Let's solve the problem step by step: --- ## **Step 1: Understanding the Problem** - The can will roll without slipping as the door is closed. - We need to find the **maximum angular velocity (\(\Omega\))** of the door so that the can is *not dislodged* (i.e., it does not lose contact with the rack). - The **distance from the hinge to the can** is \( r = 575\, \text{mm} = .575\, \text{m} \). - The **radius of the can** is \( R = 48\, \text{mm} / 2 = 24\, \text{mm} = .024\, \text{m} \). --- ## **Step 2: Forces on the Can** When the door is slammed shut with angular velocity \(\Omega\), the can experiences: 1. **Centrifugal force:** \( F_c = m r \Omega^2 \) 2. **Weight:** \( W = mg \) The can will lose contact if the normal force between the can and the rack becomes zero, i.e., when the centrifugal force equals the weight: \[ m r \Omega^2 = m g \] \[ r \Omega^2 = g \] --- ## **Step 3: Solve for Maximum Angular Velocity** \[ \Omega^2 = \frac{g}{r} \] \[ \Omega = \sqrt{\frac{g}{r}} \] Plug in known values: - \( g = 9.81\, \text{m/s}^2 \) - \( r = .575\, \text{m} \) \[ \Omega = \sqrt{\frac{9.81}{.575}} \] \[ \Omega = \sqrt{17.078} \] \[ \Omega \approx 4.134\, \text{rad/s} \] --- ## **Step 4: Consider Rolling Without Slipping (Energy Approach)** Because the can rolls without slipping, it's better to use an energy/balance approach: - The can's center of mass undergoes circular motion, so the **normal force** at the contact must not be negative. - At the threshold, the **normal force** goes to zero. For a rolling object, the **effective acceleration** at the center of mass is: \[ a_{\text{cm}} = r \Omega^2 \] The can will start to lose contact if the acceleration equals \(g\), as previously calculated. --- ## **Step 5: Final Answer** \[ \boxed{4.13~\text{rad/s}} \] --- ## **Note:** - The value of .6 rad/s shown in the screenshot is **not correct** for this problem based on the standard physics derivation. --- ## **Summary Table** | Symbol | Quantity | Value | |--------|---------------------|---------------| | \( r \) | Distance to can | .575 m | | \( g \) | Gravity | 9.81 m/s² | | \( \Omega \) | Angular velocity (max) | **4.13 rad/s** | --- ### **Final Answer** \[ \boxed{\Omega = 4.13~\text{rad/s}} \] Let me know if you need clarification on any step!