VDU_Physics_2_001

## Physics 2.- Oscillations - Simple Harmonic Motion (SHM)

Javier Montenegro Joo

## Oscillations - Simple Harmonic Motion (SHM)

(1) The vibrations of a vertical spring are mathematically modeled by

Determine: (a) the initial values of displacement, velocity and acceleration. (b) The maximum values of displacement, velocity and acceleration.

Solution

First of all get mathematical expressions for displacement, velocity and acceleration:

(2) The horizontal oscillations of a small ball are mathematically represented by

Determine: (a) the initial values of displacement, velocity and acceleration. (b) The maximum values of displacement, velocity and acceleration.

Solution

First of all get mathematical expressions for displacement, velocity and acceleration:

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(3) The vibrations of a pendulum oscillating with simple harmonic motion (SHM) are represented by

Determine:   (a) Initial amplitude   (b) Initial velocity   (c) Initial acceleration (d) length of pendulum.

Solution

In order to identify the terms in the given equation compare it with that of the pendulum oscillating with Simple Harmonic Motion (SHM) given by:

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(4) The Simple Harmonic Motion (SHM) oscillations of a spring are represented in the MKSC system by

Determine: (a)  Maximum amplitude  (b) Maximum velocity (c) Maximum acceleration (d) Initial phase   (e) frequency.

Solution

By comparing the given equation with that of a spring oscillating with Simple Harmonic Motion (SHM):

(5) When a 2.0 kg block is hung from a vertical spring, this stretches 20 cm.  Later this spring is placed horizontally on a frictionless floor with one of its extremes against the wall and, a 5 kg block is attached to its free extreme. Determine the period and the angular frequency of the oscillation of the system when it is compressed against the wall and released.

Solution.-

The data about the vertical spring is used to calculate the elastic constant of the spring from Hooke’s law:

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(6) Two pendulums ‘a’ and ‘b’ that are oscillating with simple harmonic motion have lengths Lb = 3 La  and, their maximum oscillation amplitudes are Qa and Qb  , respectively (Q : angle theta). It is known that their maximum speeds are such that   Vb-max  = 5 Va-max .  Find the relationship Qb-max  / Qa-max

Solution.-

(7) An oscillating spring coil passes through its equilibrium position every 4 s.   Determine its mass taking into account that the spring is compressed 2 cm when it is placed vertical on the floor and a block of 200 gr is put on top of it.

Solution.-

During each oscillation, the spring goes twice through its equilibrium position, then

(8) The motion of an oscillator is mathematically described by   z(t) = A Sin( 5 t + p ).    Determine its maximum amplitude of oscillation and its initial phase.  It is known that its initial position and speed are 4 and 10, respectively.

Solution .-

(9) Two pendulums ‘a’ and ‘b’ that are oscillating with simple harmonic motion (SHM) have lengths La  and Lb  respectively, so that  Lb = 3 La . Their maximum oscillation amplitudes are  Q= 2.8867 Qa,  (where Q = angle theta). Find the relationship Vb-max / Va-max.

Solution.-