Final answer:
At t = 0, the swinging door's angular position is 4.95 radians, angular speed is 10.8 rad/s, and angular acceleration is 4.02 rad/s². At t = 2.94s, the angular position is approximately 54.036 radians, angular speed is approximately 22.7692 rad/s, and angular acceleration remains at 4.02 rad/s².
Step-by-step explanation:
The given problem concerns the motion of a swinging door, which is a classical mechanics problem in physics. The door's motion is described by an angular position equation θ(t) = 4.95 + 10.8t + 2.01t², where θ is in radians and t is in seconds.
a) t=0
To find the angular position at t=0, we simply substitute t=0 into the equation:
θ(0) = 4.95 + 10.8(0) + 2.01(0)²
θ(0) = 4.95 radians
To determine the angular speed, we take the first derivative of the angular position function:
ω(t) = dθ/dt = 10.8 + 2(2.01)t
ω(0) = 10.8 rad/s
For the angular acceleration, we take the second derivative:
α(t) = dω/dt = 2(2.01)
α = 4.02 rad/s²
b) t=2.94s
Repeating the process for t=2.94s:
θ(2.94) = 4.95 + 10.8(2.94) + 2.01(2.94)²
Using a calculator, we can find:
θ(2.94) = 4.95 + 31.752 + 17.3339
θ(2.94) ≈ 54.036 radians
The angular speed at t=2.94s is:
ω(2.94) = 10.8 + 2(2.01)(2.94)
ω(2.94) ≈ 22.7692 rad/s
Angular acceleration remains unchanged as it is constant:
α = 4.02 rad/s²