Solving for the coefficient of friction:

We can use the expression for conservation of energy to solve this problem:

The problem statement tells us that the sledder is initially at rest, so we can eliminate initial kinetic energy. Also, if we assume that the bottom of the hill has a height of 0, we can eliminate final potential energy to get:

  (1)

From here we need to determine what is going to contribute to work. The only extra piece of information we have in this problem is that there is an average frictional force. Therefore that will be the only source of work. Let's began expanding each term, going from left to right:

  (2)

Note that we didn't mark the height as an initial height because we assumed that the bottom of the hill has a height of 0.

Where d is the length of the hill. We can calculate this distance using the height and angle of the hill.

Rearranging for distance:

Substituting this back into our expression for work, we get:

  (3)

Now our final term:

    (4)

Now substituting expression 2, 3, and 4 into expression 1, we get:

The reason we are subtracting friction is because it is removing energy from the system. Another way we could have written our initial expression would be to have work on the final state and then friction would be positive.

Rearranging for the frictional force:

We have all of these values, so time to plug and chug:

You simply need to know the formula for the potential energy stored in a spring to solve this problem. The formula is:

where k is the spring constant, and x is the distance from equilibrium

We can rearrange this to get:

Plugging in our values, we get:

Since the block is motionless, we know that our forces will cancel out:

There are three forces in play: one from each spring, as well as the force of gravity. If we assume that forces pointing up are positive, we can write:

Plugging in expressions for each spring force, we get:

Rearring for the displacement of the top spring, we get:

Since the block is motionless, we can assume that the forces cancel out:

If we designate any forces pointing downward as positive, we can write:

Inserting expressions for each force, we get:

Rearranging for mass, we get:

There are two ways to solve this problem: the first involves calculating the spring constant and the second does not. We'll go through both methods.

Calculating Spring Constant

We can use the expression for the force of a spring:

At equilibrium, the force of the spring equals the force of gravity:

Rearranging for the spring constant and plugging in values, we get:

Now, apply this equation when the spring is on a different planet:

Rearranging for displacement and plugging in values, we get:

Without Calculating Spring Constant

We can write the force equation for each scenario. Let the subscript 1 denote Earth, and 2 denote the other planet:

Using these equations, we can set up a proportion:

Rearranging for the displacement of scenario 2 and plugging in values:

Since the displacement of the spring is at equilibrium, we can write:

There are three forces we can account for: spring force, gravitational force, and the additional force resulting from the acceleration of the elevator. If we assume that forces pointing upward are positive, we can write:

If you are unsure whether the force resulting from the acceleration of the elevator will be positive or negative, think about the situtation from personal experience: When an elevator begins to accelerate upward, your body feels heavier. Thus, the force adds to the normal gravitational force.

Substituting expressions for each force, we get:

Rearrange to solve for displacement:

The maxmimum force applied by the spring will occur when the mass is at its maximum displacement. Since we know the energy of the system, we can calculate displacement using the following expression:

Rearranging for displacement, we get:

We can use this, along with the expression for force applied by a spring:

Substitute our first expression into our second and simplify:

We have values for each variable, allowing us to solve for the force:

Note that the mass of the block is irrelevant to the problem. Mass does not effect displacement or force applied by the spring; it only affects the velocity of the block at different points.

We simply need to alter the expression for the force of the springs to solve this problem. The following is the original expression:

Since each spring has the same costant, they actually act as one large spring with the same, original costant. Therefore the value of  in this equation is the total displacement. Multiplying the displacement of each single spring by the number of total springs will give us this total displacement:

Here,  is the number of springs, and this new displacement, , is the displacement of each spring. Rearranging for the number of springs, we get:

We can use force equilibrium to begin our derivation:

There are two vertical forces in play: gravity and total spring force. Since net force is zero, we know that these two general forces are equal to each other:

Substituting in expressions for each force, we get:

There are two things to note about the total spring force. The first is that we multiply it by 40 because there are 40 individual springs. Second, we multiply the force by the sine of the angle, because we only want to know the vertical force applied by the springs.

Rearranging for the displacement of the springs, we get:

We have values for each variable, allowing us to solve: