If a car driving in a large circle of radius 185 m has tangential acceleration of 20 m/s2, and an angular velocity of 0.5 rad/s, what is the magnitude of the total acceleration of the car in m/s2?
Question
If a car driving in a large circle of radius 185 m has tangential acceleration of 20 m/s2, and an angular velocity of 0.5 rad/s, what is the magnitude of the total acceleration of the car in m/s2?
Solution
The total acceleration of an object moving in a circle is given by the Pythagorean sum of the radial (centripetal) and tangential accelerations.
Step 1: Calculate the radial (centripetal) acceleration The radial or centripetal acceleration (a_r) can be calculated using the formula a_r = ω^2 * r, where ω is the angular velocity and r is the radius of the circle.
a_r = (0.5 rad/s)^2 * 185 m = 0.25 * 185 m = 46.25 m/s^2
Step 2: Calculate the total acceleration The total acceleration (a) is given by the formula a = sqrt(a_r^2 + a_t^2), where a_t is the tangential acceleration.
a = sqrt((46.25 m/s^2)^2 + (20 m/s^2)^2) = sqrt(2140.625) = 46.26 m/s^2
So, the magnitude of the total acceleration of the car is approximately 46.26 m/s^2.
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