This document describes an experiment to determine the spring constant and acceleration due to gravity using a spring. Two methods are used: static and dynamic. In the static method, the spring is loaded with different weights and the extension is measured. In the dynamic method, the time period of oscillations is measured for different loads. The spring constant is calculated from the slope of graphs of extension vs load and period squared vs load. The average spring constant is found to be 11.88 N/m from the static method and 12.74 N/m from the dynamic method. Gravity is calculated from the dynamic method and found to be 9.752 m/s^2 with a percentage error of 0.49% compared to the theoretical value

1. UNIVERSITY OF DELHI

2. Aim
To study the motion of the spring
and calculate
(a) spring constant (b) g

3. APPARATUS
A spring of high elasticity, a hanger with light
aluminium pointer,weight hanger with slotted
weights, stop clock , etc.

4. Theory
The helical spring is the most commonly
used mechanical spring in which a wire is
wrapped in a coil that resemble a screw
thread. The helical spring us suspended
vertically from a rigid support. The pointer is
attached horizontally to the free end of the
spring. A meter scale is kept vertically in
such a way that the tip of the pointer is over
the divisions of the scale, but doesn’t touch
the scale.

5. Spring constant can be found by two methods:
(A) static method. (B) dynamic method
STATIC METHOD
When a load F is suspended from lower end of a spring hanging
from a rigid support , it increases its length by amount x. In
equilibrium during the downward movements it executes simple
harmonic motion. During downward movement the body will
feel a restoring force acts along the initial position of the body
and Hook’s law will act i.e. Restoring force is directly
proportional to ‘x’ and opposite in direction.
F = - Kx ( K is spring constant )
K = Mg ( slope = x )
x M

6. Dynamic Method
When a loaded spring vibrates then its oscillation time period is
correlated with its spring constant.
The equation of Simple Harmonic is
K = 4π2( M2 – M1 )
( T2
2 – T2
1 )
Where ( T2 – T1) is slope of the graph
( M2 - M1)
Gravity g is calculated by,
g = 4π2x
T2

7. PROCEDURE
DYNAMIC METHOD
1. Place the load in the hook of the spring. Pull the load vertically to
give a small displacement and then release it. The spring will do
simple harmonic motion.
2. Note down the time for 10 oscillations of the spring. Repeat the
observation for the same load and find the mean time period T.
3. Increase the load and each time note down the time for 10 vibrations
and find mean time period T.
4. Plot the graph between the load M(kg) along x axis and T² (sec)
along y axis.

8. STATIC METHOD
1. Arrange the apparatus as shown in the figure. Make sure that the aluminium
pointer is close to the mirror on which the scale is etched but it does not
touch the mirror
2. Read the position of the pointer on the scale. Avoid the parallax error. Note
the reading on the scale. The apparatus should be in equilibrium
3. Increase the load on weight banger and again note the readings for different
values of load when the pointer comes at rest and one by one add load.
4. Now remove one slotted weight and note the reading when pointer comes
at rest & note in unloading column and repeat this step again.
5. After finding k, by the formula we can find g also.
PROCEDURE

9. OBSERVATION TABLE
Table for finding spring constant by STATIC METHOD
S.no Load M (kg) Reading of pointer 𝒙𝟐 (m) Mean
Readi
ng(m)
Extension
x=𝒙𝟐 − 𝒙𝟏 (m)
Spring Const.
K=
𝒎𝒈
𝒙
N𝒎−𝟏
Loading Unloading
1. 0.15 0.123 0.123 0.123 0.123 11.98
2. 0.2 0.164 0.166 0.165 0.165 11.91
3. 0.25 0.207 0.207 0.207 0.207 11.85
4. 0.3 0.250 0.250 0.250 0.250 11.76
5. 0.35 0.278 0.291 0.288 0.288 11.88
Mean value of K =
11.98+11.91+11.85+11.76+11.88
5
= 11.88 N𝑚−1

10. CALCULATION
STATIC METHOD: k=
𝑴𝒈
𝒙
N𝑚−1
𝑘1=
0.15×9.8
0,123
= 11.98 N𝑚−1
𝑘2 =
0.2×9.8
0.165
= 11.91 N𝑚−1
𝑘3 =
0.25×9.8
0.207
= 11.85 N𝑚−1
𝑘3 =
0.3×9.8
0.250
= 11.76 N𝑚−1
𝑘5 =
0.35×9.8
0.330
= 11.88 N𝑚−1

11. OBSERVATION TABLE
Table for finding spring constant by DYANIMC METHOD
S.No. Load M (kg) Extension
spring x (m)
Time for 10 oscillation (s) Period
=
𝒕
𝟏𝟎
(s)
𝑻𝟐
in 𝒔𝟐
G =
𝟒𝝅𝟐𝒙
𝑻𝟐 m𝒔−𝟏
Trial 1(s) Trial 2 (s) Mean (S)
1. 0.15 0.123 7.25 7.09 7.12 0.712 0.52 9.34
2. 0.2 0.165 8.16 8.22 8.18 0.818 0.68 9.57
3. 0.25 0.207 9.10 9.16 9.09 0.909 0.83 9.84
4. 0.3 0.250 10.00 9.9 9.93 0.993 0.99 9.96
5. 0.55 0.288 10.90 10.44 10.67 1.067 1.14 10.05
Mean value of K =
12.32+13.15+12.32+13.15
4
= 12.74 N𝑚−1
Mean value of g =
9.34+9.57+9.84+9.84+9.94+10.05
5
= 9.752

12. DYNAMIC METHOD K = 4π2(M2 – M1)/(T2
2 – T2
1)
𝑘1=
4×(3.14)2×0.05
0.68−0.52
= 12.32 N𝑚−1
𝑘1=
4×(3.14)2×0.05
0.83−0.68
= 13.15 N𝑚−1
𝑘1=
4×(3.14)2×0.05
0.99−0.83
= 12.32 N𝑚−1
𝑘1=
4×(3.14)2×0.05
1.14−0.99
= 13.15 N𝑚−1
Mean value of k =
12.32+13.15+12.32+13.15
4
= 12.74 N𝑚−1

13. Calculations
Pointer reading for no load, x = 0 m
STATIC METHOD ( K = Mg/x ) Nm-1
DYNAMIC METHOD ( K = 4π2(M2 – M1)/(T2
2 – T2
1)
Average value of spring constant, K = 12.74 Nm-1

14. RESULT
The spring constant K of the given spring,
Static method,
By calculation, K = 11.88 Nm-1
By graph, K = 11.88 Nm-1
Dynamic method,
By calculation, K = 12.74 Nm-1
By graph, K = 12.72 Nm-1
Acceleration due to gravity by Dynamic method is,
g = 9.75 ms-2

15. Percentage Error
Theoretically, the value of g is,
g = 9.8 ms -2
Percentage Error = |Theoretical value – observed value| × 100
Theoretical value
= | 9.8 – 9.752 | × 100
9.8
= 0.49 %

16. Precautions:
1. Loading and unloading of weight must be done carefully.
2. To avoid parallax, reading should be noted when the tip
of the pointer comes at rest.
3. Loading should not be done beyond elastic limit of the
spring used.
Sources of Error:
1. Parallax error may be present.
2. There may be estimation error present while measuring
the readings of stopwatch.
3. The support may not be rigid.