A small sphere with a charge of 2.40 is attached to a relaxed horizontal spring whose force constant is 90.0 . The spring extends along the axis, and the sphere rests on a frictionless surface with its center at the origin. A point charge is now moved slowly from infinity to a point on the axis. This causes the small sphere to move to the position .
To find the position of the small sphere after the point charge is moved, we can use Coulomb's law and the equation for the force in a spring.
1. Determine the force between the point charge and the small sphere using Coulomb's law:
F = k * (q1 * q2) / r^2
Where F is the force, k is the Coulomb constant, q1 and q2 are the charges, and r is the distance between the charges.
Let q1 be the charge of the small sphere (2.40 C) and q2 be the charge of the point charge (Q). Since the point charge is moved to a point on the x-axis, the distance between the charges is the x-coordinate of the point charge, which is x.
F = k * (2.40 C * Q) / x^2
2. Determine the force exerted by the spring using Hooke's law:
F = -k * x
Where F is the force, k is the spring constant, and x is the displacement from the equilibrium position.
Since the small sphere is attached to the spring, the force exerted by the spring is equal to the force between the small sphere and the point charge:
F = k * (2.40 C * Q) / x^2 = -k * x
3. Solve the equation for x:
k * (2.40 C * Q) / x^2 = -k * x
Divide both sides by k:
(2.40 C * Q) / x^2 = -x
Multiply both sides by x^2:
2.40 C * Q = -x^3
Rearrange the equation:
x^3 = -2.40 C * Q
Take the cube root of both sides:
x = (-2.40 C * Q)^(1/3)
Therefore, the position of the small sphere after the point charge is moved to a point on the x-axis is x = (-2.40 C * Q)^(1/3).
To find the position of the small sphere when the point charge is moved to a point on the x-axis, we can use Coulomb's law and Hooke's law to calculate the position of the sphere.
1. Coulomb's law states that the electrostatic force between two charged objects is given by F = k * (q1 * q2) / r^2, where F is the force, k is the Coulomb constant, q1 and q2 are the charges, and r is the distance between the charges.
2. Hooke's law relates the force exerted by a spring to its displacement. It states that F = -k * x, where F is the force, k is the force constant, and x is the displacement from the equilibrium position.
In this scenario, the force between the point charge and the small sphere exerts a force on the sphere, causing it to move. The force from the point charge is attractive since the charges have opposite signs.
To find the position of the small sphere at equilibrium, we need to set the forces from Coulomb's law and Hooke's law equal to each other:
k * (q1 * q2) / r^2 = -k * x
Since the charge q2 on the small sphere is given as 2.40 and the force constant k is given as 90.0, we can substitute these values into the equation:
90.0 * (q1 * 2.40) / r^2 = -90.0 * x
Simplifying:
(q1 * 2.40) / r^2 = -x
Now, to find the position of the small sphere when the point charge is moved to a specific point, we can substitute the values for q1 and r into the equation and solve for x.
Note: The values of q1 and r were not provided in the given information. Please provide these values in order to calculate the position of the small sphere.