Since there are two forces acting on the object, and it isn't moving, there is an equality of the two forces. This means that:

Equivalently:

Where  is the spring constant,  is the displacement of the spring,  is the equilibrium position,  is mass of the object and  is the gravitational constant given as 

Since we know that the mass on the spring spring is displaced , the mass is , and we know the gravitational constant we plug in and solve for the spring constant.

Use the equation:

for a mass on a horizontal spring (so no gravity in the direction of motion)

The stretch length will have no effect.

Plug in values:

We know that when the block is at its maximum displacement, it is not moving and therefore has no kinetic energy. Also, we know that the block has its maximum kinetic energy (thus velocity) when it is passing through the springs equilibrium and therefore has no potential energy. Therefore, we can say:

Where the potential energy of a spring is:

And the expression for kinetic energy is:

Substituting in these expressions, we get:

Canceling out the fractions:

No rearranging for mass, we get:

We have all of these values, so we can solve the problem:

The expression for the period of a spring in simple harmonic motion is:

We have the spring constant, so we just need to calculate the mass of the block. For a spring in simple harmonic motion, we know that the maxmium potential energy is equal to the maximum kinetic energy. Therefore we can say:

Rearranging for mass, we get:

Plugging in our values, we get:

Now we can use the expression for period to solve the equation:

Plugging in our values:

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

We can eliminate initial kinetic energy (block initially at rest) and final potential energy (block at equilibrium of spring) to get:

Substituting in expressions for each of these, we get:

Where initial height is simply the displacement of the spring:

Multiplying both sides of the expression by 2 and rearranging for final velocity, we get:

Plugging in values for each of these variables, we get:

The spring constant is equal to .

Using conservation of energy:

While at the maximum value of stretch (the amplitude), the velocity will be zero. While at the maximum velocity, the stretch will be zero.

Plugging in values:

Solving for 

Plugging in values:

This question deals with conservation of energy in the form of a pendulum. The equation for conservation of energy is:

According to the problem statement, there is no initial kinetic energy and no final potential energy. The equation becomes:

Subsituting in the expressions for potential and kinetic energy, we get:

We can eliminate mass to get:

Rearranging for final velocity, we get:

In order to solve for the velocity, we need to find the initial height of the ball.

The following diagram will help visualize the system:

From this, we can write:

Using the length of string and the angle it's held at, we can solve for :

Now that we have all of our information, we can solve for the final velocity:

We need to know how to calculate the period of a pendulum to solve this problem. The formula for period is:

In the problem, we are only changing the length of the string. Therefore, we can rewrite the equation for each scenario:

Dividng one expression by the other, we get a ratio:

We know that , so we can rewrite the expression as:

Rearranging for P2, we get:

We need to know the formula for the period of a pendulum to solve this problem:

We aren't given the length of the pendulum, but that's ok. We could solve for it, but it would be an unnecessary step since the length remains constant.

We can write this formula for the pendulum when it is on the student's table and when it is falling:

1 denotes on the table and 2 denotes falling. The only thing that is different between the two states is the period and the gravity (technically the acceleration of the whole system, but this is the form in which you are most likely to see the formula). We can divide the two expressions to get a ratio:

Canceling out the constants and rearranging, we get:

We know g1; it's simply 10. However, we need to calculate g2, which is the rate at which the pendulum and balloon are accelerating toward the ground. We are given enough information to use the following formula to determine this:

Removing initial velocity and rearranging for acceleration, we get:

Plugging in our values:

This is our g2. We now have all of the values to solve for T2: