Torsion Apparatus Experiment
TORSION APPARATUS EXPERIMENT
(The Modulus of Rigidity)
Student's Guide
Laboratory Manual
1 and Workbook
(2014)
Torsion Apparatus Experiment
List of Equipment
1.
Torsion Apparatus Machine
2.
Metal Test Rods (Test Specimens with Different Diameters)
2.1.
Steel
(Diameter: 3.0mm, Length:100cm)
2.2.
Steel
(Diameter: 4.0mm, Length:100cm)
2.3.
Brass
(Diameter: 3.0mm, Length:100cm)
2.4.
Brass
(Diameter: 4.0mm, Length:100cm)
3.
Wheel (Diameter:17cm)
4.
Masses (8 Pieces)
5.
4.1.
Mass:4x400gr
4.2.
Mass:2x200gr
4.3.
Mass:1x100gr
4.4.
Mass:1x50gr
String
2
Torsion Apparatus Experiment
Purpose:
The objective of this experiment is to analyze and
determine the rigidity modulus of the various
metal rods.
Introduction:
The torsion apparatus is used to determine the
modulus of rigidity (or shear modulus) of the
materials. When under application of an external
force, the shape of a solid material changes
without change in its volume, and the solid
material is said to be shared. This deformation
happens when a tangential force (F ) is applied
to one of the faces of the object.
In the case of the rigidity modulus, suppose a
rigid block whose lower face is fixed and a known
tangential force (F ) is applied to its upper face
as seen in the Figure-(1). This external force
Figure-1: Deformation of a solid object under shear stress.
causes the parallel layers to the lower face to slip
through a distance of
x from the fixed face of the
solid block such that the upper face shifts and the
Here, A is the area of the upper face of the solid
solid material takes a new form while its volume
block. The unit of stress is the Pascal (1 Pascal=1
remaining unchanged. The strain produced in this
Pa=1N/m ).
2
material by the stress is measured by the angle
Figure-(1) shows that one side of the object under
( ) called the shearing strain (or simply shear).
x
shear stress is displaced by a distance
Due to this shearing of the solid material, a
relative to the opposite side. If the displacement
tangential restoring force is developed in the solid
(x) is the deformation of the object, we can
object which is equal and opposite to the external
define
applied force.
shear
displacement
strain
as
the
ratio
of
the
x to the transverse dimension, h :
Since shear stress is a force per unit area, we
define the shear stress as the force (F ) acting
Shear strain 
tangent to the surface, divided by the area ( A) on
which it acts:
x
h
(2)
Shear strain is a dimensionless number with a
ratio of two lengths.
F
Shear stress 
A
(1)
3
Torsion Apparatus Experiment
The ratio of shear stress (tangential stress) to
shear strain is called rigidity modulus (or shear
modulus)
:

F/A
x/h
(3)
Note that the concepts of the shear stress, shear
strain
and
shear
modulus
apply
to
solid
materials only. The reason is that the shear
force in the Figure-(1) is required to deform the
(a)
solid object, and the object tends to return to its
original shape if the shear force is removed. The
corresponding elastic modulus (ratio of shear
stress to shear strain) is given by the shear
modulus.
Similar to the solid block, when a torque ( ) is
applied to a rigid rod (shaft) with length () and
(b)
radius (r ) , the torque deforms the rod by twisting
Figure-2: The shearing of a circular solid metal rod by the
it through a small angle ( ) . Torsion occurs
applied torque.
when a solid material (or shaft) is subjected to a
In the geometry given by the Figure-(2), it can be
torque ( ) . Here, torsion refers to the twisting of a
seen that one end of the rigid rod will rotate about
shaft loaded by the applied torque.
the longitudinal axis with respect to the other end
Consider a horizontal line drawn along the length
during twisting. The magnitude of this rotation is
of a given metal rod. The rod is fixed at one end
measured in terms of the angle
and twisted at the other end due to the action of
which, one end rotates relative to the other end.
torque

. This applied torque will cause the one
The angle


(in radians) by
is called the angle of twist.
face of the circular cylinder (rigid rod) to be
twisted through an angle ( ) at the length
Since the end of the metal rod is twisted, the
.
radial line on the end face of the rod rotates
When twisted, the horizontal line moves through
an angle
through the angle
 and the line BC will be shifted to AC
.
The length of the arc BA
produced at the outside edge of the rod is the
through the angle ( ) . The length of the arc BA
deformation distance (corresponding to
produced is  (Figure-2a).
x given
in the Figure-1) and will be also equal to r 
(Figure-2b).
4
Torsion Apparatus Experiment
As the external axial torque twists the rod, a
In the case of the applied torque, the relationship
section perpendicular to the axis at a given
between the torque and the twist angle is given
distance
 rotates through the angle  . If we
by the torsion equation as:
assume that the two arcs are the same, it follows

that:
BA  r  
 
where
r



J
(9)
(4)
where
(5)
J is polar moment of inertia for a solid
cylindrical specimen. The polar moment of inertia
for a solid circular shaft with the diameter (d )
 is the shear strain (the angle of the
can be calculated as:
shear) on the outer surface of the solid rod.
 d4
J
The modulus of rigidity (or shear modulus) is
given by:



(6)
(10)
32
By combining Equation-(8) and Equation-(9), we
get the general torsion equation as:
where,

:
The applied shear stress,
:
The resulting shear strain.
J





r
(11)
Substituting Equation-(5) into Equation-(6), we
Using Equation-(11), we can rewrite the shear
get:
modulus (or rigidity modulus) at the end of the
metal rod as follows:


r
(7)

(8)


J
(12)
or,



where,
:
r
Shear modulus (or rigidity modulus),
:
Length of the metal test rod (shaft) over
which the angle of twist is measured,
shear stress, shear modulus and angle of twist for
J:
Polar moment of inertia,
a given circular shaft.

Applied torque,
The Equation-(8) uses the relationship in the
:
:
5
Angle of twist in radians.
Torsion Apparatus Experiment
where,

: Applied torque,
F : The constant applied force due to gravity
acting on the hanging mass,
R : Radius of the wheel on the torsion
apparatus,
M : Hanging mass.
Figure-3: The cross-section of a solid circular shaft.
If we use the applied torque
moment of inertia
Assume that the circular shaft has a uniform

and the polar
J for a given circular shaft in
the Equation-(12), we can also rewrite the rigidity
cross-section along its length () , it is straight
modulus ( ) as a function of the mass (load):
and the torque is constant along its length. Then
we can state that a cross-section normal to the

longitudinal axis is planar before the torque
2MgR 
 r4 
(15)
application and it will remain planar after twisting
by the angle

(Figure-3).
If the torque

and the degree of rotation
where r is the radius of the circular shaft such

that
d  2r .
are
known in a given torsion experiment, the rigidity
By plotting a graph of torque ( ) versus the
 of the any test specimen can be
angle of twist ( ) , the rigidity modulus can be
determined. In the torsion apparatus, the torque
determined from the slope of the graph. So,
( ) is supplied by hanging masses (M )
Equation-(12) can be written as follows;
modulus
attached to a string wrapped around a vertical
wheel
with
radius (R) .
This
radius

is

 (Slope)
J
(16)
perpendicular to the applied force (F ) which is
the tension in the string wrapped around the
wheel. Therefore, the applied torque ( ) in the
Therefore, knowing () and (J ) of any metal
torsion apparatus can be calculated as:
test rod for a particular experiment, and then
measuring the slope of corresponding
  FR
(13)
  MgR
(14)
 
graph, Equation-(16) will give the rigidity modulus
( ) of the metal rod experimentally.
6
Torsion Apparatus Experiment
Description of the Torsion Apparatus
(a)
(b)
Figure-4: Schematic representation of the torsion apparatus (a) and the cylindrical shaft (metal rod) under applied
torsional loading (b).
7
Torsion Apparatus Experiment
The torsion apparatus consists of a wheel, metal
Experimental Procedure
test rods and masses with different weights. The
Part-I
metal rods include brass with 3.0mm and 4.0mm
in diameter (d ) ; the other test rods are steel with
The torsion experiment looks at the shear
the diameters of 3.0mm and 4.0mm.
deformation of a cylindrical shaft (metal rod)
The torsion test is performed by mounting the
under applied torsional loading.
metal rod specimen into a torsion testing machine
and then applying the twisting moment (or torque)
In the first part of the experiment, the deformation
to the material. The metal rod is held horizontal
test of the each metal rod is carried out to
and rigidly mounted into the end sockets of
determine if the deformation remains linear
torsion apparatus (Figure-4a).
elastic or not. In order to study the response of
materials
The large wheel contains a socket to hold one
under
the
torsional
force,
test
specimens are mounted between the two sockets
end of the metal rod such that this side of the rod
of the torsion testing machine and then twisted.
is subjected to a torque produced by the vertical
Each of the specimens should be elastically
wheel. A light string around the groove of the
0
deformed (0
wheel carries masses to produce the torque.
   750 ) and the recovery of
the specimen to its original shape must be
The Figure-(4b) shows a shaft fixed at one end
possible if the specimen is unloaded (
 00 ) .
and twisted at the other end due to the action of a
The deformation is measured by the angle of
torque  . The masses (M ) are chosen for each
rod so that the torsion angle (angle of twist)
twist,


at the length,
 of the metal rod.
never increases beyond 75 and the deformation
To test the deformation of the specimen:
must remain linear elastic. Different cylindrical
1.
0
rods are loaded with external torque and then the
Measure the diameter of the first specimen
to be tested as
torsion angle ( ) measured at the end () of the
2.
metal rod. Using the data set of the load (torque)
d  d1 .
Clamp the test specimen into the torsion
testing machine using the sockets and make
and the corresponding angular twist, the rigidity
sure the specimen is firmly mounted and
modulus of the different cylindrical rods is
twist angle,  is zero
determined.
(0 0 ) .
3. Load the masses one by one in order to
deform the test specimen elastically until
the rod is twisted about
  750 of
the
rotation. Note that the magnitude of torque
applied to the specimen must be such that
the resulting stress remains in the elastic
region.
8
Torsion Apparatus Experiment
4.
Remove the weights in the same order in
4.
which they were added. Now, the reading of
twist angle should be again zero (0
0
essential to make sure that the twist angle is
zero (0
) . Note
that the twist angle must be zero each time
when
the
rod
is
relaxed
(the
5.
torque
) . If not, set the angle to zero.
Now, load the rod with a torque by using
torsion angle
clockwise and anti-clockwise direction.
6.
Record

the
at the end () of the rod.
applied
torque
and
the
If the angle of twist is not zero after removing
corresponding angle of twist in the data
the added weights, the rod is not deformed
table. Note that for every load increment; you
elastically and the following conditions may
should record the torque
exist:
( ) with the
corresponding angular displacement ( ) .
5.1. The fixed rod is not straight and
4.1. Repeat until enough measurements are
horizontal along the end sockets.
taken to draw a graph of applied torque
5.2. The rod is not firmly clamped into the
as a function of twist angle.
end sockets so it is slipping.
6.
0
masses and measure the corresponding
unloaded). This procedure is done both in
5.
Before loading the masses for torque, it is
7.
Before proceeding with the experiment,
Plot a graph of the applied torque ( )
repeat the deformation test for the each
against the twist angle ( ) of rotation. In the
specimen
graph, angle of twist should be in radians.
with
different
materials
and
diameters.
8.
Draw the best straight line through the points
in the
   graph and determine the slope.
6.1. By the slope of torque-twist angle
Part-II
graph, find and record the ratio of the
applied torque to the angle of twist.
In this part of the experiment, different solid
6.2. Using the slope, the rod diameter (d1 )
cylindrical rods will be subjected to a torsional
and
load. Then, the modulus of rigidity will be
rigidity
determined for the cylindrical materials.
1.
9.
diameter (d1 ) of the test rod. Also record the
2.
3.
length () ,
modulus ( )
determine
of
the
the
test
specimen from Equation-(16).
For the first test piece, determine the
length as
rod
Compare the experimental value of the
rigidity modulus with the accepted value.
  1.0m where the angle of twist
10. Carried out the experiment on the rods of the
( ) will be measured.
steel and brass of the same diameter
Measure the radius (R) of the loading
(4.0mm) using the length,   0.5m until
wheel in the torsion apparatus.
the each rod is twisted about
Calculate and record polar moment of inertia
by using the corresponding diameter
the test piece as J
  450 .
11. Repeat the experiment to find the rigidity
modulus ( )
(d1 ) of
 J1 .
of the different metal rods
under applied torsional loading.
9
Torsion Apparatus Experiment
LABORATORY REPORT
Name
Department
Student No
Date
Material Type:
. . . . .
Length of Metal Rod, (m) :
. . . . .
Metal Rod Diameter,
d1 (m) :
Polar Moment of Inertia,
. . . . .
J 1 (m 4 ) :
. . . . .
Table-1: Experimental data values for the rigidity modulus of the metal test rod in the diameter, d 1 .
 (m)
M (kg)
 (Deg )
 (Rad )
 ( N .m)
Slope
 (N / m2 )
Measured
Measured
Measured
Experimental
Experimental
Calculated
Experimental
0.0
0.0
0.0
0.0
…..
…..
…..
…..
…..
…..
…..
1.0
…..
…..
…..
…..
…..
 (m) :
Test length of the metal rod,
M (kg) :
Hanging mass,
 (Deg ) :
Twist
angle
in
degrees
measured
experimentally at the test length () ,
 (Rad ) :
Angle of twist in radians,
 ( N .m) :
Applied torque,
 (N / m2 ) :
The modulus of rigidity.
10
Torsion Apparatus Experiment
Graph-1: Applied torque as a function of twist angle for the metal rod with diameter
(d1 ) at   1.0m .
11
Torsion Apparatus Experiment
Material Type:
. . . . .
Length of Metal Rod, (m) :
. . . . .
Metal Rod Diameter,
d 2 (m) :
Polar Moment of Inertia,
. . . . .
J 2 (m 4 ) : . . . . .
Table-2: The rigidity modulus of the metal test rod with the diameter, d 2 .
 (m)
M (kg)
 (Deg )
 (Rad )
 ( N .m)
Slope
 (N / m2 )
Measured
Measured
Measured
Experimental
Experimental
Calculated
Experimental
0.0
0.0
0.0
0.0
…..
…..
…..
…..
…..
…..
1.0
…..
…..
…..
…..
…..
Graph-2: Applied torque as a function of twist angle for the
metal rod diameter (d 2 ) at
  1.0m .
12
Torsion Apparatus Experiment
Material Type:
. . . . .
Length of Metal Rod, (m) :
. . . . .
Metal Rod Diameter,
d 3 (m) :
. . . . .
4
Polar Moment of Inertia, J 3 (m ) :
. . . . .
Table-3: The rigidity modulus of the metal test rod with the diameter, d 3 .
 (m)
M (kg)
 (Deg )
 (Rad )
 ( N .m)
Slope
 (N / m2 )
Measured
Measured
Measured
Experimental
Experimental
Calculated
Experimental
0.0
0.0
0.0
0.0
…..
…..
…..
…..
…..
…..
1.0
…..
…..
…..
…..
…..
…..
Graph-3: Applied torque as a function of twist angle for the
metal rod diameter (d 3 ) at
  1.0m.
13
Torsion Apparatus Experiment
Material Type:
. . . . .
Length of Metal Rod, (m) :
. . . . .
Metal Rod Diameter,
d 4 (m) :
Polar Moment of Inertia,
. . . . .
J 4 (m 4 ) : . . . . .
Table-4: The rigidity modulus of the metal test rod with the diameter, d 4 .
 (m)
M (kg)
 (Deg )
 (Rad )
 ( N .m)
Slope
 (N / m2 )
Measured
Measured
Measured
Experimental
Experimental
Calculated
Experimental
0.0
0.0
0.0
0.0
…..
…..
…..
…..
…..
1.0
…..
…..
…..
…..
…..
…..
Graph-4: Applied torque as a function of twist angle for the
metal rod diameter (d 4 ) at
  1.0m .
14
Torsion Apparatus Experiment
Material Type:
. . . . .
Length of Metal Rod, (m) :
. . . . .
Metal Rod Diameter, d (m) :
. . . . .
Polar Moment of Inertia,
J (m 4 ) :
. . . . .
Table-5: The rigidity modulus of the steel with the diameter,
d  4.0mm at   0.5m .
 (m)
M (kg)
 (Deg )
 (Rad )
 ( N .m)
Slope
 (N / m2 )
Measured
Measured
Measured
Experimental
Experimental
Calculated
Experimental
0.0
0.0
0.0
0.0
…..
…..
…..
…..
…..
0.5
…..
…..
…..
…..
…..
…..
Graph-5: Applied torque as a function of twist angle for the
steel rod diameter (d  4.0mm) at
  0.5m .
15
Torsion Apparatus Experiment
Material Type:
. . . . .
Length of Metal Rod, (m) :
. . . . .
Metal Rod Diameter, d (m) :
. . . . .
Polar Moment of Inertia,
J (m 4 ) :
. . . . .
Table-6: The rigidity modulus of the brass with the diameter,
d  4.0mm at   0.5m .
 (m)
M (kg)
 (Deg )
 (Rad )
 ( N .m)
Slope
 (N / m2 )
Measured
Measured
Measured
Experimental
Experimental
Calculated
Experimental
0.0
0.0
0.0
0.0
…..
…..
…..
…..
…..
0.5
…..
…..
…..
…..
…..
…..
Graph-6: Applied torque as a function of twist angle for the
brass rod diameter (d  4.0mm) at
16
  0.5m .
Torsion Apparatus Experiment
Table-7: Comparison of the experimental and expected values of the rigidity modulus corresponding
to material type.
Material
Diameter
Steel
…..
Steel
…..
Brass
…..
Brass
…..
Rigidity Modulus
Rigidity Modulus
 (N / m2 )
 (N / m2 )
Experimental
Expected
%Error ()
The test length () of the each metal rod is set to 1.0m ± 0.001m in the torsion apparatus.
Table-8: Comparison of the experimental rigidity modulus with the expected value of the metal rods
(steel and brass).
Material
Diameter
Steel
…..
Brass
…..
Rigidity Modulus
Rigidity Modulus
 (N / m2 )
 (N / m2 )
Experimental
Expected
%Error ()
The test length () of the each metal rod is set to 0.5m ± 0.001m in the torsion apparatus.
17