A 80.0 g block carrying a charge Q = 33.0 µC is connected to a spring for which k = 74.0 N/m. The block lies on a frictionless, horizontal surface and is immersed in a uniform electric field of magnitude E = 4.84 104 N/C directed away from the spring and block. The block is released from rest when the spring is unstretched (x = 0).

(a) By what maximum distance does the block move from its initial position?

(b) Find the subsequent equilibrium position of the block and the amplitude of its motion.

(c) Using conservation of energy, find a symbolic relationship giving the potential difference between its initial position and the point of maximum extension in terms of the spring constant k, the amplitude A, and the charge Q.

To find the answers to these questions, we can use the concepts of electric potential energy, spring potential energy, and conservation of energy.

(a) To find the maximum distance the block moves from its initial position, we need to consider the forces acting on the block. The electric field exerts a force on the charged block, given by F = Q * E, where Q is the charge and E is the magnitude of the electric field. Since the electric field is directed away from the spring and block, the electric force will act in the opposite direction to the spring force.

The spring force is given by F = -k * x, where k is the spring constant and x is the displacement from the equilibrium position. We can set the electric force equal to the spring force and solve for x:

Q * E = k * x

Substituting the given values: Q = 33.0 µC = 33.0 * 10^(-6) C, E = 4.84 * 10^4 N/C, and k = 74.0 N/m:

(33.0 * 10^(-6) C) * (4.84 * 10^4 N/C) = 74.0 N/m * x

Solving for x:

x = (33.0 * 10^(-6) * 4.84 * 10^4) / 74.0 = 2.15 * 10^(-3) m

Therefore, the maximum distance the block moves from its initial position is 2.15 * 10^(-3) m.

(b) To find the subsequent equilibrium position of the block and the amplitude of its motion, we need to consider the forces acting on the block when it is in equilibrium. At equilibrium, the electric force and the spring force cancel each other out, resulting in a net force of zero.

Setting the electric force equal to the spring force:

Q * E = k * x_eq

Where x_eq is the equilibrium position. Solving for x_eq:

x_eq = (Q * E) / k = (33.0 * 10^(-6) C) * (4.84 * 10^4 N/C) / 74.0 N/m = 2.04 * 10^(-3) m

Therefore, the subsequent equilibrium position of the block is 2.04 * 10^(-3) m.

The amplitude of the block's motion is half the total displacement, which is the sum of the maximum distance and the subsequent equilibrium position:

Amplitude = (2.15 * 10^(-3) m + 2.04 * 10^(-3) m) / 2 = 2.1 * 10^(-3) m

Therefore, the amplitude of the block's motion is 2.1 * 10^(-3) m.

(c) Using conservation of energy, we can find a symbolic relationship for the potential difference between the block's initial position and the point of maximum extension. The total energy of the system is the sum of the electric potential energy and the spring potential energy.

At the block's initial position, all the energy is in the electric potential energy:

Electric Potential Energy (initial) = Q * V0

At the point of maximum extension, all the energy is in the spring potential energy:

Spring Potential Energy (maximum extension) = (1/2) * k * A^2

where A is the amplitude of the block's motion.

Since energy is conserved, the two potential energies are equal:

Q * V0 = (1/2) * k * A^2

Solving for the potential difference V0:

V0 = (1/2) * k * A^2 / Q

Therefore, the potential difference between the block's initial position and the point of maximum extension is given by (1/2) * k * A^2 / Q, where k is the spring constant, A is the amplitude, and Q is the charge of the block.