What simplifications were used to derive the formula for the period of vibration of an object at the end of a spring and of a simple pendulum?

What simplifications were used to derive the formula for the period of vibration of an object at the end of a spring and of a simple pendulum?

A 5.0-g bullet traveling horizontally at an unknown speed hits and embeds itself in a 0.195-kg block resting on a frictionless table. The block slides into and compresses a 180-N/m spring a distance of 0.10 m before stopping the block and bullet. Determine the initial speed of the bullet. State any assumptions that you made to solve the problem.

You exert a 100-N pull on the end of a spring. When you increase the force by 20% to 120 N, the spring’s length increases 5.0 cm beyond its original stretched position. What is the spring constant of the spring and its original displacement?

A pendulum clock is moved from the Mississippi Delta region to the Rocky Mountains. Will the clock run faster or slower in the Rockies? Why? How can it be adjusted to compensate for this change? Describe your assumptions.

A monkey has a cart with a horizontal spring attached to it that she uses for different tricks. In one trick, the monkey sits on the vibrating cart. When the cart reaches its maximum displacement from equilibrium, the monkey picks up a 0.30-kg cantaloupe from a trainer. The mass of the monkey and the cart together is 3.0 kg. The spring constant is 660 N/m. The amplitude of horizontal vibrations is 0.24 m. Determine the ratio of the maximum speed of the monkey before and after she picks up the cantaloupe.