In an electricity experiment, a 0.900g plastic ball is suspended on a 50.0cm long string and given an electric charge. A charged rod brought near the ball exerts a horizontal electrical force F⃗ elec on it, causing the ball to swing out to a 23.0∘ angle and remain there.
4.5
To determine the magnitude of the electrical force, F⃗ elec, exerted on the plastic ball, we can use the principles of trigonometry and the given information. Here's how you can calculate it:
1. First, let's start by drawing a diagram to visualize the setup. Draw a horizontal line to represent the string, and make a vertical line from the suspending point of the string. Now, draw a line from the top of the vertical line to represent the displacement of the ball when it is at a 23.0∘ angle.
2. Now, we have a right-angled triangle. The string forms the hypotenuse of the right triangle, the vertical line represents the gravitational force acting on the ball, and the horizontal line represents the electrical force F⃗ elec.
3. From the diagram, we can identify the right triangle's sides:
- The hypotenuse (string) has a length of 50.0 cm.
- The vertical side (gravitational force) represents the weight of the ball, mg, where m is the mass of the ball (0.900 g) and g is the acceleration due to gravity (approximately 9.8 m/s²).
- The horizontal side (electrical force) is what we're trying to find.
4. To calculate the weight of the ball, we need to convert the mass from grams to kilograms: 0.900 g ÷ 1000 = 0.0009 kg. Now, calculate the weight of the ball using the formula: weight = mass × gravity, which gives us: weight = 0.0009 kg × 9.8 m/s².
5. Now, let's focus on the angle of 23.0∘. From the right triangle, we can use the tangent function to relate the angle θ and the sides of the triangle. The tangent of an angle is equal to the opposite side divided by the adjacent side. In this case, the tangent of 23.0∘ is the ratio of the vertical side to the horizontal side.
6. Substitute the values into the tangent formula: tan(23.0∘) = weight (opposite side) / F⃗ elec (adjacent side).
7. Rearrange the formula to solve for F⃗ elec: F⃗ elec = weight / tan(23.0∘).
8. Plug in the values for weight and the angle of 23.0∘ to calculate F⃗ elec.
9. Finally, perform the calculations and obtain the magnitude of the electrical force, F⃗ elec.
Remember to always use the correct units in your calculations to ensure accuracy and consistency.