each sketch below shows three objects with an electric charge. in each case, decide whether there is a net force actin is a net force, decide whether it pushes the green - ring object to the left or right. then select the appropriate button u for example, if there is a net force pushing the green - ring object in the first sketch to the left, select the left button u force on the green - ring object in the second sketch, select the middle button under the second sketch. and so on. the sketches have charges: first row +3, -3, +2; second row -2, -2, -3; third row -1, -3, -1. there are left and right arrows below each row of charges.

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To solve this, we analyze each sketch using Coulomb's law (force between charges: $ F \propto \frac{q_1 q_2}{r^2} $, attractive for opposite charges, repulsive for like charges). ### First Sketch (Top) - Charges: $ +3 $ (left of green ring), $ -3 $ (near green ring), $ +2 $ (right of green ring). - Force from $ +3 $: Repulsive (like charges), pushes **right** (distance $ r_1 $). - Force from $ -3 $: Attractive (opposite charges), pushes **left** (distance $ r_2 $, $ r_2 < r_1 $, so stronger). - Force from $ +2 $: Repulsive, pushes **left** (distance $ r_3 $, $ r_3 > r_2 $, weaker than $ -3 $’s force). - Net force: Left (since $ -3 $’s attractive force > $ +3 $ and $ +2 $’s repulsive forces). ### Second Sketch (Middle) - Charges: $ -2 $ (left), $ -2 $ (near green ring), $ -3 $ (right). - All charges are negative (like charges) relative to the green ring (assumed positive, or analyze forces): - Force from left $ -2 $: Repulsive, pushes **right** (distance $ r_1 $). - Force from middle $ -2 $: Repulsive, pushes **right** (distance $ r_2 $, $ r_2 < r_1 $, stronger). - Force from right $ -3 $: Repulsive, pushes **left** (distance $ r_3 $, $ r_3 > r_2 $, weaker than middle $ -2 $’s force). - Wait, correction: If the green ring is negative (or charge sign of green ring? Wait, the problem says “green - ring object”—assume the green ring has a charge (e.g., positive, or we analyze forces between charges). Wait, maybe the green ring’s charge is positive (common in such problems). Then: - $ -2 $ (left): Attractive, pushes **left**. - $ -2 $ (middle): Attractive, pushes **left** (closer, stronger). - $ -3 $ (right): Attractive, pushes **left** (farther, weaker). Wait, no—if green ring is positive, negative charges attract it (pull left/right? Wait, left $ -2 $: left of green ring, so attractive force pulls green ring **left** (toward $ -2 $). Middle $ -2 $: left of green ring? Wait, the middle sketch: green ring is between $ -2 $ (left), $ -2 $ (middle - left), $ -3 $ (right). Wait, positions: left $ -2 $ (left of green), middle $ -2 $ (near green, left side?), $ -3 $ (right of green). Wait, maybe the green ring is at the middle of the three circles (the three “$ \circ $” under the sketch). So: - Left $ -2 $: distance $ r_1 $, attractive (pulls left). - Middle $ -2 $: distance $ r_2 $ (closer than $ r_1 $), attractive (pulls left, stronger). - Right $ -3 $: distance $ r_3 $ (farther than $ r_2 $), attractive (pulls right, weaker than middle $ -2 $). Wait, this is confusing. Alternatively, maybe the green ring has no net force? Wait, no—let’s re - evaluate. If all charges are negative and the green ring is positive: - Left $ -2 $: pulls left (attractive). - Middle $ -2 $: pulls left (closer, stronger pull left). - Right $ -3 $: pulls left (wait, $ -3 $ is right of green ring, so attractive force pulls green ring **right** (toward $ -3 $). Oh! I made a mistake. Direction: If a negative charge is to the **right** of a positive charge, the attractive force pulls the positive charge **right** (toward the negative charge). So: - Left $ -2 $: left of green ring (positive) → pulls **left**. - Middle $ -2 $: left of green ring → pulls **left** (closer, stronger). - Right $ -3 $: right of green ring → pulls **right** (farther, weaker than middle $ -2 $’s pull left). Now, compare forces: - Force from middle $ -2 $: $ F_2 \propto \frac{(-2)(+q)}{r_2^2} $ (attractive, magnitude $ \frac{2q}{r_2^2} $, direction left). - Force from right $ -3 $: $ F_3 \propto \frac{(-3)(+q)}{r_3^2} $ (attractive, magnitude $ \frac{3q}{r_3^2} $, direction right). - Force from left $ -2 $: $ F_1 \propto \frac{(-2)(+q)}{r_1^2} $ (attractive, magnitude $ \frac{2q}{r_1^2} $, direction left). Assume $ r_2 < r_1 < r_3 $: - $ F_2 $ (left) is stronger than $ F_1 $ (left, farther). - $ F_3 $ (right) is weaker than $ F_2 $ (left, since $ r_3 > r_2 $, and $ 3/r_3^2 < 2/r_2^2 $ if $ r_3 $ is much larger). Wait, this is getting too complex. Maybe the green ring has no net force? Wait, no—all charges are negative, so if green ring is positive, all pull left? No, left $ -2 $ pulls left, middle $ -2 $ pulls left, right $ -3 $ pulls right. Wait, maybe the green ring is negative. Then: - Left $ -2 $: Repulsive, pushes **right**. - Middle $ -2 $: Repulsive, pushes **right** (closer, stronger). - Right $ -3 $: Repulsive, pushes **left** (farther, weaker). Net force: Right (since middle $ -2 $’s repulsion > right $ -3 $’s repulsion). But the problem says “select the appropriate button”—maybe the middle sketch has **no net force**? Wait, no—let’s check distances. If the green ring is at the center of the three circles, and the charges are symmetric? Wait, left $ -2 $, middle $ -2 $, right $ -3 $: not symmetric. Alternatively, maybe the forces cancel? Unlikely. Maybe I misread: the middle sketch’s charges are $ -2 $, $ -2 $, $ -3 $—if the green ring is equidistant from $ -2 $ (middle) and $ -3 $? No, the sketch shows the green ring between the three circles, so middle $ -2 $ is closer to green ring than left $ -2 $ and right $ -3 $. Alternatively, maybe the green ring has a charge such that forces balance. But this is unclear. Let’s move to the third sketch. ### Third Sketch (Bottom) - Charges: $ -1 $ (left), $ -3 $ (near green ring), $ -1 $ (right). - Green ring: assume positive (or analyze forces). - Force from left $ -1 $: Attractive, pushes **left** (distance $ r_1 $). - Force from $ -3 $: Attractive, pushes **left** (distance $ r_2 $, $ r_2 < r_1 $, stronger). - Force from right $ -1 $: Attractive, pushes **right** (distance $ r_3 $, $ r_3 > r_2 $, weaker than $ -3 $’s force). - Net force: Left (since $ -3 $’s attractive force > left and right $ -1 $’s forces). ### Final Answers (Assuming the green ring’s charge is positive, and analyzing each sketch): - **First Sketch**: Net force left → select left button. - **Second Sketch**: If forces cancel (unlikely) or net force right/left—wait, maybe the second sketch has **no net force** (middle button) if the attractive/repulsive forces balance. But without exact distances, we infer: - First: Left - Second: Middle (no net force, if forces cancel) - Third: Left (Note: The exact answer depends on the green ring’s charge and distance, but this is the logical analysis using Coulomb’s law.) # Answers (Example Selections): - First Sketch: Left button. - Second Sketch: Middle button (no net force). - Third Sketch: Left button.

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First Sketch (Top)

Second Sketch (Middle)

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