Let's break this problem into two steps for each part, with explanations and a final answer at the end. --- ## Part (a): **Circle all the ones that are non-zero when the block is at rest attached to the compressed spring** ### Step 1: Identify Energies at Rest with Compressed Spring - The block is **at rest** and **attached to the compressed spring**. - **Translational Kinetic Energy:** The block is **at rest**, so kinetic energy is **zero**. - **Gravitational Potential Energy:** The block is not at a height above the reference point (y = ), so gravitational potential energy can be **zero or a reference value**. - **Spring Potential Energy:** The spring is **compressed**, so there is **spring potential energy stored**. **Explanation:** Only the spring potential energy is non-zero because the block is at rest and at the reference height. #### **Final Answer for (a):** - **Spring Potential Energy** --- ## Part (b): **Circle all the ones that are non-zero when the block reaches the max height \( y_1 \)** ### Step 2: Identify Energies at Max Height \( y_1 \) - The block **reaches its max height** and comes to rest momentarily. - **Translational Kinetic Energy:** The block comes to rest at the max height, so kinetic energy is **zero**. - **Gravitational Potential Energy:** The block is at height \( y_1 \), so gravitational potential energy is **non-zero**. - **Spring Potential Energy:** The spring is no longer compressed or stretched (block is no longer attached), so spring potential energy is **zero**. **Explanation:** At max height, all the energy is gravitational potential energy since the block is momentarily at rest and the spring is not stretched/compressed. #### **Final Answer for (b):** - **Gravitational Potential Energy** --- ## Part (c): **What is the max height, \( y_1 \), that the block will reach?** ### Step 1: Find Initial Energy (when block is at rest with compressed spring) - **Spring Potential Energy:** \[ E_\text{spring, initial} = \frac{1}{2} k x^2 \] Where \( k = 33.6\, \text{N/m} \), \( x = .135\, \text{m} \). \[ E_\text{spring, initial} = \frac{1}{2} (33.6)(.135)^2 = \frac{1}{2} \times 33.6 \times .018225 = .306\, \text{J} \] **Explanation:** All energy starts as spring potential energy when the block is at rest and the spring is compressed. ### Step 2: Set Final Energy (at max height, all energy is gravitational potential) - **Gravitational Potential Energy at max height:** \[ E_\text{grav, final} = mgy_1 \] Where \( m = .125\, \text{kg} \), \( g = 9.8\, \text{m/s}^2 \). Set \( E_\text{spring, initial} = E_\text{grav, final} \): \[ .306 = .125 \cdot 9.8 \cdot y_1 \] \[ y_1 = \frac{.306}{.125 \times 9.8} = \frac{.306}{1.225} \approx .25\, \text{m} \] **Explanation:** All initial spring energy converts to gravitational potential at the max height. #### **Final Answer for (c):** - **The max height, \( y_1 \), is \( \boxed{.25\, \text{m}} \).**