A particle of mass 47 g and charge 54 �C is released from rest when it is 13 cm from a second particle of charge −15 �C.
Determine the magnitude of the initial acceleration of the 47 g particle.
Answer in units of m/s2
To determine the magnitude of the initial acceleration of the 47 g particle, we can use Coulomb's law and Newton's second law.
Coulomb's Law:
The electric force between two charges is given by:
F = k * |q1 * q2| / r^2,
where F is the force between the charges, k is the electrostatic constant (9 * 10^9 N·m^2/C^2), q1 and q2 are the charges of the particles, and r is the distance between them.
Newton's Second Law:
The net force on an object equals the mass of the object times its acceleration:
F = m * a,
where F is the net force, m is the mass of the object, and a is the acceleration.
Now, let's find the magnitude of the initial acceleration of the 47 g particle.
Given:
m = 47 g = 0.047 kg,
q1 = 54 μC = 54 * 10^(-6) C,
q2 = -15 μC = -15 * 10^(-6) C,
r = 13 cm = 0.13 m,
k = 9 * 10^9 N·m^2/C^2.
Using Coulomb's law, let's find the force:
F = k * |q1 * q2| / r^2
F = (9 * 10^9 N·m^2/C^2) * |(54 * 10^(-6) C) * (-15 * 10^(-6) C)| / (0.13 m)^2
Now, calculating this expression:
F ≈ 3.58 N.
Since this is the force between the charges, it is also the net force on the 47 g particle. We can set this force equal to the product of the mass and acceleration:
F = m * a
3.58 N = (0.047 kg) * a
Now, solving for the acceleration:
a ≈ 3.58 N / 0.047 kg ≈ 76.38 m/s^2.
Therefore, the magnitude of the initial acceleration of the 47 g particle is approximately 76.38 m/s^2.
To determine the magnitude of the initial acceleration of the 47 g particle, we can use the principle of electrostatic force and Newton's second law of motion.
1. Calculate the electrostatic force between the two particles:
- Given: Charge of the first particle (q1) = +54 µC = 54 * 10^(-6) C
- Given: Charge of the second particle (q2) = -15 µC = -15 * 10^(-6) C
- Given: Distance between the two particles (r) = 13 cm = 0.13 m
The electrostatic force (Fe) between two charges can be calculated using Coulomb's law:
Fe = (k * |q1 * q2|) / r^2
where:
- k is the Coulomb's constant (k ≈ 8.988 × 10^9 N m^2 C^(-2))
- |q1 * q2| is the absolute value of the product of the two charges
- r^2 is the square of the distance between the charges
Plugging in the given values:
Fe = (8.988 × 10^9 * |54 * 10^(-6) * (-15) * 10^(-6)|) / (0.13)^2
2. Calculate the net force acting on the 47 g particle:
- Given: Mass of the particle (m) = 47 g = 47 * 10^(-3) kg
The net force acting on the particle (Fnet) can be found using Newton's second law of motion:
Fnet = m * a
where:
- m is the mass of the particle
- a is the acceleration of the particle
In this case, the electrostatic force acts as the net force since there are no other external forces acting on the particle.
Hence, Fnet = Fe
3. Calculate the acceleration of the particle:
From step 1, we have the electrostatic force (Fe).
Using Newton's second law, we can rearrange the formula to solve for acceleration (a):
a = Fnet / m
Plugging in the values:
a = (Fe) / m
Now, we can substitute the value of Fe from step 1 and the value of m into the equation to get the final answer in units of m/s^2.