a car is constantly speeding up clockwise on a circular track with a radius of 100m. At a given moment, when the car is moving south, its total acceleration is 4.10m/s^2 and makes an angle of exactly 3 degrees with the radius. Suppose that the tangential acceleration is constant, how long will it take for the car to come back to exactly the same place?

Question

GPT 3.5 · Answer

To solve this problem, we need to break down the acceleration into two components: tangential acceleration (at) and centripetal acceleration (ac). Since the tangential acceleration is constant, we can calculate the time it takes for the car to reach a certain angular displacement.

First, let's find the tangential acceleration (at) using the total acceleration and the angle with the radius:

at = a * sin(θ)
at = 4.10 m/s^2 * sin(3°)
at = 0.21 m/s^2

Next, let's find the angular acceleration (α) using the centripetal acceleration (ac) and the radius (r):

ac = α * r
α = ac / r
α = (at / r) = (0.21 m/s^2) / (100m)
α = 0.0021 rad/s^2

Now we can find the time it takes for the car to complete a full circle using the equation:

θ = ω * t + (1/2) * α * t^2

Since the initial angular displacement (θ) is 360 degrees or 2π radians and the final angular displacement is 0, we can substitute those values into the equation and solve for t:

2π = 0 + (1/2) * 0.0021 rad/s^2 * t^2

Rearranging the equation to solve for t:

t^2 = (2π) / (0.0021 rad/s^2)
t^2 = 3007.96 s^2

Taking the square root of both sides:

t = sqrt(3007.96 s^2)
t ≈ 54.84 s

Therefore, it will take approximately 54.84 seconds for the car to come back to exactly the same place.