Let's solve the problem step by step: --- ## **Given Data** - **Gear radii:** \( r_A = 15\, \text{mm} \) \( r_B = 30\, \text{mm} \) \( r_C = 20\, \text{mm} \) \( r_D = 45\, \text{mm} \) - **Angular acceleration of gear A:** \( \alpha_A = 2\, \text{rad/s}^2 \) (clockwise) - **Initial condition:** Starts from rest (\( \omega_{A} = \)) - **Time:** \( t = 2\, \text{s} \) --- ## **(a) Find the magnitude of angular velocities of gears B, C, and D at \( t = 2s \)** ### **Step 1: Angular velocity of Gear A at 2 s** Use: \[ \omega_A = \omega_{A} + \alpha_A t \] \[ \omega_A = + (2)(2) = 4\, \text{rad/s} \] --- ### **Step 2: Gear Relationships** - **Gears A & B are in mesh:** \( r_A \omega_A = r_B \omega_B \) (opposite directions) - **Gears B & C are on the same shaft:** \( \omega_B = \omega_C \) - **Gears C & D are in mesh:** \( r_C \omega_C = r_D \omega_D \) (opposite directions) --- ### **Step 3: Calculate \( \omega_B \) and \( \omega_C \)** \[ r_A \omega_A = r_B \omega_B \] \[ \omega_B = \frac{r_A}{r_B} \omega_A = \frac{15}{30} \times 4 = 2\, \text{rad/s} \] \[ \omega_C = \omega_B = 2\, \text{rad/s} \] --- ### **Step 4: Calculate \( \omega_D \)** \[ r_C \omega_C = r_D \omega_D \] \[ \omega_D = \frac{r_C}{r_D} \omega_C = \frac{20}{45} \times 2 = \frac{40}{45} = .8889\, \text{rad/s} \] --- ### **Step 5: Final Answers (Magnitudes)** - **\( \omega_B = 2\, \text{rad/s} \)** - **\( \omega_C = 2\, \text{rad/s} \)** - **\( \omega_D = .8889\, \text{rad/s} \)** --- ## **(b) Angular displacement of gear D from to 2 s** ### **Step 1: Angular displacement of Gear A** \[ \theta_A = \omega_{A} t + \frac{1}{2} \alpha_A t^2 \] \[ \theta_A = + \frac{1}{2} \times 2 \times (2^2) = 4\, \text{rad} \] --- ### **Step 2: Relate Displacement between A and D** For meshed gears: \[ r_A \theta_A = r_B \theta_B \] \[ \theta_B = \frac{r_A}{r_B} \theta_A = \frac{15}{30} \times 4 = 2\, \text{rad} \] Similarly, \( \theta_C = \theta_B = 2\, \text{rad} \). For C and D: \[ r_C \theta_C = r_D \theta_D \] \[ \theta_D = \frac{r_C}{r_D} \theta_C = \frac{20}{45} \times 2 = \frac{40}{45} = .8889\, \text{rad} \] --- ## **Final Answers** ### **(a) Magnitude of angular velocities at \( t = 2\,\text{s} \):** - \( \omega_B = 2.\, \text{rad/s} \) - \( \omega_C = 2.\, \text{rad/s} \) - \( \omega_D = .89\, \text{rad/s} \) (rounded to 2 decimal places) ### **(b) Angular displacement of gear D from to 2 s:** - \( \theta_D = .89\, \text{rad} \) (rounded to 2 decimal places) --- **Let me know if you need clarification or further explanation on any step!**