A particle of charge Q is fixed at the origin of an xy coordinate system. At t = 0 a particle (m = 0.690 g, q = 4.75 µC) is located on the x axis at x = 20.0 cm, moving with a speed of 50.0 m/s in the positive y direction. For what value of Q will the moving particle execute circular motion? (Neglect the gravitational force on the particle
R = 0.30 m and V = 50 m/s.
If V^2/R = k Q q/R^2 , there will be circular motion.
Solve for Q
Here R = 0.20 m
mV^2 /R = kQq/R^2
Am I right?
To find the value of Q that will make the moving particle execute circular motion, we can equate the centripetal force to the electrostatic force between the two particles.
Step 1: Calculate the mass of the moving particle in kilograms.
Given: m = 0.690 g = 0.690 × 10^(-3) kg
Step 2: Calculate the charge of the moving particle in Coulombs.
Given: q = 4.75 µC = 4.75 × 10^(-6) C
Step 3: Calculate the distance between the two particles.
Given: The moving particle is located on the x-axis at x = 20.0 cm = 0.20 m. The fixed particle is at the origin (0,0).
Distance between the particles = x = 0.20 m
Step 4: Calculate the electrostatic force between the two particles using Coulomb's Law.
The electrostatic force between two point charges q1 and q2 separated by a distance r is given by: F = (k * |q1| * |q2|) / r^2
where k is the electrostatic constant (k = 8.99 × 10^9 N m^2/C^2).
Electrostatic force between the particles = F_electric = (k * |Q| * |q|) / r^2
Step 5: Calculate the centripetal force required for circular motion.
The centripetal force required for circular motion is given by: F_centripetal = m * a
where a is the centripetal acceleration.
Since the particle is moving in a circle, the centripetal acceleration is given by a = v^2 / r, where v is the speed of the particle. Given v = 50.0 m/s.
Centripetal force = F_centripetal = m * (v^2 / r)
Step 6: Equate the electrostatic force and centripetal force.
Setting F_electric = F_centripetal, we have:
(k * |Q| * |q|) / r^2 = m * (v^2 / r)
Step 7: Solve for Q.
|Q| = (m * v^2 * r) / (k * |q|)
Substituting the values:
|Q| = (0.690 × 10^(-3) kg * (50.0 m/s)^2 * 0.20 m) / (8.99 × 10^9 N m^2/C^2 * 4.75 × 10^(-6) C)
Calculating |Q| will give you the value of charge Q that will make the moving particle execute circular motion.
To determine the value of charge Q for which the moving particle will execute circular motion, we need to find the net force acting on the particle and then set it equal to the centripetal force required for circular motion.
Let's break it down step by step:
Step 1: Calculate the momentum of the particle.
The momentum of a particle is given by the formula:
p = m * v
where p is the momentum, m is the mass, and v is the velocity. In this case, the mass (m) is given as 0.690 g, which is equivalent to 0.000690 kg, and the velocity (v) is given as 50.0 m/s. Calculate the momentum using these values.
Step 2: Calculate the net force acting on the particle.
The net force acting on a charged particle in an electric field is given by the formula:
F_net = q * E
where F_net is the net force, q is the charge of the particle, and E is the electric field. In this case, the electric field is created by the fixed particle at the origin.
Step 3: Set the net force equal to the centripetal force required for circular motion.
For an object in circular motion, the net force acting on it must be equal to the centripetal force, which is given by the formula:
F_c = (m * v^2) / r
where F_c is the centripetal force, m is the mass, v is the velocity, and r is the radius of the circular path.
In this case, the radius of the circular path is the distance between the two particles (20.0 cm) since the moving particle is constrained to move in a circle centered at the fixed particle at the origin.
Step 4: Set the net force equal to the centripetal force and solve for Q.
Equate the net force (F_net) calculated in Step 2 to the centripetal force (F_c) calculated in Step 3:
q * E = (m * v^2) / r
Substitute the given values into the equation and solve for Q.
By following these steps, you can find the value of charge Q for which the moving particle will execute circular motion.