# MHS exercises

## Periodic and oscillatory motion

1. The earth takes 1 year to complete a loop around the sun. This is called a periodic movement and 1 year is the period of movement. How often is the earth moving around the sun? Consider 1 year = 365 days.

First we must transform the year unit to the one used inversely in frequency, that is, second. Being the frequency equal to the inverse of the period, we have to: 2. A pendulum takes 0.5 seconds to restore its initial position after going through all the oscillation points, what is its frequency?

As the given time is equivalent to the complete movement of the pendulum, this is considered its period of oscillation, ie: As the frequency is the inverse of the period we have: ## MHS Time Functions

1. A spring-mass oscillator has a 2mm range of motion, 2π pulsation, and no phase lag. When t = 10s, what is the elongation of motion?

Being the time function of the elongation: Substituting the given values ​​we have: Remembering that the resulting unit will be mm, because the values ​​were not passed to SI. Since cosine of 20π is a maximum value (+1), the elongation will be maximum, ie equal to amplitude.

2. Given the time function of the elongation: Knowing that all values ​​are in SI units answer:

The) What is the range of motion?

Removing the value of the equation, with SI units we have:

A = 3m

B) What is the pulse of movement?

Removing the value of the equation, with SI units we have: ç) What is the period of the movement?

Knowing the pulse and knowing that: Equating the values: d) What is the initial phase of the movement?

Removing the value of the equation, with SI units we have: and) When t = 2s what will be the elongation of the motion?

Applying the value in the equation we have: 3. A harmonic oscillator has its elongation described by the following equation: Being all units found in SI. What is the speed of movement at times t = 1s, t = 4s and t = 6s?

Recalling that the equation used for speed in mhs is: Using the values ​​found in the elongation equation we will have: Overriding the requested time values ​​we have:

For t = 1s: For t = 4s: For t = 6s: 4. What is the acceleration of a body that describes mhs when its elongation is x = 0 and when x = A?

Using the equation: Knowing that the pulse has a fixed value, regardless of the elongation, it is easy to see that:

At x = 0, the acceleration will be zero (a = 0) and

At x = A, the acceleration will be maximum (or minimum, depending on the sign of A).