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# An Experiment on the Vertical Oscillation of a Spiral Spring

## Objective:

• 1. To show how the time of vertical oscillation depends on the load
• 2. To determine the spring constant
• 3. To determine the effective mass of the spring

## Introduction:

Let the force constant of the spring, i.e. the force to produce unit extension, be k. Consider the spring in equilibrium under a load m. If, now, the spring is pulled down a further down a further distance x, the extra restoring force called into play is, by Hooke’s Law, equal to kx. When the spring is released, the equation of motion is therefore

kx= -mx

Where x represent the acceleration towards the equilibrium position

?= –

Hence the motion is simple harmonic and the periodic time T is

pp

Thus, the graph of T2 against the straight line.

At first sight it would appear that this straight line ought to go through the origin, whereas the actual line obtained in the experiment does not. This is because the effective mass (mo) of the spring has been neglected and the above equation ought to be written

## Material:

Spiral spring, stands and clamps, slotted masses and hanger, stop watch.

## Method:

• 1. The spring is suspended from a firm support and is loaded, by means of slotted masses is attached to the free lower end, until it is possible to measure the time taken by the suspended load to execute 20 complete vertical oscillations.

• 2. The timing for two times is repeated to get the mean time of the oscillations.
• 3. The load of 50g is increased and repeated for 20 oscillations.
• 4. Continue until the times for five different loads have been taken.
• 5. The reading is recorded and the result is tabulated into the table below.

Plot a graph with values of T?/s? as ordinates against the corresponding values of m/kg

## From which it follows

Slope from which l may be calculated

Since when T=0, the magnitude of m0 is equal to be the negative intercept OC on the load axis of the graph

## Discussion:

The definition of spring constant is characteristic of a spring which is defined as the ratio of the force affecting the spring to the displacement caused by it. The value of spring constant that we can get from the calculation above is, k=24. The definition of effective mass is a quantity that is used to simplify band structures by constructing an analogy to the behaviour of a free particle with that mass. The value of effective mass can be determined from plotting graph where x-intercept is equal to the value of effective mass. The value of effective mass of the spring that we get from the graph is kg.

## Conclusion:

The value of spring constant, k=24. The value of effective mass of the spring, kg.

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