Abstract
This lab study was undertaken to analyze the effects of torque forces and shear modulus of rigidity on three circular solids namely, half-hard brass (60% Cu and 40% Zn), 0.1% and 0.4% carbon steel. This was done by subjecting the samples to a shearing stress and by noting down its angular displacement. The maximum torque of each sample was calculated and compared. Their polar moment of inertia was also calculated to assist in analysis. The value of shear modulus of rigidity of each of the samples was determined using a graph which related torque and angular displacement. It was found that when the torque was increased, the value of shear modulus of rigidity decreased. The angular displacement of each sample was found to increase with increase in torque. This showed that bodies that were subjected to high torque showed greater displacement in their plane. Torque affects the tensile strength of a material. This study can signify the effect of torque on materials which are subjected to great forces like in automobile parts. Effect use of torque can increase the life of materials.
Introduction
The purpose of this study is to assess the effects of torsion forces on solids, and thus study its relation to the rigidity modulus of the material, in order to understand the capacity of a solid to withstand force. Torque is a measure of how much force is acting on an object that causes the object to rotate. (University of Guelph, 2016). Its SI unit is Newton-meter (Nm). Torque is an important feature in automobile engineering. It determines engine speed, capacity of gears, and overall acceleration. However, a high rate of torque can be dangerous to a wet, slippery road or during snowfall. Horsepower is a unit which expresses the power of an engine. It is a fraction of torque. When torque acts on a body, the body tends to twist. When a number of torque forces are acting on a body, the total torque, under which the body is subjected, is given by the sum of all individual torque forces. Torque is a vector quantity. The maximum torque τmax acting on a body can be calculated using the equation,
Τmax = Tr/Ip
Where T is the shear stress,
r is the radius of circular cross section
and Ip is the polar moment of inertia for a solid circular cross section.
It is used to determine how much an object can resist torsion or how much it has been displaced in its plane. The shear modulus of rigidity is the coefficient of elasticity for a torsion force (Engineering Toolbox, 2016). It can be determined experimentally from the slope of a stress-strain curve.
Equipment
Half-hard Brass (60% Copper, 40% Zinc)
0.1% Carbon Steel
0.4% Carbon Steel
Torque Testing Equipment
Methodology
The diameter and gauge length of the three samples were measured and noted.
A line was drawn parallel to the long axis to assist in observing the specimen.
The specimen was then mounted in the hexagonal drive sockets.
Clear guard was applied to the specimen for protection.
The software was turned on and the diameter and the gauge length were introduced.
The test was started by turning the wheel to maintain constant velocity.
The amount of torque and the angle of twist of the specimen were noted.
Observed Readings
Readings obtained were tabulated as follows.
The diameter and length of each of the sample were used to calculate the maximum stress τmax achieved and shear modulus of elasticity, G, in each case.
Maximum stress τmax = Tr/Ip
Where T is the shear stress in N/mm2
r is the radius of solid circular cross section in mm
and Ip is the polar moment of inertia for a solid circular cross section in mm4. Polar moment of inertia Ip is given by,
Ip = π d4/32
where d is the diameter of the solid circular cross section. Shear modulus of rigidity of the material, G, is given by,
G = T/α*L/Ip
Sample 1 (Half-hard Brass)
Diameter, d = 6.02mm, radius, r = 3.01mm, Length, L = 73.71mm, T = 14.24Nm
Ip = π d4/32 = π*(6.02)4/32 = 128.94mm4
Τmax = Tr/Ip = (14240*3.01)/128.94 = 332.42Nmm
Using linear part of the graph, the following values were obtained.
T = 5.24Nm
Α = 0.27698 rad
G = T*L/α*Ip = 5240*73.71/0.27698*128.94= 10,815.12N/mm2
Thus, the half-hard brass sample was subjected to a maximum torque of 332.42Nmm and had shear modulus rigidity of 10,815.12N/mm2.
Sample 2 (0.1% Carbon Steel)
Diameter, d = 6.05mm, radius = 3.025mm, Length, L = 73.54 mm
Torque, T = 21.30Nm
Ip = π d4/32 = π*(6.05)4/32 = 131.58mm4
Τmax = Tr/Ip = 21300*3.025/131.58 = 489.68Nmm
Using the linear part of the graph, the following values were obtained
T = 8.04Nm
Α = 0.745974rad
G = T*L/α*Ip = 8040*73.54/0.745974*131.58 = 6023.75N/mm2
Thus, 0.1% carbon steel was subjected to a maximum torque of 489.68 Nmm2 and had shear modulus rigidity of 6023.75N/mm2.
Sample 3 (0.4% Carbon Steel)
Diameter, d = 6.03mm, radius = 3.015mm
Length, L = 72.04mm
Torque, T = 28.61Nm
Ip = π d4/32 = π*(6.03)4/32 = 129.85mm4
Τmax = Tr/Ip = 28610*3.015/129.85 = 4872.27Nmm.
Thus, 0.4% carbon steel was subjected to a maximum torque of 665.29 Nmm2 and had shear modulus rigidity of 4872.27N/mm2.
Discussion
Analysis of torque forces acting on the three samples of solids revealed the following. When a sample of half-hard brass (60% Cu and 40% Zn) was subjected to a shear stress of 5.24 Nm, its angular displacement was found to be 0.2769 rad. The maximum torque acting on it was 332.42 Nmm. When a sample of 0.1% carbon steel was subjected to a torque of 21.30 Nm, it exhibited an angular displacement of 0.745974 rad. Maximum torque acting on it was 489.68 Nmm. Sample 3 of 0.4% carbon steel was subjected to torsion of 15.53 Nm. It showed an angular displacement of 1.76837 rad. A maximum torque value of 664.29 Nmm acted on it. These values clearly indicate that an increase in torsion forces acting on a body causes a decrease in its rigidity modulus. Results also show that increase in torque causes increase in angular displacement. This implies that more the torsion forces acting on a solid, greater the body gets displaced. Equations with shear stress, polar moment of inertia and rigidity modulus establish a clear interrelationship between it and torsion forces.
Conclusion
The present study indicates the significance of torsion and its effect on the rigidity modulus of a material. The study also establishes a relationship between torsion and angular displacement. These are important concepts in determining the tensile strength of materials, thus guiding its effective use. Effect of torsion was analyzed using different values of shear stress. Instead a fixed value of shear stress can be used to make comparison of materials easier.
References:
Anon, (2016) Boundless: Torque. [Online] [Accessed 29th December 2016], Available at:
http://boundless.com/physics/textbooks/boundless-physics-textbook/static-equilibrium-elasticity-and-torque-8/introduction-73/torque-307-746
Anon, (2016) The Engineering Toolbox: Torsion of Shafts. [Online] [Accessed 30th
December 2016], Available at:
http://engineeringtoolbox.com/torsion-shafts-d_947.html
Anon, (2016) University of Guelph: What is Torque? [Online] [Accessed 29th December 2016], Available at:
http://www.physics.uoguelph.ca/tutorials/torque/Q.torque.intro.html
