Which quantities decrease as the distance between a planet and the Sun increases? Check all that apply.
2 answers:
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(a) The angular position of the door is described by
![\theta(t)=5+10t+2t^2 [rad]](https://tex.z-dn.net/?f=%5Ctheta%28t%29%3D5%2B10t%2B2t%5E2%20%5Brad%5D)
The angular velocity is given by the derivative of the angular position:
![\omega(t)=10+4t [rad/s]](https://tex.z-dn.net/?f=%5Comega%28t%29%3D10%2B4t%20%5Brad%2Fs%5D)
While the angular acceleration is given by the derivative of the angular velocity:
![\alpha(t)=4 [rad/s^2]](https://tex.z-dn.net/?f=%5Calpha%28t%29%3D4%20%5Brad%2Fs%5E2%5D)
We want to find the values of these quantities at time t=3.00 s, so we must substitute t=3.00 s into the expressions for
:
(b) The door starts from rest, so its initial angular velocity is
, and it reaches a final angular velocity of 
with an angular acceleration of 
. We can find the angular distance covered by the door by using the following relationship:
from which we find
The angular velocity is given by the derivative of the angular position:
While the angular acceleration is given by the derivative of the angular velocity:
We want to find the values of these quantities at time t=3.00 s, so we must substitute t=3.00 s into the expressions for
(b) The door starts from rest, so its initial angular velocity is
from which we find