Two rods are pinned to a fixed point at one end, and connect to each other at the other end using a slider joint. Point C can freely move along rod DE but is pinned to rod AB at C. Rod AB rotates clockwise with angular velocity ω_AB = 4 rad/s and angular acceleration α_AB = 5 rad/s² when θ = 45^o. Find the angular acceleration of rod DE at this instant. The collar at C is pin connected to AB and slides over rod DE.

Since the distance between point D and C varies, two frames of reference is useful to solve the problem. Attach the local x-y coordinate to the rotating AE rod, and the global X-Y coordinate to the non-moving background.

Velocities:
Before accelerations can be found, the velocity of point C, v_C/D, and angular velocity of Ω_DE (= Ω_DC) needs to be determined.
     v_C = v_D + Ω × r_C/D + ( v_C/D)_rel
     v_C = v_D + Ω_DC× r_C/D + v_C/D
     v_D = 0
     ω_AB = 4k
     r_C/D = 0.5 m i
     Ω_DC = ω_DC

Since distance from A and C does not change, velocity of C can be determined from bar AB
     v_C = ω_AB × r_C/A = -4k × (0.5i + 0.5j)
          = 2i - 2j m/s

Putting all terms together gives,
     2i - 2j = 0 + (-ω_DC k) × (0.5i) + v_{C/D rel}i
     2i - 2j = -0.5ω_DC j + v_{C/D rel} i

Comparing the i and j terms give
     v_C/D = 2 m/s   and   ω_DC = 4 rad/s

Accelerations:
     a_C = a_D + dΩ_DC/dt × r_C/D + Ω_DC× (Ω_DC × r_C/D)
              + 2Ω_DC × v_C/D + a_C/D
     a_C = a_D + α_DE × r_C/D + ω_DE× ( ω_DE × r_C/D)
              + 2ω_DE × v_C/D + a_C/D
where,
     a_D = 0
     r_C/D = 0.5i m
     α_DC = (-α_DCk) × 0.5i = -0.5α_DEj m/s²
     ω_DC × ( ω_DC × r_C/D) = (- 4)² (0.5 i) = -8i rad/s
     2ω_DC × v_C/D = 2 (-4k) × 2i = 16j m/s²
     a_C/D = a_C/Di

Acceleration of C can be determined from bar AC
     a_C = α_AC × r_C/A - ω² r_C/A
         = -5k × (0.5i + 0.5j) - (4)² (0.5i + 0.5j)
         = -5.5i - 10.5j m/s²

-5.5i - 10.5j = 0 + (-α_DEk)× 0.5i - (- 4)² (0.5 i)
       + 2 (-4k)×2i + (a_C/D)i

Putting all terms together gives,
     -5.5i - 10.5j = 0 - 0.5α_DCj - 8i - 16j + a_C/Di

Comparing the j terms
     -10.5 = -16 - 0.5α_DE
     α_DC = -11 rad/s²

α_DC = 11k rad/s² (counter-clockwise)