To answer this question, we need to separately consider the component forces acting in the horizontal and vertical direction.

First, let's consider the vertical component. Since the box is only moving in the horizontal direction, we know that there are no net forces acting in the vertical direction. Consequently, the net force in the y-direction is zero.

Next, let's look at the forces acting in the horizontal direction. Since we need to figure out the force in the string necessary to make the box move at a constant speed, we're looking for a situation in which the box is not accelerating in the horizontal direction. Thus, we're looking for a case in which the net force in the x-direction is zero.

Now, let's rewrite the expression for the force of kinetic friction.

Now, plugging in the expression for the normal force, we obtain the following.

Next, let's go ahead and plug in the values given to us in the question stem.

The swinging back and forth motion is a part of a circle, thus

Combining equations

There will be the force of gravity and the string acting on the coin

Solving for tension:

Converting and plugging in values:

Start with the formula for force and solve for acceleration. Then subtract the acceleration of gravity because the elevator is moving up.

There are two forces contributing to tension in the rope: the weight of the mass and the force of the spring. When the tension of the rope is zero, we know that these two forces are equal and opposite. Furthermore, the weight of the mass will always be pointed down, so the force of the spring (relative to the rope) will be pointed upward. This occurs in a moment of compression of the spring. With that said, we just need to solve the following expression:

Rearranging for displacement:

Plugging in our values:

If the diver stays at a constant depth, we know that all the forces on him must cancel out:

We can write out all of the forces on the diver, using force of the flippers to denote the vertical force he exerts to stay at constant depth:

The force of gravity and bouyancy counteract each other, so they will have opposite signs. If we assume that the force of flippers will be down, we can write:

The bouyancy force is simply the weight of the water displaced by the diver. Plugging in expressions for the latter two forces:

We have values for all of these, allowing us to solve:

Since the spring is initially in equilibrium, we can write:

There are three forces in play: spring force, gravitational force, and the additional force resulting from the acceleration of the box. If we say that any forces pointing downward are positive, we can write:

The force resulting from the additional acceleration must be subtracted since all forces cancel out and both of the other two forces are postive. This means that the resulting force from acceleration is pointing upward; Thus, the box is accelerating downward (think about riding in an elevator).

Substituting expressions for each force, we get:

Rearrange to solve for the acceleration of the box: