Two equal sphere each weighting 1g hang by 2 equal silk thread from the same point the sphere are charged with in Contact and comes to rest with their centre 2 cm apart and 20 cm vertically below the point of support .find charge on sphere? solutions of this question
Figure shows forces acting on charged sphere. Weight mg is acting downward. Repulsive electrostatic force is shown as Fe .
Tension force T acting along string is resolved as ( T cosθ ) and ( T sinθ ) .
(Tcosθ) component balances weight mg. (T sinθ) component balances Electrostatic force Fe .
T cosθ = mg ...................... (1)
T sinθ = Fe ........................(2)
If we eliminate tension T in above equations, we get , Fe = m g tanθ .......................(3)
Eqn.(3) is written as
K × ( q2 / d2 ) = ( m g tanθ ) ........................... (4)
where K = 1/( 4 π εo ) = 9 × 109 N m2 C-2 Coulomb's constant and d = 2 cm is the distance between charged spheres.
m is mass of sphere , g is acceleration due to gravity and θ is the angle made by the string with vertical
Solve this if you have a brain
We are given that each sphere weighs 1g, so their combined weight is 2g or 0.002kg. The distance between their centers is 2cm or 0.02m. We need to find the charge on each sphere.
From equation (1), we have:
T cosθ = mg
Since the spheres are hanging vertically, θ = 90° and cosθ = 0. Substituting values, we get:
T * 0 = 0.002 * 9.8
T = 0
This means there is no tension in the silk threads holding the spheres.
From equation (2), we have:
T sinθ = Fe
Substituting values, we get:
T * 1 = Fe
Fe = T
But we just found that T = 0, so Fe = 0 as well. This means there is no electrostatic force between the spheres, and they must be neutral (i.e. have no charge).
Therefore, the charge on each sphere is 0.
To solve this question, we can use the concept of equilibrium of forces. Let's break down the problem step by step:
Step 1: Identify the forces acting on each sphere.
In this case, there are three forces acting on each sphere:
1. Gravitational force (mg): This force is equal to the weight of the sphere, which is 1g. Considering the acceleration due to gravity (g) as 9.8 m/s^2, the gravitational force can be calculated using the formula F = mg.
2. Tension in the silk thread (T): The silk threads are assumed to be inextensible and massless, so their tension is the same throughout. We can consider that each sphere is subject to half of the total tension. Therefore, the tension acting on each sphere can be calculated using the Pythagorean theorem.
3. Electric force (Fe): The charged spheres exert an electric force on each other. Since the spheres are in contact initially, they acquire equal charges when the contact is broken.
Step 2: Calculate the gravitational force (mg).
Given that the weight of each sphere is 1g, we can calculate the gravitational force acting on each sphere using the formula F = mg. Substituting the mass (m) as 1g (which is equal to 0.001 kg) and acceleration due to gravity (g) as 9.8 m/s^2, we find that the gravitational force acting on each sphere is 0.001 kg * 9.8 m/s^2 = 0.0098 N.
Step 3: Calculate the tension in the silk thread (T).
We need to consider the forces acting vertically and horizontally.
Vertically:
Given that the spheres come to rest 20 cm vertically below the point of support, we can use the concept of equilibrium to calculate the tension in the silk thread vertically.
The net vertical force acting on each sphere is the sum of the gravitational force (mg) acting downwards and the tension force (T) acting upwards. Since the spheres are at rest, the net force vertically should be zero. Therefore, the tension in the silk thread vertically can be calculated as T = mg.
Horizontally:
Since the spheres are at rest horizontally, the net force horizontally should also be zero. Therefore, the tension in the silk thread horizontally can be calculated as T = Fe (electric force).
Step 4: Find the charge on each sphere.
Let's denote the charge on each sphere as q. Since the spheres are identical, they have the same charge.
The electric force (Fe) between the spheres can be calculated using Coulomb's law: Fe = k * (q^2) / d^2, where k is the electrostatic constant and d is the distance between the centers of the spheres.
In this case, the distance between the centers of the spheres (d) is given as 2 cm, which is equal to 0.02 m.
The tension in the silk thread (T) can be equated to the electric force (Fe) horizontally: T = Fe.
By equating the two equations (T = mg and T = Fe), we can find the value of q (the charge on each sphere).
mg = Fe
mg = k * (q^2) / d^2
Substituting the known values:
0.0098 N = k * (q^2) / (0.02 m)^2
Now, you can use the value of the electrostatic constant (k = 9 x 10^9 N m^2/C^2) and solve the above equation to find the charge on each sphere (q).