Two particles are fixed on an x axis. Particle 1 of charge 70.5 μC is located at x = -28.5 cm; particle 2 of charge Q is located at x = 21.2 cm. Particle 3 of charge magnitude 37.4 μC is released from rest on the y axis at y = 28.5 cm. What is the value of Q if the initial acceleration of particle 3 is in the positive direction of (a) the x axis and (b) the y axis?

a) P3 initially accelerating in the + x direction means the net force on P3, in the y direction, is 0:

[(kq1q3)/(r13^2)]sin45 + [(kq2q3)/(r23^2)]sin53.36 = 0
Divide out common factors k and q3, and sub in known quantities to solve for q2:
[(70.5 * 10^-6)/(28.5*^2 *2]sin45 + [q2/(21.2^2 + 28.5^2)]sin 53.36 = 0
q2= -4.83 x 10^-5 C
(Draw a FBD of P3 to see why q2 must be negative for this case).

b) net force in the x direction is 0, so sum the forces on P3 in this direction to find q2. (which will be positive this time; again a FBD is useful for understanding this)

To find the value of Q in both cases, we need to use Coulomb's law, which states that the electrical force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

Let's start with case (a), where the initial acceleration of particle 3 is in the positive direction of the x-axis.

Step 1: Determine the distance between particle 3 and particle 1.
The distance between particle 3 (located at y = 28.5 cm) and particle 1 (located at x = -28.5 cm) can be found using the Pythagorean theorem. Since the x and y axes form a right triangle, the distance is the hypotenuse:
Distance = sqrt((-28.5 cm)^2 + (28.5 cm)^2) = sqrt(2 * (28.5 cm)^2)

Step 2: Calculate the electrical force between particle 3 and particle 1.
The electrical force between particle 3 and particle 1 can be calculated using Coulomb's law:
F = k * (q1 * q3) / r^2,

where F is the electrical force, k is the electrostatic constant (k = 8.99 x 10^9 N * m^2/C^2), q1 and q3 are the charges of particles 1 and 3, respectively, and r is the distance between them.

Step 3: Calculate the net force on particle 3 in the x direction.
Since particle 3 is at rest initially and only experiences the electrical force from particle 1, this force will be the net force acting on particle 3 in the x direction:
Net Force (x) = F * cos(theta),

where theta is the angle between the x-axis and the line connecting particles 1 and 3. In this case, theta will be the angle whose cosine is the adjacent side (x-axis) divided by the hypotenuse (distance).

Step 4: Apply Newton's second law to find Q.
The net force on particle 3 in the x direction causes its acceleration, which is the derivative of its velocity with respect to time:
Net Force (x) = m * a,

where m is the mass of particle 3, and a is its acceleration in the x direction.

Equating the expressions for the net force on particle 3 in the x direction from step 3 and the net force (m * a) from step 4, we can solve for Q.

Repeat the above steps for case (b), where the initial acceleration of particle 3 is in the positive direction of the y-axis. In this case, calculate the net force on particle 3 in the y direction and find Q using Newton's second law.

By following these steps for both cases, you can calculate the value of Q.