## Step 1: **Conceptual Argument (Part a)** **Explanation:** All objects are **rolling without slipping** down the same ramp of height 40 cm, starting from rest. The **final speed at the bottom** depends on how the initial potential energy is split between translational and rotational kinetic energy. For rolling without slipping, the distribution depends on the shape (moment of inertia), **not on mass or radius**. - A **penny and a nickel** are both solid disks (same type of object). - A **marble** is a solid sphere. **Solid spheres** have a smaller moment of inertia relative to their mass than solid disks, so more energy goes into translation (linear speed). Thus, the **marble will have the highest final speed**, and the penny and nickel will have the same (but lower) final speeds. **Conclusion for Step 1:** - Marble > Penny = Nickel (in terms of final speed) --- ## Step 2: **Numerical Calculation (Part b)** ### **Energy Conservation:** Total mechanical energy is conserved: \[ mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2 \] For rolling without slipping, \( v = \omega r \). ### **Moment of Inertia:** - **Solid disk (penny, nickel):** \( I = \frac{1}{2}mr^2 \) - **Solid sphere (marble):** \( I = \frac{2}{5}mr^2 \) ### **Plug in for each shape:** #### **Solid Disk:** \[ mgh = \frac{1}{2}mv^2 + \frac{1}{2} \left(\frac{1}{2}mr^2\right)\left(\frac{v^2}{r^2}\right) \] \[ mgh = \frac{1}{2}mv^2 + \frac{1}{4}mv^2 = \frac{3}{4}mv^2 \] \[ gh = \frac{3}{4}v^2 \implies v = \sqrt{\frac{4}{3}gh} \] #### **Solid Sphere:** \[ mgh = \frac{1}{2}mv^2 + \frac{1}{2} \left(\frac{2}{5}mr^2\right)\left(\frac{v^2}{r^2}\right) \] \[ mgh = \frac{1}{2}mv^2 + \frac{1}{5}mv^2 = \frac{7}{10}mv^2 \] \[ gh = \frac{7}{10}v^2 \implies v = \sqrt{\frac{10}{7}gh} \] ### **Plug in \( h = .40 \) m, \( g = 9.8 \) m/s²:** - **Solid disks (penny, nickel):** \[ v_{\text{disk}} = \sqrt{\frac{4}{3} \cdot 9.8 \cdot .40} = \sqrt{\frac{4}{3} \cdot 3.92} = \sqrt{5.227} \approx 2.29\, \text{m/s} \] - **Solid sphere (marble):** \[ v_{\text{sphere}} = \sqrt{\frac{10}{7} \cdot 9.8 \cdot .40} = \sqrt{\frac{10}{7} \cdot 3.92} = \sqrt{5.6} \approx 2.37\, \text{m/s} \] **Conclusion for Step 2:** - **Marble:** ~2.37 m/s - **Penny:** ~2.29 m/s - **Nickel:** ~2.29 m/s --- ## Step 3: **Check (Part c)** **Was the conceptual answer correct?** Yes. The marble (solid sphere) moves fastest, and the penny and nickel (both solid disks) have the same, but slightly lower, speed. --- ## **Final Answers** - **(a)** The marble will be fastest; penny and nickel will be equally fast but slower than the marble. - **(b)** Marble: **2.37 m/s**; Penny/Nickel: **2.29 m/s** - **(c)** Yes, the conceptual answer was correct.