A 0.30-m-radius automobile tire rotates how many radians after starting from rest and accelerating at a constant 2.0 rad/s^2 over a 5.0-s interval?

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a. 0.50 rad

b. 12.5 rad
c. 2.0 rad
d. 25 rad

To find the number of radians through which the tire rotates, we need to determine the angular displacement, which can be calculated using the formula:

θ = ω0 * t + (1/2) * α * t^2

where:
θ = angular displacement (in radians)
ω0 = initial angular velocity (in radians per second)
α = angular acceleration (in radians per second squared)
t = time (in seconds)

In this case, the tire starts from rest, so the initial angular velocity is zero (ω0 = 0). The angular acceleration is given as 2.0 rad/s^2. The time interval is 5.0 seconds.

Substituting these values into the formula:

θ = 0 * 5 + (1/2) * 2.0 * 5^2
= 0 + (1/2) * 2.0 * 25
= 0 + 1 * 25
= 25 radians

Therefore, the tire rotates 25 radians after starting from rest and accelerating at a constant 2.0 rad/s^2 over a 5.0-s interval.