We also can calculate the total number of seconds in a day.

There are seconds in one day.

Therefore we want our clock to swing a certain number of times with a period of to equal .

We know that our current clock has a certain number of swings with a period of to equal

So we have

We can calculate the current period of the pendulum using the equation

We can set up a ratio of each of these two periods to determine the missing length.

Notice that 2, pi and g are all in both the numerator and denominator and therefore fall out of the problem.

We can now solve for our missing piece.

Square both sides to get rid of the square root.

We should lengthen the pendulum by

The formula for the potential energy in a spring is:

Use the given spring constant and displacement to solve for the stored energy.

We also can calculate the total number of seconds in a day.

There are seconds in one day.

Therefore we want our clock to swing a certain number of times with a period of to equal .

We know that our current clock has a certain number of swings with a period of to equal

So we have

We can calculate the current period of the pendulum using the equation

We can set up a ratio of each of these two periods to determine the missing length.

Notice that 2, pi and g are all in both the numerator and denominator and therefore fall out of the problem.

We can now solve for our missing piece.

Square both sides to get rid of the square root.

We should lengthen the pendulum by

The formula for the potential energy in a spring is:

Use the given spring constant and displacement to solve for the stored energy.

The mass has no bearing on the relationship between frequency and period. This relationship is given by the equation:

Given the period, the frequency will be equal to its reciprocal.

Conservation of energy dictates that the total mechanical energy will remain constant. Initially, the mass will not be moving and will be at its highest height. When released, it will begin to travel downward (lose potential energy) and gain velocity (gain kinetic energy). When the mass reaches the bottommost point in the swing, the potential energy will be at a minimum and the kinetic energy will be at a maximum. This point corresponds to the string being perpendicular to the ground.

The mass has no bearing on the relationship between frequency and period. This relationship is given by the equation:

Given the period, the frequency will be equal to its reciprocal.

Conservation of energy dictates that the total mechanical energy will remain constant. Initially, the mass will not be moving and will be at its highest height. When released, it will begin to travel downward (lose potential energy) and gain velocity (gain kinetic energy). When the mass reaches the bottommost point in the swing, the potential energy will be at a minimum and the kinetic energy will be at a maximum. This point corresponds to the string being perpendicular to the ground.

For this problem, use Hooke's law:

In this formula, is the spring constant, is the compression of the spring, and is the necessary force. We are given the spring constant and the force, allowing us to solve for the displacement.

Plug in our given values and solve.

Note that both the force and the displacement are positive because the stretching force will pull in the positive direction. If the spring were compressed, the change in distance would have been negative.

We will need to start at the end of the situation and work backwards in order to determine the velocity of the bullet. At the very end, the pendulum with the bullet reaches its maximum height and therefore comes to a stop. It has gravitational potential energy. At the bottom of the pendulum right after the bullet collides with it, it has kinetic energy due to the velocity of the bullet. With the law of conservation of energy we can set the kinetic energy of the pendulum right after the collision equal to the gravitational potential energy of the pendulum at the highest point.

To determine the height of the pendulum we will need to use trig and triangles to find the height. We know that the pendulum makes a 30 degree angle with the equilibrium position at its maximum height. The length of the pendulum is provided which is the hypotenuse of this triangle. We need to find the adjacent side of this triangle. We can use cosine to determine this.

We can now subtract this value from the length of the pendulum to determine how high off the ground the pendulum is at its highest point.

We can now set the kinetic energy of the pendulum right after the collision equal to the gravitational potential energy of the pendulum at the highest point.

The mass is the same throughout so it falls out of the equation.

The pendulum with the bullet was moving after the collision. We can now use momentum to determine the speed of the bullet before the collision. Conservation of momentum states that the momentum before the collision must equal the momentum after the collision.

They both move together after the collision

Since the pendulum was not moving at the beginning

We can now plug in these values and solve for the missing piece.