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Experiment-339 F

MEASUREMENT OF SPRING CONSTANT BY STATIC AND DYNAMIC

METHODS

Sarmistha Sahu Dept. of Physics, Maharani Lakshmi Ammanni College for Women, Bengaluru- 560012. INDIA.

E-mail: sarmistha.sahu@gmail.com

Abstract Using a 2 inch diameter silky spring, its effective mass and the spring constant are determined, both in the static and dynamic modes.

Introduction

A spring is an elastic object which stores mechanical energy. Springs are usually made of hardened steel. Small springs can be wound from pre-hardened steel, while larger ones are made from annealed steel and hardened after fabrication. Some non-ferrous metals, such as phosphor bronze and titanium, are also used in making spring [1].

The spring constant, k, of an ideal spring is defined as the force per unit length and is different from one spring to another. Spring constant is represented in Newton/meter (N/m). It can be determined both in static (motionless) as well as dynamic (in motion) conditions. Two different techniques are used for determination of the spring constant. In the static method, Newtons second Law is used for the equilibrium case, and laws of periodic motion are applied for determining the spring constant in the dynamic case.

Static mode

In this mode of determination of spring constant, a weight is added to the spring and its extension is measured. The spring is fixed at one end and a weight is added in equal amounts one by one. The extension produced in the length of the spring is noted from the meter scale fixed. After adding a weight the spring will attain a stationary position after some time. At equilibrium, there are two equal and opposite forces, acting upward and downward. In the static mode the spring constant obtained by this method is denoted by ks; the subscript s indicates that the static method has been employed for determination of the spring constant.

In the equilibrium condition,

Upward force, Fup = Downward force, Fdown

Fup = ks e

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Fdown = mg

Equating the RHS of the above two equations

ks e = mg, or

ks =

1

where m is the mass of the load applied ;

g is the acceleration due to gravity; ks is the spring constant in the static condition ; and e is the extension of the massless spring.

However, the spring has a finite mass, denoted by Ms, which adds to the load, hence m in Equation-1 is replaced by (m+Ms), giving

ks = ( )

...2

Equation-2 represents a straight line in which spring constant ks and acceleration due to gravity g are constants. A graph of e versus (m+MS) is a straight with slope g/ks and Y-axis intercept of (gMS/ks). Hence ks can be determined from the values of extension of the spring with variation in the applied load.

Slope = g/k ...3

Y-intercept = = Ms x (slope of the line) 4

Figure-1: Silky spring used in this experiment

Dynamic mode

If the spring is made to oscillate by pulling the weight applied to it downward, it executes a simple harmonic motion; the equation representing its motion is written as

= ...6 The angular velocity is given by:

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= ...7

Therefore, the time period of the oscillation of the spring is:

T = 2 ...8

If the dynamic spring has an effective mass Md, then its time period is :

T = 2 ...9

A graph of T2 on the y-axis and the mass (m+Md) on the x-axis will result in a straight line with: Slope =

...10

Y-intercept =

= Md x (slope of the line) 11

where m: mass of the weight hanging Md: effective dynamic mass of the spring e: extension of the spring T: time period of oscillation kd: spring constant g: acceleration due to gravity

Apparatus used

Spring mass apparatus consisting of a very thin spring of about 5 cm (2 inch) diameter, fitted on a stand, and a digital stop clock.

Figure-2: Spring mass apparatus

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Figure-1 shows the silky spring used in experiment. It always remains in the compressed position because of its light weight. Each coil of the springs rests on the other hence it is fully compressed. Figure-2 shows the spring mass apparatus used in this experiment. One can tie a few spring coils together which act like a weight hanger. By increasing the number of turns, the mass (m) hanging can be varied.

Experimental procedure

Figure-3: Measurement of spring length

1. The spring length is measured using a scale as shown in Figure-3, and its mass is determined using a digital balance.

Spring length (X) =5.36cm = 0.0536m Spring mass (M) = 46.72g = 46.72x10-3Kg Mass per unit length = (M\X)=46.72x10-3/0.0536= 0.871Kg/m Total number of turns in the spring =72 Hence weight per turn of the spring (m0) = 46.72x10-3/72=0.00065Kg/turn

2. About one third of the coil is separated, held together and fixed on to a stand as shown in the Figure-4. The rest of the spring coils hang downward because of their own weight as shown in Figure-4.

3. The length of the spring coils hanging in air is measured using a scale and extension of the spring is calculated as

x = 14.4cm

The spring is now compressed by pushing it back and its compressed length or the relaxed length xo is found as

x0 = 3.3cm

The extension e = x-xo = 14.4-3.3=3.1cm = 3.1x10-2m

4. Now five turns (n=5) of the hanging coil are tied to form the mass as shown in Figure-2. This tied mass forms the weight. The weight is pulled down slightly and released which makes it to oscillate. The time period for 50 oscillations is counted using a digital stop clock and the period of the simple harmonic oscillation is determined. The readings are tabulated in Table-1.

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Time for 50 oscillations = 30.7 s Hence period, T = 30.7/50 = 0.614 s.

Figure-4: Static position of the spiral spring

5. To repeat the trial, five more turns of the coil tied at the top are released and mass is increased by tying 10 turns together, keeping the total number of coil turns hanging in air the same.

6. Extension of the spring is calculated again and its period of oscillation is determined and recorded in Table-1.

7. A graph is drawn taking mass of the spring along X-axis and its extension on the Y-axis as shown in Figure-5. From the graph the slope and Y-intercepts are noted as

Slope = = (.)!"(#$.)%!"& = 8.203

ks =,.-

-.!$ = 1.194 N/m

The Y-intercept =

The effective mass of the spring in the static condition

Ms= 1234567584

9:;85 =.%!"

-.!$ =0.0136kg = 13.6g

8. A second graph showing the variation T2 with mass is also drawn as shown in Figure-6 from which the slope and intercept are calculated as

Slope = !.#(-.-

Therefore, dynamic spring constant kd =4x3.142/34.18=1.15N/m

Y-intercept = 0.38 =Md slope

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Md= !.$-

$.- =0.0111Kg =11.1g

Table-1 No of turns tied (n)

Mass nm0

X10-3 (Kg)

Free hanging length x (10-2)

(m)

Relaxed length of

the spring

(x0)x10-2

Extension e 10-2 (cm)

Time for 50 oscillations

(sec)

Period T (sec)

T2

0 0 14.4 3.0 11.42 30.7 30.9 0.61 0.38 5 3.25 17.3 3.3 14.05 34.1 34.1 0.68 0.47

10 6.50 20.2 3.5 16.67 38.2 37.9 0.76 0.58 15 9.75 23.1 3.7 19.30 41.6 41.8 0.83 0.69 20 13.0 26.4 4.0 22.33 44.7 44.8 0.90 0.80 25 16.25 30.6 4.2 26.36 48.6 48.1 0.97 0.93 30 19.5 32.0 4.6 27.38 50.8 50.5 1.01 1.03

Variation of period of oscillation and extension of the spring with mass

Figure-5: Variation of mass with extension of the spring in the static mode

Figure-6: Variation of mass with square of the period in the dynamic mode

0

5

10

15

20

25

30

0 2 4 6 8 10 12 14 16 18 20 22

Exte

nsio

n (cm

)x10-

2

Mass (X10-3Kg)

0

0.2

0.4

0.6

0.8

1

1.2

0 5 10 15 20 25

T2

Mass (X10-3Kg)

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Results

The results obtained are tabulated in Table-2.

Table-2 Parameters Experimental

Spring constant in static mode (N/m) 1.19 Spring constant dynamic mode (N/m) 1.15 Average spring constant (N/m) 1.17 Effective mass of spring in static mode (g) 13.6 Effective mass of spring in dynamic mode (g) 11.1 Average effective mass (g) 12.3

Experimental results

Discussion

A new experiment is presented here using which basic properties of a spring are determined. As no slotted weight is used, the weight of the spring itself acts like a hanging weight. The spring used is very thin, or silky, hence all the turns (layers) of it is in contact with each other, which makes it a compressed spring. The spring constant and effective mass obtained by the static and the dynamic methods are nearly the same indicating that both the methods give reasonably correct result.

References

[1] http://en.wikipedia.org/wiki/Spring_%28device%29