A 625-kg racing car completes one lap in 14.3 s around a circular track with a radius of 50.0 m. The car moves at a constant speed.
(a) What is the acceleration of the car?
(b) What force must the track exert on the tires to produce this acceleration?
To find the acceleration of the car, we can use the formula for centripetal acceleration:
ac = (v^2) / r
where ac is the acceleration, v is the velocity, and r is the radius of the circular track.
(a) First, let's find the velocity of the car using the given information. We know that the car completes one lap in 14.3 seconds, and the distance around the circular track can be calculated using the circumference formula:
C = 2πr
where C is the circumference and r is the radius.
C = 2π(50.0 m) = 100π m
Since the car completes one lap in 14.3 seconds, the average speed can be found by dividing the distance by the time:
v = C / t = (100π m) / (14.3 s)
Now, we can calculate the velocity:
v ≈ 21.97 m/s
Substituting the values into the formula for centripetal acceleration:
ac = (v^2) / r = (21.97 m/s)^2 / 50.0 m
Now, let's calculate the acceleration:
ac ≈ 9.56 m/s^2
Therefore, the acceleration of the car is approximately 9.56 m/s^2.
(b) To find the force that the track must exert on the tires to produce this acceleration, we can use Newton's second law of motion:
F = ma
where F is the force, m is the mass of the car, and a is the acceleration.
Given that the mass of the car is 625 kg and the acceleration is approximately 9.56 m/s^2, we can calculate the force:
F = (625 kg) * (9.56 m/s^2)
Therefore, the force that the track must exert on the tires to produce this acceleration is approximately 5965 N.
To find the acceleration of the car, we can use the formula for centripetal acceleration:
(a) The formula for centripetal acceleration is given by:
ac = (v^2) / r
where ac is the centripetal acceleration, v is the velocity, and r is the radius.
Since the car moves at a constant speed, we can find the velocity by dividing the distance traveled by the time taken:
v = 2πr / t
where v is the velocity, r is the radius, and t is the time taken for one lap.
Substituting the values, we have:
v = 2 * π * 50.0 m / 14.3 s
v ≈ 219.2 m/s
Now, let's substitute the velocity and radius into the formula for centripetal acceleration:
ac = (219.2 m/s)^2 / 50.0 m
ac ≈ 956.16 m/s^2
Therefore, the acceleration of the car is approximately 956.16 m/s^2.
(b) The force required to produce this acceleration can be calculated using Newton's second law:
F = m * ac
where F is the force, m is the mass of the car, and ac is the acceleration.
Substituting the values, we have:
F = 625 kg * 956.16 m/s^2
F ≈ 597,600 N
Therefore, the track must exert a force of approximately 597,600 N on the tires to produce this acceleration.