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

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

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**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

**Oscillations of a Loaded Spring**

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

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 K_{1} and K_{2} 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 F_{1} and F_{2 }be the restoring forces acting on the springs.

Total force = F_{1}+ F_{2}

ma = -K_{1} + (-K_{2} )

ma =

Thus, the springs will execute S.H.M

From

T_{P = Periodic time for parallel connection }

If (for identical spring)

**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 S_{1}and S_{2} of force constants k_{1} and k_{2} attached to a mass m and two fixed supports as shown in figure 11.

When the mass pulled downward, then length of the spring S_{1} will be extended by x while that of spring S_{2} will be compressed by x

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

Let F_{1} and F_{2} be the restoring forces exerted by springs S_{1} and S_{2} 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**

**2. SERIES CONNECTION**

Consider two springs S_{1} and S_{2} of force constant K_{1}, and K_{2} connected in series as shown below

Fig. 12

If x is the total displacement

Where x_{1} is extension due to spring of spring constant k_{1} and x_{2} to that of k_{2}

F_{1} =

F_{2} =

F_{1} = F_{2} =_{ } F

The effective spring constant

= periodic time for series connection.

** **

** **For identical spring

**Also since,**