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Answer the following with simple explanation 

Let's break down the two questions based on the image provided.

### 1. **Which statement best describes gravitational potential energy \( U_{\text{grav}} \)?**
   - **I.** \( U_{\text{grav}} \) is always positive.
   - **II.** \( U_{\text{grav}} \) is always negative.
   - **III.** \( U_{\text{grav}} \) can be negative or positive.

   **Answer:**
   The best description of gravitational potential energy is **Statement III: \( U_{\text{grav}} \) can be negative or positive**.

   **Explanation:**
   Gravitational potential energy is defined based on the choice of a reference point. In many cases, it's defined as zero at infinity, making the gravitational potential energy negative near a planet or star (since gravity is an attractive force). However, in other contexts (e.g., close to the Earth's surface), it might be defined as positive. So, depending on the system, \( U_{\text{grav}} \) can be either positive or negative.

### 2. **Consider the plot of \( U(x) \) for a particle in 1D motion.**

   - **1. Rank regions AB, BC, and CD according to the magnitude of force on the particle, in decreasing order.**

     **Answer:**
     The magnitude of the force is related to the slope of the potential energy function, since \( F(x) = -\frac{dU(x)}{dx} \). The steeper the slope, the greater the force.

     **Ranking:**
     - **AB** (steepest slope) → Largest force
     - **CD** (next steepest) → Moderate force
     - **BC** (almost flat) → Smallest force

     So, the ranking is: **AB > CD > BC**.

   - **2. What is the direction of force when the particle is in region CD?**

     **Answer:**
     The force is given by \( F(x) = -\frac{dU(x)}{dx} \). In region **CD**, the slope is negative, so the derivative \( \frac{dU(x)}{dx} \) is negative, which means the force will be positive (to the right).

     **Direction of force in region CD: To the right**.

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### Do you want more details or have any questions?

Here are 5 related questions to expand your understanding:
1. How does gravitational potential energy change with distance from the Earth?
2. Why is the gravitational potential energy negative near massive bodies like planets?
3. How do we calculate the gravitational potential energy near the Earth's surface?
4. What would happen to the force if the potential energy plot had a steeper slope in the BC region?
5. Can the gravitational potential energy of a system ever be zero? If so, under what conditions?

**Tip:** In physics, the force is always directed towards lower potential energy, hence the force points in the opposite direction of the gradient (slope) of \( U(x) \).

Which statement best describes gravitational potential energy Ugrav? Consider plot of U(x) for particle in 1D motion. Rank regions AB, BC, and CD according to magnitude of force on particle, in decreasing order. What is the direction of force when particle is in CD?

Explore gravitational potential energy and force in 1D motion through a plot of U(x). Learn how to rank force magnitude in regions AB, BC, and CD, and understand the direction of force in region CD. Suitable for Grades 11-12.

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