Question
6. Three identical small Styrofoam balls (m=1.86g) are suspended from a fixed point by three nonconducting threads, each with a length of 53.7cm and with negligible mass.At equilibrium the three balls form an equilateral triangle with sides of 31 .6cm. What is the common charge q carried by each ball? square square Incorrect. Tries 1/11 Previous Tries
Solution
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Justin
Master · Tutor for 5 years
Answer
Here's how to solve this problem:**1. Understand the Forces**Each Styrofoam ball experiences three forces:* **Gravity (downward):** Fg = mg* **Tension (along the thread):** FT* **Electrostatic repulsion (horizontally outward from the triangle's center):** Fe = kq²/r² (where k is Coulomb's constant, q is the charge, and r is the distance between balls)**2. Analyze the Geometry*** Since the triangle is equilateral, the angle between any two threads is 60°.* We can analyze the forces on just one ball. The tension force can be resolved into vertical and horizontal components.* The horizontal component of the tension balances the electrostatic force.* The vertical component of the tension balances the gravitational force.**3. Set up the Equations*** **Vertical equilibrium:** FTcosθ = mg* **Horizontal equilibrium:** FTsinθ = Fe = kq²/r²Where θ is the angle between the thread and the vertical. We can find this angle using trigonometry. Imagine a line drawn from the fixed point to the center of the equilateral triangle formed by the balls. This line bisects the 60° angle between the threads, so θ is half of 60°, or 30°. Alternatively, we can use the triangle formed by the thread, the vertical line from the suspension point, and half the distance between two balls.**4. Solve for q*** From the vertical equilibrium equation: FT = mg/cosθ* Substitute this into the horizontal equilibrium equation: (mg/cosθ)sinθ = kq²/r²* Simplify: mgtanθ = kq²/r²* Solve for q: q = sqrt((r²mgtanθ)/k)**5. Calculate*** m = 1.86 g = 0.00186 kg* g = 9.8 m/s²* r = 31.6 cm / 2 = 0.158 m (half the distance between balls)* θ = 30° (or calculate using trigonometry and the given lengths)* k = 8.99 x 10⁹ N⋅m²/C²Substitute these values into the equation for q and calculate.**Important Note:** The distance 'r' is the distance between the centers of the balls, which is half the side length of the equilateral triangle. Make sure all units are consistent (SI units are recommended).**Example Calculation:**q = sqrt(((0.158 m)² * 0.00186 kg * 9.8 m/s² * tan(30°)) / (8.99 x 10⁹ N⋅m²/C²))q ≈ 7.13 x 10⁻⁸ C (This is an approximate value. The final answer will depend on the exact values used and the precision of the calculation.)Therefore, the common charge q carried by each ball is approximately 7.13 x 10⁻⁸ C. Remember to double-check the calculations and units.