A race car travels with a constant tangential speed of 74.1 m/s around a circular track of track of radius 628 m. Find the magnitude of the total acceleration.
The magnitude of the total acceleration can be found using the following equation:
a_total = sqrt(a_tangential^2 + a_radial^2)
where a_tangential is the tangential acceleration and a_radial is the radial acceleration.
The tangential acceleration can be calculated using the formula:
a_tangential = r * ω^2
where r is the radius of the circular track and ω is the angular velocity.
The angular velocity can be calculated using the formula:
ω = v / r
where v is the tangential speed.
Plugging in the given values, we have:
ω = 74.1 m/s / 628 m = 0.118 s^-1
Plugging this value into the formula for tangential acceleration, we have:
a_tangential = (628 m) * (0.118 s^-1)^2 = 8.06 m/s^2
To find the radial acceleration, we can use the formula:
a_radial = r * ω^2
Plugging in the given values, we have:
a_radial = (628 m) * (0.118 s^-1)^2 = 8.06 m/s^2
Now we can plug these values into the formula for total acceleration:
a_total = sqrt((8.06 m/s^2)^2 + (8.06 m/s^2)^2) = sqrt(129.87 m^2/s^4 + 129.87 m^2/s^4) ≈ 18.07 m/s^2
Therefore, the magnitude of the total acceleration is approximately 18.07 m/s^2.