A race car travels on a flat circular track at the maximum possible speed without slipping. If the radius of the track is 39m and the coefficient of friction is 0.73, how long (seconds) does it take the race car to complete 5 laps?
force friction=centripetal force
mg*mu=mv^2/r
solve for v
then time=distance/velocity=5*2PI*r/v
To solve this problem, we can break it down into smaller steps. First, let's calculate the maximum speed at which the race car can drive without slipping.
The maximum speed before slipping occurs is determined by the centripetal force, which is provided by the frictional force between the tires and the track. The formula for centripetal force is:
F = m * a
where F is the centripetal force, m is the mass of the car, and a is the centripetal acceleration. The centripetal acceleration, in turn, can be calculated using the formula:
a = v^2 / r
where v is the velocity of the car and r is the radius of the circular track.
To determine the maximum speed, we need to find the point at which the frictional force reaches its maximum value, which is given by the formula:
F_friction = μ * N
where μ is the coefficient of friction and N is the normal force. In this case, the normal force is equal to the weight of the car, which can be calculated using the formula:
N = m * g
where g is the acceleration due to gravity (approximately 9.8 m/s^2).
By equating the centripetal force with the maximum frictional force, we can solve for v:
m * a = μ * N
m * (v^2 / r) = μ * (m * g)
v^2 = μ * r * g
v = √(μ * r * g)
Now that we have the maximum speed, we can determine how long it takes for the car to complete 5 laps.
First, we need to calculate the time it takes for the car to complete one lap at the maximum speed. The formula for the circumference of a circle is:
C = 2 * π * r
With 5 laps, the total distance traveled by the car is:
d = 5 * C
The time it takes to complete this distance can be calculated using the formula:
t = d / v
Now, let's plug in the values and calculate the answer step by step:
1. Calculate the maximum speed:
v = √(0.73 * 39 * 9.8)
v ≈ 15.57 m/s
2. Calculate the distance of one lap:
C = 2 * π * 39
C ≈ 245.04 m
3. Calculate the total distance traveled:
d = 5 * 245.04
d ≈ 1225.2 m
4. Calculate the time taken for 5 laps:
t = 1225.2 / 15.57
t ≈ 78.67 seconds
Therefore, it takes approximately 78.67 seconds for the race car to complete 5 laps on the circular track at the maximum possible speed without slipping.