Lab Experiments 146
KamalJeeth Instrumentation and Service Unit
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: [email protected]
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, Newton’s 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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Lab Experiments 147
KamalJeeth Instrumentation and Service Unit
Fdown = mg
Equating the RHS of the above two equations
ks e = mg, or
ks =
୫୥
…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 =
(୫ ା ୑౩ ) ୥
...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 Yaxis 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 ୱ
Y-intercept =
...3
௚ெೞ
௞ೞ
= 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
ୢ²୷
ୢ୶²
=
୩୷
...6
୫
The angular velocity is given by:
Vol-11, No-2, June-2011
Lab Experiments 148
KamalJeeth Instrumentation and Service Unit
ω=ට
௞
...7
௠
Therefore, the time period of the oscillation of the spring is:
୫
T = 2πට ୩
...8
If the dynamic spring has an effective mass Md, then its time period is :
T = 2πට
୫ା୑ౚ
...9
୩
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
=
ସπమ
Y-intercept
=
୑ౚ ସ஠మ
where m:
Md:
e:
T:
kd:
g:
mass of the weight hanging
effective dynamic mass of the spring
extension of the spring
time period of oscillation
spring constant
acceleration due to gravity
...10
୩
୩
= Md x (slope of the line)
…11
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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Lab Experiments 149
KamalJeeth Instrumentation and Service Unit
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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Lab Experiments 150
KamalJeeth Instrumentation and Service Unit
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
(ଶହିଵସ.ହ)ଵ଴షమ
௚
Slope = ௞ = (ଵ଺ିଷ.ଶ)௫ଵ଴షయ = 8.203
ೞ
ଽ.଼
ks =଼.ଶ଴ଷ = 1.194 N/m
The Y-intercept =
௚ெೞ
௞ೞ
The effective mass of the spring in the static condition
Ms=
௒ି௜௡௧௘௥௖௘௣௧ ଵଵ.ଶ௫ଵ଴షమ
ௌ௟௢௣௘
=
଼.ଶ଴ଷ
=0.0136kg = 13.6g
8. A second graph showing the variation T2 with mass is also drawn as shown in Figure6 from which the slope and intercept are calculated as
ଵି଴.଺
଴.ସ
Slope = (ଵ଼.଼ି଻)௫ଵ଴షయ = ଵଵ.଻௫ଵ଴షయ = 34.18 =
ସగ మ
௞೏
Therefore, dynamic spring constant kd =4x3.142/34.18=1.15N/m
Y-intercept = 0.38 =Md slope
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Lab Experiments 151
KamalJeeth Instrumentation and Service Unit
଴.ଷ଼
Md= ଷସ.ଵ଼ =0.0111Kg =11.1g
No of
turns
tied
(n)
Free
Relaxed
hanging length of
length
the
x (10-2)
spring
(m)
(x0)x10-2
Time for 50
oscillations
(sec)
T2
Period
T (sec)
0
14.4
3.0
11.42
30.7
30.9
0.61
3.25
17.3
3.3
14.05
34.1
34.1
0.68
6.50
20.2
3.5
16.67
38.2
37.9
0.76
9.75
23.1
3.7
19.30
41.6
41.8
0.83
13.0
26.4
4.0
22.33
44.7
44.8
0.90
16.25
30.6
4.2
26.36
48.6
48.1
0.97
19.5
32.0
4.6
27.38
50.8
50.5
1.01
Variation of period of oscillation and extension of the spring with mass
0.38
0.47
0.58
0.69
0.80
0.93
1.03
30
Extension (cm)x10-2
25
20
15
10
5
0
0
2
4
6
8
10
12
14
16
18
20
22
Mass (X10-3Kg)
Figure-5: Variation of mass with extension of the spring in the static mode
1.2
1
0.8
T2
0
5
10
15
20
25
30
Mass
nm0
X10-3
(Kg)
Table-1
Extension
e 10-2
(cm)
0.6
0.4
0.2
0
0
5
10
Mass
15
20
25
(X10-3Kg)
Figure-6: Variation of mass with square of the period in the dynamic mode
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Lab Experiments 152
KamalJeeth Instrumentation and Service Unit
Results
The results obtained are tabulated in Table-2.
Table-2
Parameters
Spring constant in static mode (N/m)
Spring constant dynamic mode (N/m)
Average spring constant (N/m)
Effective mass of spring in static mode (g)
Effective mass of spring in dynamic mode (g)
Average effective mass (g)
Experimental results
Experimental
1.19
1.15
1.17
13.6
11.1
12.3
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
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