a 0.30-m radius automobile tire rotates how many rad after starting from rest and accelerating at a constant 2.0 rad/s^2 over a 3.54-s interval
To find the number of radians the tire rotates, we can use the kinematic equation:
θ = θ₀ + ω₀t + (1/2)αt²
Where:
θ = final angle (in radians)
θ₀ = initial angle (in radians) -- In this case, since the tire starts from rest, θ₀ = 0
ω₀ = initial angular velocity (in radians per second) -- In this case, since the tire starts from rest, ω₀ = 0
α = angular acceleration (in radians per second squared) -- Given as 2.0 rad/s²
t = time interval (in seconds) -- Given as 3.54 s
Plugging in the values:
θ = 0 + 0*t + (1/2)*2.0*t²
Simplifying the equation:
θ = t²
Now we can substitute the given time interval (3.54 s) into the equation to find the number of radians the tire rotates:
θ = (3.54 s)²
θ = 12.5316 radians
Therefore, the 0.30-m radius automobile tire rotates approximately 12.5316 radians during the 3.54-second interval.