The period and frequency of a pendulum depend only on its length and the gravity force constant, . Changing the mass of the pendulum does not affect the frequency, and since the student released the new pendulum from the same displacement as the old, the amplitude and phase remain the same, and the equation of motion is the same for both pendula.

The expression for the period of a pendulum is:

Therefore, the period of a pendulum is proportional to the square root of the length of the pendulum (assuming they are both on earth, or the same planetary body). Thus, the pendulum with the longer length will have the longer period.

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:

Since we are told that the block is 1m above its lowest point (half way between its low point and horizontal), we can calculate the angle that the pendulum makes with the vertical:

Then we can use the following expression to determine the net force acting on the block in the direction of its motion:

Then we can use Newton's 2nd law to determine the instantaneous acceleration:

Canceling out mass and rearranging for acceleration:

Since the maximum angle achieved by the pendulum is very small, we can use the follow expression to determine the angle of the pendulum at any time :

Note how we are using the cosine function since the pendulum began at its highest point. We already have all of these values, so we can simply plug and chug:

Since we are given the length of the pendulum and told that it begins at rest and in the horizontal position, we can calculate the maximum velocity of the block as it travels through its lowest point using the expression for conservation of energy:

We can eliminate initial kinetic energy since the pendulum begins at rest. We can also eliminate final potential energy if we assume that the height at the lowest point of rotation is equal to 0.

Substituting in expressions for each of these:

Where the initial height is just the length of the pendulum:

Rearranging for final velocity, we get:

We can then calculate centripetal acceleration from this:

Where the radius is the length of the pendulum:

The frequency of a pendulum is given by:

Where is the length of the pendulum and is the gravity constant. Notice how the frequency is independent of mass.

Plugging in values:

We can use the equation for conservation of energy to solve this problem.

If the initial state is when the mass is at its highest position and the final state is when the mass is at its lowest position, then we can eliminate initial kinetic energy and final potential energy:

Substituting expressions in for each term, we get:

Canceling out mass and rearranging for height, we get:

Thinking about a pendulum practically, we can write the height of the mass at any given point as a function of the length and angle of the pendulum:

Think about how this formula is written. The second term gives us how far down the mass is from the top point. Therefore, we need to subtract this from the length of the pendulum to get how high above the lowest point (the height) the mass currently is.

Substituting this into the previous equation, we get:

Rearrange to solve for the angle:

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

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:

To find the answer one must manipulate the equation

Where represents the period of the motion, the length of the pendulum, and the gravity or acceleration the system is under.

To solve this for we will start by dividing both sides by . Next will with square both sides and finally multiply by , to come to the form below

Now plugging in our numbers

Keep in mind that the most accurate method is to round numbers at the very end of calculations (above this isn't done from the start of pi for simplicity).

The unit calculation above will end with meters as we are taking which will leave as the final unit of your answer.