Two equally charged particles, held 2.8 x 10-3 m apart, are released from rest. The initial acceleration of the first particle is observed to be 6.8 m/s2 and that of the second to be 6.8 m/s2. If the mass of the first particle is 5.3 x 10-7 kg, what are (a) the mass of the second particle and (b) the magnitude of the charge (in C) of each particle?
(a) m2=m1=5.3 x 10-7 kg=m.
(b) F=m•a
F= k•q1•q2/r²
m•a= k••q² /r²,
q=sqrt(m•a• r²/k)
where k =9•10^9 N•m²/C²
To find the mass of the second particle and the magnitude of the charge of each particle, we can use Coulomb's law and Newton's second law.
Coulomb's law states that the electrostatic force between two charged particles is given by:
F = k * (|q1| * |q2|) / r^2
Where F is the electrostatic force, k is the electrostatic constant, |q1| and |q2| are the magnitudes of the charges on the particles, and r is the distance between the particles.
Newton's second law states that the force on a particle is equal to its mass multiplied by its acceleration:
F = m * a
We can equate these two equations to find an expression for the mass of the second particle and the magnitude of the charge:
m2 * a2 = k * (|q1| * |q2|) / r^2
Now, let's substitute the given values:
k = 8.99 x 10^9 N m^2/C^2 (electrostatic constant)
m1 = 5.3 x 10^-7 kg (mass of the first particle)
a1 = a2 = 6.8 m/s^2 (acceleration of both particles)
r = 2.8 x 10^-3 m (distance between the particles)
Substituting these values into the equation, we get:
m2 * (6.8 m/s^2) = (8.99 x 10^9 N m^2/C^2) * (|q1| * |q2|) / (2.8 x 10^-3 m)^2
Simplifying and rearranging the equation, we have:
m2 = (8.99 x 10^9 N m^2/C^2) * (|q1| * |q2|) / (6.8 m/s^2) * (2.8 x 10^-3 m)^2
Now, to find the charge, we can use the equation:
F = k * (|q1| * |q2|) / r^2
Since the initial acceleration is the same for both particles, we can substitute the value of a1 (6.8 m/s^2) into the equation:
m1 * a1 = k * (|q1| * |q2|) / r^2
Substituting the values:
(5.3 x 10^-7 kg) * (6.8 m/s^2) = (8.99 x 10^9 N m^2/C^2) * (|q1| * |q2|) / (2.8 x 10^-3 m)^2
Rearranging the equation, we have:
(|q1| * |q2|) = ((5.3 x 10^-7 kg) * (6.8 m/s^2) * (2.8 x 10^-3 m)^2) / (8.99 x 10^9 N m^2/C^2)
To find the magnitude of the charge of each particle, we take the square root of the product:
|q1| = |q2| = sqrt(((5.3 x 10^-7 kg) * (6.8 m/s^2) * (2.8 x 10^-3 m)^2) / (8.99 x 10^9 N m^2/C^2))
Calculating this expression will give us the magnitude of the charge of each particle.
To solve this problem, we can use Newton's second law of motion, which states that the force acting on an object is equal to the mass of the object multiplied by its acceleration (F = m*a).
Let's start with part (a) and find the mass of the second particle.
Given:
Initial acceleration of the first particle (a1) = 6.8 m/s^2
Initial acceleration of the second particle (a2) = 6.8 m/s^2
Mass of the first particle (m1) = 5.3 x 10^-7 kg
We need to find the mass of the second particle (m2).
Since the particles are equally charged, they experience the same electrostatic force (F) between them. The electrostatic force between two charged particles is given by Coulomb's law:
F = k * (q1 * q2) / r^2
Where:
- F is the electrostatic force
- k is the electrostatic constant (9 x 10^9 N * m^2/C^2)
- q1 and q2 are the charges of the particles
- r is the distance between the particles
Since the particles are released from rest, their initial velocity is zero. This means that there are no other forces acting on them besides the electrostatic force. Therefore, the electrostatic force is responsible for the observed acceleration of the particles.
Using Newton's second law, we can express the electrostatic force in terms of the mass and acceleration:
F = m1 * a1 = m2 * a2
Rearranging the equation, we can solve for m2:
m2 = (m1 * a1) / a2
Substituting the given values:
m2 = (5.3 x 10^-7 kg * 6.8 m/s^2) / 6.8 m/s^2 = 5.3 x 10^-7 kg
So, the mass of the second particle is also 5.3 x 10^-7 kg.
Now, let's move on to part (b) to find the magnitude of the charge of each particle.
We can use the electrostatic force formula again to solve for the charges. Rearranging Coulomb's law, we get:
F = (k * q1 * q2) / r^2
q1 * q2 = (F * r^2) / k
Since the particles have the same charge magnitude (q1 = q2 = q), we can write:
q^2 = (F * r^2) / k
Now, we can substitute known values to calculate the charge:
Given:
F = m1 * a1 = (5.3 x 10^-7 kg) * (6.8 m/s^2)
r = 2.8 x 10^-3 m
k = 9 x 10^9 N * m^2/C^2
Substituting the values:
q^2 = [(5.3 x 10^-7 kg * 6.8 m/s^2) * (2.8 x 10^-3 m)^2] / (9 x 10^9 N * m^2/C^2)
Calculating the expression on the right side of the equation:
q^2 ≈ 2.4 x 10^-21 C^2
To find the magnitude of the charge, we take the square root:
q ≈ √(2.4 x 10^-21 C^2) ≈ 1.55 x 10^-11 C
Thus, the magnitude of the charge of each particle is approximately 1.55 x 10^-11 C.