A car at the Indianapolis-500 accelerates uniformly from the pit area, going from rest to 320 km/h in a semicircular arc with a radius of 190 m.
1.) Determine the tangential acceleration of the car when it is halfway through the turn, assuming constant tangential acceleration.
2.) Determine the radial acceleration of the car at this time.
3.) 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?
Rrfd
To determine the tangential and radial accelerations, we need to use the formulas for centripetal acceleration and tangential acceleration.
1.) Determine the tangential acceleration:
The tangential acceleration can be calculated using the equation:
at = (vf - vi) / t
where:
- at is the tangential acceleration
- vf is the final velocity
- vi is the initial velocity
- t is the time taken to reach the final velocity
In this case, the car starts from rest, so the initial velocity (vi) is 0 km/h. The final velocity (vf) is 320 km/h.
First, we need to convert the velocities from km/h to m/s by dividing by 3.6 (since 1 km/h = 1/3.6 m/s).
vi = 0 km/h = 0 m/s
vf = 320 km/h = 320/3.6 m/s = 88.9 m/s
As the car travels in a semicircular arc, halfway through the turn means it has covered half the distance of the circumference of the arc. The formula to calculate the circumference of a circle is 2πr, where r is the radius.
Circumference = 2π(190m) = 1193.04m
Distance halfway = 1193.04m / 2 = 596.52m
Now, we need to calculate the time taken to cover this distance. We can use the formula:
d = vit + (1/2)at²
Here, d is the distance traveled, vi is the initial velocity, t is the time taken, and a is the tangential acceleration.
Since the car starts from rest, vi = 0 m/s, and the above equation simplifies to:
d = (1/2)at²
Rearranging the equation, we get:
at = √(2d / t²)
Substituting the values:
at = √(2 * 596.52m) / t²
2.) Determine the radial acceleration:
The radial acceleration can be calculated using the formula:
ar = (v²) / r
where:
- ar is the radial acceleration
- v is the velocity of the car
- r is the radius of the arc
Substituting the values:
ar = (88.9 m/s)² / 190m
3.) Determine the coefficient of static friction:
If the curve were flat, the radial acceleration would be due to the friction between the tires and the roadbed. The maximum friction force before slipping or skidding occurs is given by the formula:
Fmax = μs * N
where:
- Fmax is the maximum friction force
- μs is the coefficient of static friction
- N is the normal force
The normal force (N) is equal to the weight of the car (mg), where m is the mass of the car and g is the acceleration due to gravity (9.8 m/s²).
Since we need the maximum static friction force to match the radial acceleration, Fmax = mar.
Substituting the values:
mar = μs * m * g
Simplifying, we get:
μs = ar / (m * g)
We can calculate the coefficient of static friction using the radial acceleration (ar) calculated in step 2, the mass of the car, and the acceleration due to gravity.
Please provide the mass of the car for a complete calculation.