Without knowing the specific geometry and dimensions of the system involving arm AB, collar D, and midpoint G of bar BD, it is not possible to provide an exact solution to this problem. However, we can provide a general method for finding the accelerations of collar D and midpoint G using the equations for angular velocity, angular acceleration, linear velocity, and linear acceleration.
Let's assume that the arm AB is rotating counterclockwise about a fixed point at a constant angular velocity of 24 rad/s. At the instant when theta = 90 degrees, we can use the following equations to find the linear velocities and accelerations of collar D and midpoint G:
Angular velocity: w = d(theta)/dt = 24 rad/s (given)
Angular acceleration: alpha = dw/dt = 0, since the angular velocity is constant
Linear velocity: v = r*w, where r is the distance from the fixed point to the point of interest (collar D or midpoint G)
Linear acceleration: a = ralpha + vdw/dt, where alpha = 0
For collar D, let's assume that r is the distance from the fixed point to the point where the arm AB is attached to collar D. Let's also assume that the distance from collar D to the fixed point is L. Then, we can write:
r = L/2, since collar D is at the midpoint of bar BD
v = rw = (L/2)(24) = 12L
alpha = 0
a = ralpha + vdw/dt = (L/2)*0 + (12L)*0 = 0
Therefore, the acceleration of collar D is 0 in/s^2.
For midpoint G, let's assume that r is the distance from the fixed point to midpoint G, and that the distance from midpoint G to the fixed point is d. Then, we can write:
r = sqrt((L/2)^2 + d^2), since midpoint G is at the midpoint of the hypotenuse of right triangle ABD
v = r*w = sqrt((L/2)^2 + d^2)*24
alpha = 0
a = ralpha + vdw/dt = sqrt((L/2)^2 + d^2)0 + sqrt((L/2)^2 + d^2)dw/dt(-sin(theta)) = -24sqrt((L/2)^2 + d^2)*cos(theta)
At theta = 90 degrees, the acceleration of midpoint G is:
a = -24*sqrt((L/2)^2 + d^2)cos(90) = -24sqrt((L/2)^2 + d^2)
Note that the accelerations of collar D and midpoint G depend on the dimensions and geometry of the system, which are not specified in the problem statement. Therefore, we cannot provide numerical values for the accelerations without additional information.