## 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).*