Hooke’s Law
Rentech
Hooke’s Law Experimental Set
(Spring Constant and Oscillation Period Measurements)
Student’s Guide
1
Laboratory Manual and Workbook
(2016)
Hooke’s Law
Rentech
List of Equipment
1.
Hooke’s Law Experimental Set
1.1.
Springs with Different Spring Constants
1.2.
Different Masses (m=50g, m=100g, m=200g)
1.3.
Holders
2.
Chronometer
3.
Height Finder
4.
Student and Teacher Manuals
2
Hooke’s Law
Rentech
Table of Contents
Section
Page
1.
Purpose
......................................................................
4
2.
Hooke’s Law
......................................................................
4
....................................................
7
.................................................
9
................................................
10
............................................................
11
3.
4.
2.1.
Simple Harmonic Motion
2.2.
Springs Connected in Series
2.3.
Springs Connected in Parallel
Experimental Procedures
EXPEIMENT-1:
Hooke’s Law (Determination of Spring Constant)
EXPEIMENT-2:
Springs Connected in Series (Determination of Spring Constant)
EXPEIMENT-3:
EXPERIMENT-4:
Laboratory Report
......................
11
..........
13
Springs Connected in Parallel (Determination of Spring Constant)
......
15
Simple Harmonic Motion (Determination of Period of Oscillation)
........
17
..............................................................
19
3
Hooke’s Law
Rentech
1.
F  kx
Purpose
(External Force on Spring)
(1)
The purpose of this experiment is to;
Here, the proportionality constant
1. To verify Hooke’s Law and find the spring constant
using Hooke’s Law,
is the spring
constant for the spring and is a measure of the
2. Connect the springs in series and parallel,
stiffness of that particular spring. Spring constant ( ) is
3. Study the simple harmonic motion of a mass attached
always positive and the unit is force per length (
).
to the end of a spring,
4. Learn the relationship between period and frequency,
Figure-(1a) shows a spring in unstretched (normal)
5.
position. To stretch this spring a distance , one has to
Determine the spring constant by using the period of
oscillation for a mass on the end of a spring.
exert an external force on the free end of the spring
with a magnitude at least equal to
2.
Hooke’s Law
1b). The greater the value of
(Figure-
, the greater the force
needed to stretch a spring a given distance. This
means that the stiffer the spring, the greater the spring
constant .
Moreover, the spring itself exerts a force in the
opposite direction;
Fx  kx
(Hooke’s Law)
(2)
This force exerted by spring is called “restoring
force” because the spring exerts its force in the
direction opposite the displacement (that is, the minus
sign) as seen in the Figure-(1b).
Equation-(2) is known as Hooke’s Law and is
accurate for springs as long as “ ” is not too great and
no permanent deformation occurs. Since the spring
applies a force in the opposite direction to the
movement of the spring at a distance “ ” from its
equilibrium position, the spring wants to come back to
Figure-1: Spring in normal (unstretched) position (a). Spring
-
its normal length (reference point) due to the restoring
direction from its normal position (reference point). The
force. Note that the equilibrium position is chosen at
is stretched by a force ( ) applied along the positive
spring pulls back with a force
spring is compressed (
with the force
where
(b). If the
and the minus sign indicates that the restoring
), the spring is pushed back
force is always in the direction opposite to the
where,
) (c).
displacement. For example, if you choose the positive
direction to the right,
is positive when the spring is
When a force is applied to a spring, it is stretched or
stretched, but the direction of the restoring force is to
compressed. To hold a spring either stretched or
the
compressed an amount
compressed,
from its normal length
left
(negative
direction).
4
the
spring
is
is negative (to the left) but the force
acts toward the right (Figure-1c).
(reference point) requires a force ( );
If
Hooke’s Law
Rentech
Therefore, in the mass-spring system, the forces on
the mass ( ) attached to the spring will be:
1. The gravitational force (
) acting on the object
due to the gravity and,
2. The restoring force (
) applied by the spring
in the direction opposite the displacement.
In the mass-spring system, the restoring force
acting on the mass to the upwards by the
Figure-2: When a mass ( ) is attached to a spring which is
spring
at rest, the mass changes its position at a distance ( ) and
gravitational force (
stops at the equilibrium position. When the system is in
a spring, the spring will stretch to the point where the
the vertical equilibrium position, the hanging mass
(
as
much
as
the
magnitude
of
the
). If a mass is attached to
two forces in the opposite direction on the mass are
) is balanced with the restoring force of the spring
(
is
equal to each other. This point will give the
).
equilibrium point. This means simply that the net
force on the object ( ) is zero, so it remains at rest.
According to Hooke’s Law, in order to stretch a spring
at a distance
As we now know, the weight attached to the spring is
from its equilibrium length, we need to a
force defined as
the external force that stretches the spring. The
. The ratio of the applied force
magnitude of the restoring force is equal to the
to the amount of stretch is constant. Therefore, spring
weight (
constant or ratio constant “ ” can be found by
) at the equilibrium point. If the
forces is in equilibrium, the force down (the weight) is
applying an external force to the spring and
equal and opposite to the force acting upward (the
measuring the amount of stretch of the spring.
restoring force of the spring);
As shown in Figure-(2), an external force (
) to
stretch a spring is created by attaching a mass ( ) to
mg  kx
this spring in the vertical position. Weight ( ) is the
(EXPERIMENTAL)
(4)
external force of gravity on the object ( ) and so, the
force stretching the spring will be the gravitational force
(
) acting on the mass. Gravitational force is
Equation-(4) gives a general expression between the
downward and the force applied by the spring to the
force acting on the spring (weight) and the stretching of
mass is upward in the vertical position. The spring can
the spring (that is, the change in the length,
). Unless
any additional external force is applied, the system
stretch until these two forces are equal to each other.
consisting of spring and the mass will be in the
The position of the mass ( ) at this point is called the
equilibrium position.
equilibrium position.
As the limit of flexibility is not exceeded, the force
The magnitude of the gravitational force ( ) on the
acting on the spring is directly proportional to the
object with a mass ( ) pulls the mass to the
amount of stretch of the spring ( ). For instance, if an
downward;
object with a mass of
is attached to a spring
and the spring stretches at a distance of
F  W  mg
(Gravitational Force)
same spring will stretch at a distance of
(3)
an object with a mass of
spring.
2
Here, g is the gravitational acceleration (g=9.80 m/s ).
5
, the
as
is attached to that
Hooke’s Law
Rentech
To verify Hooke’s Law for a spring, we must show
The spring constant ( ) can be experimentally determined
that the force exerted by spring and the stretch of the
by applying different external forces supplied by the hanging
spring ( ) are proportional to each other and that the
masses to stretch the spring different distances. When the
proportionality constant is spring constant ( ). In the
force on the spring (the weight of the mass on the end of
experiment, the external force (
) is provided by
spring) is plotted versus distance ( ) the spring is stretched,
attaching a mass ( ) to the free end of the spring.
the SLOPE of the straight line will be equal to spring
When a mass ( ) is attached to the end of spring, the
constant ( ). This is the first way to determine the spring
spring will stretch until it reaches the equilibrium
constant ( ) if the slope is known.
position where the two forces of gravitational force
(
) and the restoring force (
) are
equal but in opposite directions. At this equilibrium
position, the spring and mass system will stay at the
equilibrium point as long as no additional external
forces are exerted on it.
The force (
) on the spring is the weight of the
object (mass) which can be found by multiplying the
mass ( ) by the acceleration due to gravity (g). If we
plot the force ( ) produced by different masses (
)
as a function of the stretch distance ( ) from
EQUILIBRIUM POINT, the data should be a straight
line and the slope of this line will be equal to the spring
constant ( ). Comparing Equation-(4) with the form of
the equation of a straight line, we can see that the
graph is defined simply as the linear function;
y  ax
(EXPERIMENTAL)
(5)
Here;
y:
The force ( ) applied to the spring,
a:
Spring constant ( ) as a slope,
x:
The change in the spring length ( ).
The amount of stretch of the spring ( ) can be
measured by observing the position of the spring
according to a convenient reference point near the
bottom of the spring before and after applying an
external force ( ) to the spring. The SLOPE of the
Force versus Spring Stretch (displacement) graph
is known as the spring constant ( ).
6
Hooke’s Law
Rentech
2.1.
When the mass
Simple Harmonic Motion
is released at a distance
from its
vertical equilibrium point, the restoring force of the
spring accelerates back to the equilibrium point and
thus the mass starts to make a simple harmonic
motion.
An oscillating object with mass
undergoes simple
harmonic motion if the restoring force is proportional
to the displacement as “
”. The maximum
displacement from equilibrium is called amplitude. The
Figure-3: The mass
period ( ) is the time required for one complete cycle
which is stretched at a distance
from its equilibrium position and then released makes a
(back and forth). Another term in the harmonic motion
simple harmonic motion between A and B.
is frequency ( ) and it is the number of cycles per
second. The relation between the period ( ) and the
frequency ( ) is given by:
If the string is stretched beyond its equilibrium point by
pulling it down and then releasing it, the spring exerts a
T
force on the mass ( ) that pushes it toward the
1
f
(6)
equilibrium position (point) and the mass will then
continue to move upward. Above the equilibrium point,
The period ( ) of oscillation for a mass ( ) on the end
the motion of mass will slow because the net force
of a uniform spring is given by;
acting on the mass is now downward. As the mass
reaches the equilibrium position, the net force on it
T  2
decreases to zero, but its velocity at this point is a
maximum. It then repeats the motion, moving upward
and downward symmetrically between
m
k
(EXPERIMENTAL)
(7)
and
. So, the mass will oscillate around the equilibrium
It is noted that the frequency and period do not depend
position.
on the amplitude. If the Equation-(7) is rewritten for a
mass ( ) attached to the end of a simple spring with
When the object oscillates upward and downward
the spring constant ( ), we get;
over the same path, each oscillation taking the same
amount of time, then the motion is periodic. The
m
k
(8)
k 2
T
4 2
(9)
T 2  4 2
simplest form of periodic motion can be represented
by an object oscillating on the end of a uniform spring.
The motion which repeats itself at a certain time around
m
a fixed point under the effect of restoring force is
called simple harmonic motion.
The simple harmonic motion of a mass attached to a
spring is shown in Figure-(3). While the mass
attached to the spring is stretched at an amount
downward from its vertical equilibrium position
(
) and then released, it starts to make a
simple harmonic motion between A and B.
7
Hooke’s Law
Rentech
Equation-(9) is similar to the standard equation of a
straight line;
y  ax
(10)
Here;
y:
The mass (
a:
Slope (
x:
Square of Period (
) attached to the spring,
),
).
As can be seen from the Equation-(10),
graph
is linear and the SLOPE ( ) of the graph is related to
the spring constant by:
a
k
4 2
(EXPERIMENTAL)
(11)
In the mass-spring system, if we pull the mass ( )
downward from its equilibrium point and then release
it, the mass ( ) starts to make harmonic motion
along the vertical direction. If the period ( ) of the
harmonic motion for different masses is measured, the
spring constant ( ) can be found experimentally from
the
graph.
The spring constant can be determined by measuring the
period of oscillation for different hanging masses on the
end of a spring. In the experiment, you will measure the
period of oscillation for various hanging masses ( ) on
the end of a spring, then plot
versus
graph and use
Equation-(11) to determine the spring constant ( )
experimentally. This is the second way to determine the
experimental spring constant ( ).
8
Hooke’s Law
Rentech
2.2.
If the two springs with spring constants
Springs Connected in Series
and
are
connected end to end, this connection is called series
connection. In this case, the force ( ) acting on each
spring connected in series is the same (Figure-4b).
The total extension or compression of the spring
system depends on the extension (or compression) of
the springs individually.
If the force ( ) is acting on the spring system
connected in series, the same force act on the each
spring. If the forces acting on each spring are given as
and
, the relation can be written as;
F  F1  F2
(a)
(Series
(12)
Connection)
If the amount of stretch of the system is
amount of stretch of each spring is
and
and the
, we can
write;
x  x1  x2
(Series
Connection)
(13)
Therefore, the spring constant ( ) of the spring system
in series is given as;
(b)
F F F
 
k k1 k 2
(14)
1 1
1
 
k k1 k 2
(15)
Figure-4: Series connection of the two springs with spring
constants
and
(a). When the mass ( ) is attached to
k
the spring system in series, the tension forces of the two
springs are equal to each other. The object ( ) moves
downward
at
a
distance
of
from
k1 k 2
k1  k 2
(EXPERIMENTAL)
(16)
the
unstretched position (b).
As a result, if the two springs with spring constants
By connecting the springs in series or parallel,
and
mechanical spring systems with different spring
constant ( ) of the system is varied according to the
constants
Equation-(16).
can be constructed. As shown in the
Figure-(4a), suppose that you have two different
springs each with spring constant
an object of mass
and
from which
is suspended.
9
are connected in series, the equivalent spring
Hooke’s Law
Rentech
2.3.
If the force acting on the spring system connected in
Springs Connected in Parallel
parallel is “ ” and the force acting on each spring is
given as
and
, then the sum of the forces on the
springs is equal to the weight of the object:
F  F1  F2
(Parallel
Connection)
(17)
As can be seen from Figure-(5b), in the spring system
connected in parallel, two springs stretch at a same
amount of :
x  x1  x2
(a)
(Parallel
Connection)
(18)
Because of this reason, the sum of force acting on
each spring will be equal to the force acting on the
system, and the amount of stretch or compression of
the springs will be equal to each other. By using
Equation-(17) and Equation-(18), the total spring
constant ( ) of the system is given by;
kx  k1 x  k 2 x
(19)
(b)
k  k1  k 2
Figure-5: Parallel connection of the two springs with spring
constants
and
(a). In the spring system connected in
parallel, two springs extend at an equal amount of
mass m moves downward at the same amount of
(EXPERIMENTAL)
(20)
and the
(b).
Therefore, two springs with spring constants
and
are connected in parallel, the spring constant of the
If the two springs with spring constant
and
are
system is varied as
connected side by side, this connection is called
parallel
connection
(Figure-5a). When
is,
) is applied to the system and the system
stretches at a distance
. As seen in Equation-
(20), the equivalent spring constant of the system is
(that
bigger than the spring constants of each spring.
from the unstretched
position, the amount of stretch (displacement) at
each spring will be .
10
Hooke’s Law
Rentech
3.
Experimental Procedures
Experimental Note
When the mass ( ) is attached to the free end of the
spring, the spring will stretch until it reaches the point
EXPERIMENT-1:
Hooke’s Law
where the two forces on the spring and mass system
Determination of Spring Constant
(the force exerted by the spring and force of gravity) will
be equal but pointing in opposite directions (that is,
). This point where the two forces
balance each other out is known as the vertical
equilibrium position (equilibrium point).
2.
Before hanging mass ( ) to the spring, a
reference point (
) is chosen to the free end
of the spring when the spring is in normal
(unstretched) position. (Figure-6a).
3.
When mass
is attached to the spring, the
amount of stretch (displacement) of the spring
according to the reference point is measured. For
this process;
3.1. Mass
(a)
(
) is attached to the
spring.
(
3.2. When the mass
)
is attached to the spring, the
mass-spring system is waited to come to its
equilibrium position.
Experimental Note
The weight (
) of the object hanged on the
spring is the force that stretches the spring. Before
measuring the stretch of the spring, mass-spring
system is waited to come to its vertical equilibrium
position.
(b)
3.3. In the equilibrium position, the amount of
Figure-6: Experimental set-up to determine the spring
stretch ( ) of the spring from the vertical
constant (a) and measurement of displacement ( ) from the
equilibrium position is determined by the height
vertical equilibrium position.
finder.
3.4. Note that the displacement ( ) of the reference
1.
The spring that will be used in the experiment is
point on the spring is also the amount of stretch
hanged to a fixed point (diameter of spring wire,
of the spring (Figure-6b).
).
3.5. The amount of stretch ( ) in the spring is recorded
in Table-(1).
11
Hooke’s Law
Rentech
4.
6.3. The best line is drawn that most fits the data
In a similar way, hanging masses with
and
points and the equation of this line is shown on
, the new amount of stretch ( ) on
the graph.
the spring by each mass is measured.
6.4. It is important that the graph should be linear and
the best line should pass through the origin of the
Experimental Note
graph.
A spring will return to its rest (original) length when
the force (weight) is removed. If too much force is
applied,
the
spring
will
become
6.5. Use your graph to verify Hooke’s Law. The
permanently
“EXPERIMENTAL” spring constant (
deformed such that the original length is altered.
) will
be the SLOPE of the best line (straight line).
6.6. Record the value of the spring constant (
5.
),
By the each mass ( ) attached to the spring, the
based on Hooke’s Law, as determined from the
force ( ) applied to the spring is determined.
slope of the best-fit straight line in Table-(1).
5.1. If a mass
7.
is attached to the free end of a
spring, it exerts a force
on the spring and
The experimental setup is reconstructed by using
the same spring.
the length of the spring is changed by .
7.1. Mass
5.2. Determine the applied force
7.2. The amount of stretch ( ) in the spring is
weight of the mass ( ). Remember that the
is given by
) is attached to the spring.
which the
mass exerts on the spring by calculating the
weight
(
measured.
g and “g” is the
acceleration due to gravity,
7.3. According to Hooke’s Law, the EXPECTED
.
spring constant ( ) value is calculated.
8.
Experimental Note
The spring constant found from the slope
The applied force ( ) by the weight ( ) of the each
“experimentally”
mass
“expected”
is found by using the equation:
F  mg
Here, g is the gravitational acceleration (g=
value
is
compared
and
the
with
“difference”
the
is
determined:
).
Difference (%) 
Expected  Experiment al
100
Expected
5.3. Record these forces in the data table.
Difference (%) 
k  k
x100
k
6. Now, the force ( ) applied by the each mass ( )
to the spring as a function of the displacement ( )
9.
will be plotted:
The same procedure is repeated for different
springs with different spring constants.
6.1. Plot the applied force ( ) to the spring as a
10. Is the amount of stretch of the springs
function of its displacement ( ) .
6.2. On the graph, the amount of stretch
proportional to the hanging mass? Explain briefly.
is
Unless exceeding the flexibility limit, the weight (the
represented on the horizontal axis (x-axis) and
force acting on the spring) is directly proportional to
the force
the amount of stretch of the spring.
supplied by the mass is represented
on the vertical axis (y-axis).
12
Hooke’s Law
Rentech
EXPERIMENT-2:
Springs Connected in Series
4.
Determination of Spring Constant
In a similar way, hanging masses with
and
, the new amount of stretch ( ) on
the spring system by each mass is measured and
recorded in Table-(6).
5.
By calculating the weight of the each mass
attached to the system connected in series, the
force (
) applied to the spring system is
determined;
6.
Figure-7: Experimental setup of the determination of the
.....
.....
.....
.....
The force ( ) applied to the spring system as a
spring is plotted.
6.1. On the graph, the amount of stretch
Two springs with different spring constants are
is
represented on the horizontal axis ( -axis) and
connected in series (diameter of spring wire
and
.....
function of the change of the length ( ) of the
spring constant for the spring system connected in series.
1.
.....
the applied force
).
is represented on the vertical
axis ( -axis).
2.
Before hanging mass
to the spring, a
6.2. The best line that most fits the data points is
reference point is chosen at the free end of the
drawn and the equation of this line is shown on
two-spring system (Figure-7).
the graph.
3.
Mass
is attached to the free end of the
6.3. It is important that the best line should pass
spring system.
through the origin of the
(
)
6.4. The
slope
of
the
graph.
graph
will
EXPERIMENTAL spring constant (
3.1. When mass
is attached to the free end of the
give
the
) value of
the spring system connected in series.
spring, the mass-spring system is waited to come
its equilibrium position.
6.5. EXPERIMENTAL spring constant (in other
words, equivalent spring constant of the system)
3.2. When the spring system comes to its equilibrium
is recorded in Table-(6).
position, the amount of stretch ( ) of the system
according to the reference point (
) is
measured.
3.3. Measured amount of stretch ( ) is recorded in
Table-(6).
13
Hooke’s Law
Rentech
Figure-8: In the mass-spring system, the determination of
Figure-9: For the second spring, the determination of the
spring constant for the first spring.
spring constant.
7.
8.
Now, these two springs connected in series are
setup, the spring constant (
separated from each other.
spring (
7.1. One of the two springs is chosen (
9.
) is determined. (Figure-9).
Each spring constant value is recorded in Table(7) and (8) as
(Figure-8).
and
.
10. By using these spring constants (
7.2. Before hanging mass ( ) to the spring, a
reference point (
) for the second
)
and it is located in the experimental setup
and
), the
EXPECTED spring constant ( ) value of the
) is chosen at the free end
of the spring.
7.3. Mass
In the same way, by using the same experimental
spring system connected in series is calculated:
(
) is attached to the free
k
end of the single spring.
7.4. When the mass
k1 k 2
k1  k 2
is attached to the spring,
mass-spring system is waited to come to its
This equation will give the expected equivalent spring
equilibrium position.
constant of the spring system connected in series.
7.5. In the vertical equilibrium position; the spring is
stretched at a distance
11. For the spring system, the calculated (expected)
from the reference
spring
point.
constant
value
is
compared
with
experimental spring constant value found from
7.6. Now, the amount of stretch (
) of the
the slope and the difference is determined.
spring is determined by the height finder.
7.7. The force ( ) applied to the spring by the hanging
mass ( ) is calculated by:
F  mg
7.8. Using the amount of stretch (
applied force ( ), the spring constant (
) and the
) of the
single spring is determined by:
F  kx  mg
14
Hooke’s Law
Rentech
EXPERIMENT-3:
Springs Connected in Parallel
4.
Similarly, hanging masses with
Determination of Spring Constant
and
, the new amount of stretch ( ) on the
spring system by the each mass is measured and
recorded in Table-(10).
5.
By calculating the weight of the each mass
attached to the system connected in parallel, the
applied force (
) to the spring system is
determined;
Figure-10: Experimental setup of the determination of the
6.
spring constant for the spring system connected in
.....
.....
.....
.....
.....
.....
The force ( ) applied to the spring system as a
function of the change of the length ( ) is plotted.
parallel.
6.1. On the graph, the amount of stretch ( ) is
1.
Two springs with different spring constants are
represented on the horizontal axis ( -axis) and the
connected in parallel (diameter of spring wire
applied force
and
2.
is represented on the vertical axis
( -axis).
).
6.2. The best line that most fits the data points is
Before hanging mass ( ) to the spring system, a
drawn and the equation of this line is shown on
reference point is chosen at the free end of the
the graph.
two-spring system (Figure-10).
6.3. Note that the best line should pass through the
3.
Mass
is attached to the free end of the
origin of the
graph.
spring system.
6.4. The
slope
of
the
graph
will
is attached to the spring, the
EXPERIMENTAL spring constant (
mass-spring system is waited to come to its
spring system connected in parallel.
3.1. When the mass
give
the
) of the
vertical equilibrium position.
6.5. EXPERIMENTAL
spring
constant
value
(equivalent spring constant of the system ) is
3.2. In the spring system connected in parallel, these
recorded in Table-(10).
two springs will stretch at a same amount of .
3.3. When the spring system comes to its equilibrium
position, the amount of stretch ( ) of the
system according to the reference point is
measured.
3.4. Measured amount of stretch (displacement) is
recorded in Table-(10).
15
Hooke’s Law
Rentech
Figure-11: The determination of spring constant (
) for the
Figure-12: The determination of the spring constant (
first spring (1.spring).
8.
7.
) for
the second spring (2.spring).
Now, these two springs connected in parallel are
In the same way, by using the same experimental
setup, the spring constant (
separated from each other.
second spring (
) value for the
) is determined
(Figure-12).
7.1. One of the two springs is chosen (
)
and it is located in the experimental setup
9.
(Figrue-11).
Each spring constant value is recorded in Table(11) and (12) as
7.2. Before hanging mass
10. By using these spring constants (
single spring.
and
), the
EXPECTED spring constant ( ) value of the
spring system connected in parallel is calculated:
is attached to the free end of the
the single spring.
7.4. When the mass
.
to the spring, a
reference point is chosen at the free end of the
7.3. Mass
and
k  k1  k2
is attached to the spring, the
mass-spring system is waited to come to its
This equation will give the expected equivalent spring
equilibrium position.
constant of the spring system connected in parallel.
7.5. In the vertical equilibrium position, the amount
of stretch (
11. For the spring system, the calculated (expected)
) of the spring is determined by
spring constant is compared with experimental
the height finder.
spring constant value found from the slope and
7.6. The force ( ) applied to the single spring by the
then the difference is determined.
hanging mass ( ) is determined:
12. The comparison results are recorded in Table-
F  mg
7.7. Using the amount of stretch (
applied force ( ), the spring constant (
(13).
) and the
13. Two springs with spring constants
) of the
and
with
equal lengths are first connected in series and
single spring is calculated by:
then in parallel. A mass is attached to the free
end of the spring system. When the springs are
F  kx  mg
connected in series, the period is
and when
connected in parallel, the period is
. What is
the ratio
16
?.
Hooke’s Law
Rentech
EXPERIMENT-4:
Simple Harmonic Motion
4.5. Dividing time reading from chronometer (
Determination of Period of Oscillation
) to
the number of the one complete oscillation (10),
PERIOD ( ) is found:
T
t10
10
Experimental Note
In this part of experiment, an object with the mass
is
hanged to the free end of the spring and the spring is
stretched a distance
from its vertical equilibrium
position (point) and then released. The object ( ) attached
to the spring accelerates as it moves back towards the
equilibrium position. The object oscillates back and forth. It
Figure-13: Experimental set-up to determine the period of an
executes simple harmonic motion from the lowest position (a
oscillating mass ( ) attached to the free end of a spring. At
, the hanging mass is released at
fully stretched spring) to the highest position (a fully
from the vertical
compressed spring) in the vertical direction. When the object
equilibrium position.
oscillates around its equilibrium point, the time passing for
one complete oscillation is defined as period ( ).
1.
The spring (diameter of the spring wire with
) is hanged to a fixed point.
4.6. The value found for one complete oscillation is
2.
The mass,
(
recorded as period ( ) (Table-14).
) is attached to the
free end of the spring and it is waited to return to
5.
the vertical equilibrium position.
Similarly, using the masses
and
on the same spring, period ( ) of
3.
Mass ( ) is pulled to
oscillation of each of the masses in a simple
in downward
harmonic motion is determined.
direction from its vertical equilibrium position.
4.
The mass ( ) is released at
(
6.
)
Square of period (
) of each of the masses is
calculated and then recorded in the Table-(14).
from the vertical equilibrium position and at this
moment the chronometer is started.
7.
4.1. Mass (
The hanging mass ( ) versus square of period
(
) attached to the spring will begin to
) is plotted.
oscilalte up and down in simple harmonic motion
7.1. In the
with a period (Figure-13).
graph, square of the period, (
) is
represented on the horizontal axis (x-axis) and
4.2. Starting
from
the
lowest
position
(a
fully
value of the mass
stretched spring), the time required to come
vertical axis (y-axis).
is represented on the
back to the lowest position is expressed as one
complete oscillation.
7.2. The best line that most fits the data points is
drawn and the equation of this line (straight line) is
4.3. When the mass completes ten (10) oscillation,
shown on the graph.
chronometer is stopped and the time reading
(
) from this device is recorded.
4.4. The time taken for one complete oscillation is
called the PERIOD ( ).
17
Hooke’s Law
Rentech
7.3. There is a relation between slope of the graph
14. In the spring system, what is the velocity of the
and spring constant ( ) as;
Slope 
oscillating mass in a simple harmonic motion
k
4
when it passes from the equilibrium position?.
2
The velocity of the mass ( ) which makes a simple
7.4. Using
this
constant (
slope,
EXPERIMENTAL
harmonic motion is in maximum value at its vertical
spring
equilibrium position. Moreover, the velocity of the
) is calculated (Table-14).
object with mass m hanged to the spring is zero at the
8.
The experimental set-up is prepared again using
top and lowest points (at the positions where the object
the same spring.
momentarily stops) in a harmonic motion.
8.1. A mass of
(
) is hanged to the
15. How does the value of the period ( ) change if
free end of the spring.
the mass ( ) attached to the spring in the simple
8.2. Amount of stretch ( ) on the spring is measured.
harmonic motion increases?
8.3. According to Hooke’s law, the spring constant is
If the mass of the object hanged to the spring increases in
calculated.
its simple harmonic motion, the period ( ) of the
oscillating mass attached to the spring increases.
8.4. This calculated spring constant is recorded in the
Table-(15) as the EXPECTED value ( ).
9.
16. Two similar spring having
Percent difference is determined by comparing
the
experimental spring constant with
spring constant
is connected in series and a mass
the
is
hanged to the free end of the spring system. If the
expected value.
mass is pulled down in a certain amount from its
vertical equilibrium point (position) and then
10. The result is recorded to Table-(16).
released, how is the period ( ) of the mass
oscillating on the spring system found?.
11. The experiment is repeated following the same
experimental procedures for different springs.
At first, the equivalent spring constant ( ) of the spring
system in series is found. Writing this value ( ) and the
12. What is the period of the simple harmonic
mass ( ) of the object in the period equation, period of
motion?
oscillating mass is calculated.
It is the time passing to complete one full oscillation.
13. If the same mass ( ) is hanged to the free ends
of two different springs which made of hard and
soft materials, respectively but in the same size
and then if it starts to oscillate, which motion has
the large period ( )?.
The spring constant ( ) is larger for the hard spring
and smaller for the soft spring. Therefore, the period
( ) of the object hanged to the soft spring is larger
than in the case of the hard spring.
18
Hooke’s Law
Rentech
4.
Laboratory Report
Name and Surname:
__________________________________
Department:
__________________________________
Student ID:
__________________________________
Date:
__________________________________
EXPERIMENT-1: Hooke’s Law
Determination of Spring Constant
Table-1: Using different masses, determination of spring constant from the slope of the graph
Measured
Used
Calculated
Measured
Diameter of
Mass
Force applied to the spring
The amount of
spring wire
(mm)
stretch
m(kg)
x(m)
F(N)
F ( N )  mg
.....
Graph
Slope
a(N/m)
Experimental
Spring constant
k(N/m)
Slope = k
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
Graph-1: The amount of stretch of the spring versus the force acting on the spring.
19
.....
Hooke’s Law
Rentech
Table-2: The calculated (expected) spring constant value for the first spring.
Measured
Used
Calculated
Measured
Diameter of
Mass
Force applied to the spring
The amount of
spring wire
(mm)
stretch
m(kg)
Expected
Spring constant
x1(m)
F(N)
k1(N/m)
F ( N )  mg
.....
.....
.....
F ( N )  kx
.....
.....
Expected
Table-3: The calculated (expected) spring constant value for the second spring.
Measured
Used
Calculated
Measured
Diameter of
Mass
Force applied to the spring
The amount of
spring wire
(mm)
stretch
m(kg)
Spring constant
x2(m)
F(N)
k2(N/m)
F ( N )  mg
.....
.....
.....
F ( N )  kx
.....
.....
Expected
Table-4: The calculated (expected) spring constant value for the third spring.
Measured
Used
Calculated
Measured
Diameter of
Mass
Force applied to the spring
The amount of
spring wire
(mm)
stretch
m(kg)
Spring constant
x3(m)
F(N)
k3(N/m)
F ( N )  mg
.....
.....
.....
F ( N )  kx
.....
.....
Table-5: The comparison of experimental spring constant value with the expected spring constant value for the first spring.
First Spring
Diameter of
spring wire
(mm)
.....
Spring Constant
Spring Constant
(Experimental)
(Expected)
Slope
Calculated
Calculated
k(N/m)
k(N/m)
k(±%)
Slope = k
F ( N )  kx
.....
.....
20
Percentage Difference
Difference (%) 
.....
k  k
 100
k
Hooke’s Law
Rentech
EXPERIMENT-2:
Springs Connected in Series
Determination of Spring Constant
Table-6: For the spring system connected in series, the determination of the spring constant from the slope of the graph.
Used
Calculated
Measured
Mass
Force applied to the spring
The amount of stretch of
the system
m(kg)
x(m)
F(N)
F ( N )  mg
Graph
Experimental
Spring constant
Slope
a(N/m)
k(N/m)
Slope = k
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
Graph-2: The amount of stretch of the spring system connected in series versus the force acting on the
spring system.
21
Hooke’s Law
Rentech
Table-7: The calculated (expected) spring constant value for the first spring in the spring system connected in series.
Measured
Used
Diameter of
spring wire
(mm)
Calculated
Measured
The force applied to the
The amount of
system
stretch
F(N)
x1(m)
Mass
m(kg)
Expected
Spring constant
k1(N/m)
F ( N )  kx
F ( N )  mg
.....
.....
.....
.....
.....
Table-8: The calculated (expected) spring constant value for the second spring in the spring system connected in series.
Measured
Used
Diameter of
spring wire
(mm)
Calculated
Measured
The force applied to the
The amount of
system
stretch
F(N)
x2(m)
Mass
m(kg)
Expected
Spring constant
k2(N/m)
F ( N )  mg
.....
.....
.....
F ( N )  kx
.....
.....
Table-9: The comparison of experimental spring constant value with the expected spring constant value for the spring
system connected in series.
The Spring Constant of the
System
(Experimental)
The Spring Constant of the System
Percentage Difference
(Expected)
Slope
Calculated
Calculated
k(N/m)
k(N/m)
k(±%)
Slope = k
.....
k
k1 k 2
k1  k 2
.....
22
Difference (%) 
.....
k  k
 100
k
Hooke’s Law
Rentech
EXPERIMENT-3: Springs Connected in Parallel
Determination of Spring Constant
Table-10: For the spring system connected in parallel, the determination of the spring constant from the slope of the graph.
Used
Calculated
Measured
Mass
Force applied to the spring
The amount of stretch of
the system
m(kg)
F(N)
x(m)
F ( N )  mg
Graph
Experimental
Spring constant
Slope
a(N/m)
k(N/m)
Slope = k
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
Graph-3: The amount of stretch of the spring system connected in parallel versus the force acting on
the spring system.
23
Hooke’s Law
Rentech
Table-11: The calculated (expected) spring constant value for the first spring in the spring system connected in parallel.
Measured
Used
Calculated
Mass
The force applied to the system
m(kg)
F(N)
Diameter of
spring wire
(mm)
Measured
Expected
The amount of
Spring constant
stretch
x1(m)
k1(N/m)
F ( N )  mg
.....
.....
.....
F ( N )  kx
.....
.....
Table-12: The calculated (expected) spring constant value for the second spring in the spring system connected in parallel.
Measured
Used
Diameter of
spring wire
(mm)
Calculated
Measured
The force applied to the
The amount of
system
stretch
F(N)
x2(m)
Mass
m(kg)
Expected
Spring constat
k2(N/m)
F ( N )  mg
.....
.....
.....
F ( N )  kx
.....
.....
Table-13: The comparison of the spring constants for the spring system connected in parallel.
The Spring Constant of the
System
(Experimental)
The Spring Constant of the System
Percentage Difference
(Expected)
Slope
Calculated
Calculated
k(N/m)
k(N/m)
k(±%)
Slope = k
k  k1  k 2
.....
.....
Difference (%) 
.....
24
k  k
 100
k
Hooke’s Law
Rentech
EXPERIMENT-4: Period
Determination of Period of Oscillation
Table-14: By using the period of oscillation, the determination of the spring constant from the slope of the graph.
Measured
Diameter of
spring wire
(mm)
Used
Chronometer
Calculated
Graph
Experimental
Mass
Period
Square of period
Slope
Spring constant
m(kg)
T(s)
T (s )
a(N/m)
k(N/m)
2
2
Slope =
.....
.....
.....
.....
.....
.....
.....
.....
.....
.....
k
4 2
.....
Graph-4: The mass of the oscillating object versus the square of the oscillation period.
25
.....
Hooke’s Law
Rentech
Table-15: Expected (calculated) spring constant value of the spring.
Measured
Diameter of
spring wire
(mm)
Used
Calculated
Measured
The force applied to the
The amount of
system
stretch
F(N)
x1(m)
Mass
m(kg)
Expected
Spring constant
k1(N/m)
F ( N )  mg
.....
.....
.....
F ( N )  kx
.....
.....
Table-16: The comparison between the experimental and the expected spring constant.
First Spring
Diameter of
spring wire
(mm)
Spring Constant
Spring Constant
(Experimental)
(Expected)
Slope
Calculated
Calculated
k(N/m)
k(N/m)
k(±%)
Slope =
.....
k
4
2
F ( N )  kx
.....
.....
26
Percentage Difference
Difference (%) 
.....
k  k
 100
k