Question

Два маленьких шарика массами m1 = m2 = 5 г подвешены на шелковых нитях длиной l = 0,6 м каждая, прикрепленных к одному крючку. После того как шарикам сообщили равные одноименные заряды q1 = q2 = q0, они разошлись так, что каждая нить отклонилась от вертикали на угол α = 30. Определите заряды шариков (срочно с решением)

Answer 1

Дано
m₁=m₂=5г=5*10⁻³кг
l = 0,6 м
α=30°
q₁ = q₂ = q₀
Найти q₀
При отклонении о вертикали каждый шарик отклонился на угол α. При этом он приподнялся на высоту h=l-lcosα=l(1-cosα)
При этом потециальная энергия шарика станет Eп=mgh=mgl(1-cosα)
потенциальная энергия двых шариков 2Еп
По горизотали каждый шарик отклониться на lsinα, и расстояние между шариками станет r=2lsinα
Энергия взаимодействия двух зарядов
U=kq₁q₂/r
U=kq₀²/(2lsinα)
U=2Eп

kq₀²/(2lsinα)=2mgl(1-cosα)

q₀²=4mgl²(1-cosα)sinα/k=4*5*10⁻³кг * 10H/кг*0,6²м²(1-cos30)sin30/(9*10⁹ Н·м²/Кл²)= 2*10⁻¹кг * H/кг*0,36 м²(1-√3/2)*0,5/(9*10⁹ Н·м²/Кл²)= 3,6*10⁻² * H*м²(2-√3)/(2*9*10⁹ Н·м²/Кл²)=5,4*10⁻¹³Кл²

q₀=7,3*10⁻⁷Кл

Answer 2

Problem Analysis

We have two small spheres with masses m1 = m2 = 5 g, suspended on silk threads of length l = 0.6 m each, attached to a single hook. After the spheres were given equal charges q1 = q2 = q0, they moved apart in such a way that each thread deviated from the vertical by an angle α = 30°. We need to determine the charges of the spheres.

Solution

To solve this problem, we can use the concept of electrostatic forces and the principle of equilibrium.

Let's consider one of the spheres, say sphere 1. The electrostatic force between the two spheres can be given by Coulomb's law:

F = k * (q1 * q2) / r^2,

where F is the electrostatic force, k is the electrostatic constant, q1 and q2 are the charges on the spheres, and r is the distance between the spheres.

Since the spheres are in equilibrium, the net force acting on each sphere must be zero. This means that the electrostatic force acting on sphere 1 must be balanced by the tension in the silk thread.

Let's analyze the forces acting on sphere 1: 1. The weight of sphere 1 acts vertically downward with a magnitude of m1 * g, where m1 is the mass of sphere 1 and g is the acceleration due to gravity. 2. The tension in the silk thread acts along the thread and is directed towards the center of the circle formed by the motion of the sphere. Let's call this tension T1.

Since the thread is at an angle α = 30° with the vertical, we can resolve the tension into two components: - The vertical component of tension, T1 * cos(α), balances the weight of sphere 1. - The horizontal component of tension, T1 * sin(α), balances the electrostatic force between the spheres.

Similarly, for sphere 2, the forces can be analyzed in the same way.

Now, let's write the equations for the forces acting on each sphere:

For sphere 1: 1. Vertical equilibrium: T1 * cos(α) = m1 * g 2. Horizontal equilibrium: T1 * sin(α) = k * (q1 * q2) / r^2

For sphere 2: 1. Vertical equilibrium: T2 * cos(α) = m2 * g 2. Horizontal equilibrium: T2 * sin(α) = k * (q1 * q2) / r^2

Since the masses of the spheres are equal (m1 = m2 = 5 g) and the charges on the spheres are equal (q1 = q2 = q0), we can simplify the equations:

1. Vertical equilibrium: T1 * cos(α) = T2 * cos(α) = 5 g * g 2. Horizontal equilibrium: T1 * sin(α) = T2 * sin(α) = k * (q0^2) / r^2

Dividing the second equation by the first equation, we get:

(T1 * sin(α)) / (T1 * cos(α)) = (k * (q0^2) / r^2) / (5 g * g)

Simplifying further, we get:

tan(α) = (k * (q0^2) / r^2) / (5 g * g)

Now, we can substitute the known values: - α = 30° - k = electrostatic constant - q0 = charges on the spheres (to be determined) - r = distance between the spheres (to be determined) - g = acceleration due to gravity

We can rearrange the equation to solve for q0:

(q0^2) = (5 g * g * r^2 * tan(α)) / k

Taking the square root of both sides, we get:

q0 = sqrt((5 g * g * r^2 * tan(α)) / k)

Now, we can substitute the known values: - g = acceleration due to gravity - r = length of the silk thread = 0.6 m - α = 30° - k = electrostatic constant

Let's calculate the value of q0.