# PHYSICS FORM FIVE-SIMPLE HARMONIC MOTION-APPLICATION OF S.H.M

PHYSICS FORM FIVE-SIMPLE HARMONIC MOTION-APPLICATION OF S.H.M

UNAWEZA JIPATIA NOTES ZETU KWA KUCHANGIA KIASI KIDOGO KABISA:PIGA SIMU: 07872327719

### PHYSICS FORM FIVE-SIMPLE HARMONIC MOTION-APPLICATION OF S.H.M

#### APPLICATION OF S.H.M

We shall consider the following cases of S.H.M

i)   Oscillation of a Loaded Spring

ii)  Oscillation of a Simple Pendulum

iii  Oscillation of a Liquid in a U – tube

iv)  Oscillation of a Floating Cylinder

v)    Body Dropped in a funnel along earth diameter

vi)    Oscillation of a ball placed in the Neck of Chamber Containing air

PHYSICS FORM FIVE-SIMPLE HARMONIC MOTION-APPLICATION OF S.H.M

If load attached to a spring is pulled a little from its mean position and then released the load will execute S.H.M

We shall consider the following two cases

1.      Vibrations of a Horizontal spring

2.      Vibrations of a Vertical spring

VIBRATIONS OF A HORIZONTAL SPRING

Consider a block of mass M attached to one end of a horizontal spring whi9le the other end of the spring is fixed to a rigid support Fig. 7

The Block is at rest but is free to move along a friction less horizontal surface

In figure 7 is displaced through a small distance x to the right, the spring gets stretched Fig.8

According to Hooke’s law, the spring exerts a restoring force F to the left given by

F  = ………………………………. (i)

Here k is the force constant (spring constant) and is the displacement of mass m from the mean position.

Clearly equation (i) satisfies the condition to produce S.H.M

If the block is released from the displaced position and left , the block will execute S.H.M

The time period (T) and frequency (f) of the vibrations can be obtained from

PHYSICS FORM FIVE-SIMPLE HARMONIC MOTION-APPLICATION OF S.H.M

F = Ma =  =     From  2.      VIBRATION OF A VERTICAL SPRING

Consider unloaded vertical spring of spring constant k Fig. 9

Suppose the spring is loaded with a body of mass m and extended from its original length to an extension ‘e’

By Hookes law

mg = Now suppose the load is displaced down to distance x and then released. The applied force is given by

F = When realized the applied force is opposed by gravity force (weight)

Net result force = F – W

Ma = Ma = Ma = a =  2 =    From  The period of oscillation depends on mass of the loaded body and the spring constant.

In many practical situation springs are connected in series as well as in parallel.

SERIES AND PARALLEL CONNECTION OF SPRINGS

1.  PARALLEL

Consider two  springs of spring constant K1 and K2 arranged in parallel and then both loaded with a body of mass m as shown in fig. 10 Fig. 10

Suppose this body is displaced from its equilibrium position and the extension is x for the system to remain horizontal   Let F1 and F2 be the restoring forces acting on the springs.

Total force = F1+ F2

ma = -K1 + (-K2 )

ma = Thus, the springs will execute S.H.M

From    TP = Periodic time for parallel connection

If (for identical spring) PHYSICS FORM FIVE-SIMPLE HARMONIC MOTION-APPLICATION OF S.H.M

From  OR

For (for identical spring) Alternative arrangment of parallel connection formula Therefore if two identical sprinngs are arranged in parallel , their frequency increases by the factor of ; since is  the frequency for a single spring  for n-identical springs their frequency increases by a factor of .

Consider two springs S1and S2 of force constants   k1 and k2 attached to a mass m and two fixed supports as shown in figure 11.

When the mass pulled downward, then length of the spring S1 will be extended by x while that of spring S2 will be compressed by x

Since the force constants of the two springs are different the restoring force exerted by each spring

Let F1 and F2 be the restoring forces exerted by springs S1 and S2 respectively  Both the restoring forces will be directed upward (opposite to displacement)

The resultant restoring force F

Fig .11  Effectively force constant of the system PHYSICS FORM FIVE-SIMPLE HARMONIC MOTION-APPLICATION OF S.H.M

2.    SERIES CONNECTION

Consider two springs S1 and S2 of force constant  K1, and K2 connected in series as shown below Fig. 12

If x is the total displacement Where x1 is extension due to spring of spring constant k1 and x2 to that of k2

F1 = F2 = F1 = F2  =   F       The effective spring constant  PHYSICS FORM FIVE-SIMPLE HARMONIC MOTION-APPLICATION OF S.H.M       = periodic time for series connection. For identical spring Also since,       