(i) Two charged particles having vharge `4.0 xx 10^(-6) C` and mass `24 xx 10^(-3) Kg` each are joining by an insulating string of length 1 m and the system is kept on a smooth horizontal table. Find the tension in the string.
(ii) If suddenly string is cut then what is the acceleration of each particle ?
(iii) Are they having equal acceleration ?

Answer

Let's solve the question step by step.

### Given Data:
- Charge of each particle, \( q = 4.0 \times 10^{-6} \, \text{C} \)
- Mass of each particle, \( m = 24.0 \times 10^{-3} \, \text{kg} \)
- Distance between the particles (length of the string), \( r = 1.0 \, \text{m} \)
- Coulomb's constant, \( k = 9.0 \times 10^9 \, \text{N m}^2/\text{C}^2 \)

### (i) Finding the tension in the string:
The tension in the string is equal to the electrostatic force of repulsion between the two charged particles. The formula for the electrostatic force \( F \) is given by Coulomb's law:

\[ F = k \frac{q_1 q_2}{r^2} \]

Since both charges are the same, we can write:

\[ F = k \frac{q^2}{r^2} \]

Substituting the values:

\[ F = 9.0 \times 10^9 \cdot \frac{(4.0 \times 10^{-6})^2}{(1.0)^2} \]

Calculating \( (4.0 \times 10^{-6})^2 \):

\[ (4.0 \times 10^{-6})^2 = 16.0 \times 10^{-12} \, \text{C}^2 \]

Now substituting back into the equation for \( F \):

\[ F = 9.0 \times 10^9 \cdot \frac{16.0 \times 10^{-12}}{1} \]

Calculating \( F \):

\[ F = 9.0 \times 10^9 \cdot 16.0 \times 10^{-12} = 144.0 \times 10^{-3} \, \text{N} = 0.144 \, \text{N} \]

Thus, the tension in the string is:

\[ \text{Tension} = 0.144 \, \text{N} \]

### (ii) Finding the acceleration of each particle if the string is cut:
When the string is cut, each particle will experience the same force \( F \) and will have the same mass \( m \). The acceleration \( a \) of each particle can be calculated using Newton's second law:

\[ a = \frac{F}{m} \]

Substituting the values:

\[ a = \frac{0.144 \, \text{N}}{24.0 \times 10^{-3} \, \text{kg}} \]

Calculating \( a \):

\[ a = \frac{0.144}{0.024} = 6.0 \, \text{m/s}^2 \]

Thus, the acceleration of each particle is:

\[ \text{Acceleration} = 6.0 \, \text{m/s}^2 \]

### (iii) Are they having equal acceleration?
Since both particles have the same mass and experience the same force when the string is cut, they will have equal acceleration.

Thus, the answer is:

\[ \text{Yes, they have equal acceleration.} \]

### Summary of Answers:
(i) Tension in the string = \( 0.144 \, \text{N} \)
(ii) Acceleration of each particle = \( 6.0 \, \text{m/s}^2 \)
(iii) Yes, they have equal acceleration.