A car travels at a constant speed around a circular track whose radius is 2.59 km. The car goes once around the track in 380 s. What is the magnitude of the centripetal acceleration of the car?
a = r omega^2
omega = 2pi/380
To find the magnitude of the centripetal acceleration of the car, we can use the formula:
a = v^2 / r
Where:
a = centripetal acceleration
v = velocity of the car
r = radius of the circular track
First, we need to find the velocity of the car. Since the car travels at a constant speed, we can find the velocity by dividing the distance traveled by the time taken:
Distance = 2πr (because the car goes once around the circular track)
Time = 380 s
Therefore, the velocity of the car is:
v = (2πr) / Time
v = (2π * 2.59 km) / 380 s
v = 16.30 km/h
Now we can substitute the values of velocity and radius into the centripetal acceleration formula:
a = (16.30 km/h)^2 / 2.59 km
To find the magnitude of the centripetal acceleration, we need to convert the velocity from km/h to m/s:
1 km/h = 1000 m/3600 s = 0.2778 m/s
Therefore, the velocity in m/s is:
v = 16.30 km/h * 0.2778 m/s = 4.52 m/s
Substituting the values into the formula:
a = (4.52 m/s)^2 / 2.59 km
To convert the radius from km to meters, we need to multiply by 1000:
r = 2.59 km * 1000 m/km = 2590 m
Substituting the new values into the formula:
a = (4.52 m/s)^2 / 2590 m
Calculating the magnitude of the centripetal acceleration:
a = 0.079 m/s^2
Therefore, the magnitude of the centripetal acceleration of the car is 0.079 m/s^2.