to 290 km/h in a semicircular arc with a radius of 200 m.
Determine the tangential acceleration of the car when it is halfway through the turn, assuming constant tangential acceleration.
Correct: Your answer is correct.
5.16 m/s2
Determine the radial acceleration of the car at this time.
Correct: Your answer is correct.
16.22 m/s2
If the curve were flat, what would the coefficient of static friction have to be between the tires and the roadbed to provide this acceleration with no slipping or skidding?
Only struggling with this question ^^
an explanation would be great, thanks
Since we have tangential and radial acceleration our total acceleration is going to be:
a = sqrt(tangential^2 + radial^2)
Since the car doesn't slip or skid the total force on the car is zero:
fmg = ma <-- where f is friction force
Mass cancels, so our final f is
thank you very much but is this only the Ff correct? I would still need to find the coefficient of Ff, or no?
Nevermind I figured it out. Thanks again!
To find the coefficient of static friction between the tires and the roadbed, we can use the formula for centripetal acceleration:
Centripetal acceleration (ac) = Radial acceleration (ar)
The radial acceleration of the car can be calculated by the following equation:
ar = v^2 / r
where v is the velocity and r is the radius of the curve. In this case, the velocity (v) is given as 290 km/h, which needs to be converted to m/s. There are 1000 meters in a kilometer and 3600 seconds in an hour, so we can convert km/h to m/s as follows:
v = 290 km/h × (1000 m/km) / (3600 s/h) = 80.56 m/s
Now we can calculate the radial acceleration:
ar = (80.56 m/s)^2 / 200 m = 32.22 m/s^2
Since the radial acceleration also represents the centripetal acceleration, we can say that centripetal acceleration is equal to the tangential acceleration of the car (at). Therefore, at = ar.
So the tangential acceleration of the car when it is halfway through the turn is 32.22 m/s^2.
To find the coefficient of static friction, we need to use the formula for static friction:
Static friction (fs) = μs × Normal force (N)
In this case, the normal force is equal to the weight of the car, which can be calculated using the formula:
Weight (W) = mass (m) × acceleration due to gravity (g)
The mass of the car is not given, but we can cancel out the mass by dividing both sides of the equation by the mass:
fs / m = μs × g
Now we can substitute the known values:
fs / m = 32.22 m/s^2 × g
The acceleration due to gravity is approximately 9.8 m/s^2, so:
fs / m ≈ 32.22 m/s^2 × 9.8 m/s^2
Simplifying further:
fs / m ≈ 316.09 m^2/s^2
Finally, to find the coefficient of static friction (μs), we multiply both sides of the equation by mass (m) and divide by the normal force (fs):
μs ≈ (fs / m) / (W / g)
Since fs / m is approximately equal to 316.09 m^2/s^2 and W / g is the mass (m) of the car, the coefficient of static friction becomes:
μs ≈ (316.09 m^2/s^2) / m
Therefore, to provide the given acceleration with no slipping or skidding, the coefficient of static friction between the tires and the roadbed would need to be approximately 316.09 m^2/s^2.