To solve the given problem, let’s break it into steps.

Problem Details:

• Radius of the circular disk: $r = 3 \, \text{m}$
• Angular velocity: $\omega = 2t^2 + 3 \, \text{rad/s}$
• Time of interest: $t = 3 \, \text{s}$
• Quantities to find:
  1. Radial acceleration ($a_r$)
  2. Tangential acceleration ($a_t$)
  3. Net acceleration ($a_{\text{net}}$)

Step 1: Angular velocity ($\omega$) at $t = 3 \, \text{s}$

The angular velocity is given by: $\omega = 2t^2 + 3$ Substitute $t = 3$: $\omega = 2(3)^2 + 3 = 2(9) + 3 = 18 + 3 = 21 \, \text{rad/s}$

Step 2: Angular acceleration ($\alpha$)

The angular acceleration is the time derivative of angular velocity: $\alpha = \frac{d\omega}{dt}$ From $\omega = 2t^2 + 3$, differentiate: $\alpha = \frac{d}{dt}(2t^2 + 3) = 4t$ Substitute $t = 3$: $\alpha = 4(3) = 12 \, \text{rad/s}^2$

Step 3: Tangential acceleration ($a_t$)

The tangential acceleration is related to angular acceleration: $a_t = \alpha r$ Substitute $\alpha = 12 \, \text{rad/s}^2$ and $r = 3 \, \text{m}$: $a_t = 12(3) = 36 \, \text{m/s}^2$

Step 4: Radial acceleration ($a_r$)

The radial acceleration is given by: $a_r = \omega^2 r$ Substitute $\omega = 21 \, \text{rad/s}$ and $r = 3 \, \text{m}$: $a_r = (21)^2(3) = 441(3) = 1323 \, \text{m/s}^2$

Step 5: Net acceleration ($a_{\text{net}}$)

The net acceleration is the vector sum of radial and tangential accelerations: $a_{\text{net}} = \sqrt{a_r^2 + a_t^2}$ Substitute $a_r = 1323 \, \text{m/s}^2$ and $a_t = 36 \, \text{m/s}^2$: $a_{\text{net}} = \sqrt{(1323)^2 + (36)^2}$ First, calculate the squares: $(1323)^2 = 1750329, \quad (36)^2 = 1296$ Add the results: $a_{\text{net}} = \sqrt{1750329 + 1296} = \sqrt{1751625}$ Take the square root: $a_{\text{net}} \approx 1323.49 \, \text{m/s}^2$

Final Answers:

1. Radial acceleration: $a_r = 1323 \, \text{m/s}^2$
2. Tangential acceleration: $a_t = 36 \, \text{m/s}^2$
3. Net acceleration: $a_{\text{net}} \approx 1323.49 \, \text{m/s}^2$

Would you like further clarification or more detailed steps? Here are related questions to expand your understanding:

1. What is the significance of radial acceleration in circular motion?
2. How is angular acceleration derived from the angular velocity equation?
3. What is the relationship between angular and tangential acceleration?
4. How does the radius of the disk affect the radial and tangential accelerations?
5. Can we calculate the velocity of a point on the disk's edge at $t = 3 \, \text{s}$?

Tip: Always differentiate carefully when dealing with time-dependent quantities like angular velocity or position to avoid errors in acceleration calculations.