4 same conductor balls with charge q = 40 ìC and mass m = 30g mare located at the corners of a square a = 30cm. Now let the charged ball move freely from the initial position. What the final speed the ball can reach?
To find the final speed the charged ball can reach, we can use the principle of conservation of energy. The electric potential energy of the ball at the initial position will be converted to kinetic energy at the final position.
First, let's calculate the electric potential energy at the initial position. The electric potential energy of a charged object in the presence of another charged object can be calculated using the formula:
U_initial = k * (q^2) / r_initial
where:
- U_initial is the electric potential energy at the initial position
- k is the Coulomb's constant (8.99 x 10^9 Nm^2/C^2)
- q is the charge on each ball (40 μC or 40 x 10^-6 C)
- r_initial is the distance between the balls at the corners of the square (a = 30 cm or 0.3 m)
Let's substitute the values into the formula:
U_initial = (8.99 x 10^9 Nm^2/C^2) * ((40 x 10^-6 C)^2) / 0.3 m
U_initial = 3833333.33 Nm
Now, at the final position, all four balls will repel each other and move away from each other. Since the system is symmetrical, we consider one ball moving away from the corner towards the center of the square. The work done by electric force will be equal to the change in electric potential energy:
ΔU = U_final - U_initial
Since the balls started from rest, the initial kinetic energy is zero. Therefore, the change in potential energy will be equal to the final kinetic energy.
Let's consider the final kinetic energy as K_final.
ΔU = K_final - 0
K_final = ΔU
Next, let's calculate the change in electric potential energy:
U_final = k * (q^2) / r_final
where:
- U_final is the electric potential energy at the final position
- r_final is the distance between the balls at the final position. Since the balls are moving towards the center of the square, the distance between them can be calculated using Pythagoras theorem.
r_final = a / √2 (since the diagonal of a square is a√2)
Substituting the values:
U_final = (8.99 x 10^9 Nm^2/C^2) * ((40 x 10^-6 C)^2) / (0.3 / √2) m
U_final = 215844115.4 Nm
Now, let's calculate the change in electric potential energy:
ΔU = U_final - U_initial
ΔU = 215844115.4 Nm - 3833333.33 Nm
ΔU = 211010782.1 Nm
Finally, the final kinetic energy and speed can be calculated using the formula:
K_final = ΔU
K_final = (1/2) * (m * v_final^2)
where:
- K_final is the kinetic energy at the final position
- m is the mass of the sphere (30 g or 0.03 kg)
- v_final is the final speed of the sphere
Substituting the values:
211010782.1 Nm = (1/2) * (0.03 kg) * v_final^2
Now, solve for v_final:
v_final = √[(2 * 211010782.1 Nm) / (0.03 kg)]
v_final ≈ 10260.267 m/s
Therefore, the final speed the ball can reach is approximately 10260.267 m/s.