TORSION LABORATORY
Objective
The objective of this experiment was to determine the relationship between load and angular deflection of a mild steel bar, the maximum shear stress applied to the bar and its modulus of rigidity.
Apparatus
- Torsion test rig
- Weight hangers
- Masses
- Rule
- Calipers
Procedure
- A distance L was set between two angle indicators. Then the fixings were tightened to make sure that it was not moving. The indicators were set to zero on both the scales.
- The effective radius of the pulley and the diameter of the steel shaft were measured.
- The weight hanger was attached to the string that was wound around the pulley.
- The masses of 0.5kg were added up to 4 kg. Further, for each mass added, the angles on the two angle indicators were recorded. The repeatability was checked.
- The procedure was repeated for one more length
Theory of the Experiment
For a shaft subject to a load producing a torque, the angle of twist will be dependent on diameter of the shaft, its length, and the material it is made from. The general equation for the torsion of circular cross section shafts:
TJ=τr=GθL
Where: t= torque applied to shaft (Nm)
J= polar second moment of area (m4) (J=πd432 for a circular solid shaft
τ= Shear stress (N/m2)
R= Radius of shaft (m)
G= Modulus of Rigidity of Shaft (N/m2)
θ=Angle of twist of shaft (radians)
L=Length of shaft (m)
Results
J=πd432=π*4.764=32=50.3997mm4
Diameter of bar = 4.76mm
Length 1=400mm
Diameter of bar = 4.76mm
Length 2 = 300mm
Diameter of bar = 4.76mm
Length 3 = 200mm
Formulas
Pulley radius =60.02mm
torque=mass*9.81*pulley radius
angle of twist in radians=θ1*θ2360*2π
Torque 1 Results (Length 1=400mm)
y=11806x-103.04
slope=11806
modulus of rigidity, G=slope of the graph*LJ
modulus of rigidity, G=11806*400mm50.3997mm4=9.369*104N/m²
shear stress τ=TrJ
shear stress τ=TrJ
Torque 2 Results (Length =300mm)
y=15063x-240.46
slope=21047
modulus of rigidity, G=slope of the graph*LJ
modulus of rigidity, G=15063*300mm50.3997mm4=8.9667*104N/m²
Torque 3 Results (Length =200mm)
y=21047x-341.16
slope=21047
modulus of rigidity, G=slope of the graph*LJ
modulus of rigidity, G=21047*200mm50.3997mm4=8.352*104N/mm²
Shear Stress Values
Maximum shear Stress= 2103.56 N/mm²
Discussion
Using the line of best fit, the graphs tend to have linear relationship between the Torque and the angle of twist. The lines of best fit do not pass through the origin. The applied torque tends to increase the angular deflection Further Gambhir (2009) notes that the max shear stress will be experienced at when the radial distance is long. The published value for the modulus of rigidity according to Rajput (2005) is 80 GPa. From the calculated values, the average G is 88.959 GPa.
Conclusion
SIMPLE HAROMINIC MOTION OF A SPRING LABORATORY
Objective
Apparatus
- Retort stand
- Clamp
- Balance
- Spring
- Stopwatch
- Weight hanger
- 20g/50g masses
- Meter rule
Procedure
- The spring was placed on the balance and the mass of the spring recorded.
- The clamp was attached to the retort stand and the spring firmly clamped so that it oscillated vertically.
- The height from the workbench to the end of the spring was measured using a meter rule. Then the displacement after the weight hanger was added was recorded.
- The weight hanger was pulled slightly to allow the spring to oscillate vertically. 20 oscillations were timed using the stopwatch. This was repeated twice to obtain a reliable mean.
- The 20g mass was added to the weight hanger and the displacement recorded. Step 4 was then repeated.
- This procedure above was repeated for four more masses.
Theory
Simple harmonic motion is a periodic motion that occurs when the restoring force on a body that has been displaced from its equilibrium position causes oscillations. The magnitude of the restoring force is proportional to the displacement of the body about a fixed equilibrium position.
For a spring system, the time period for one oscillation can be found:
T=2πmassforcedisplacement
For a spring with mass, m, oscillating with attached mass, M and spring stiffness, k:
T=2πM+m3k
Notes
T=2πM+m3k
T2=4π2Mk+4π2m3k
y=T2=;x=M;c= 4π2m3k;
m=4π2k=slope of the graph
Results
Spring stiffness=3791.6
X intercept is when y= 0, thus y=1.9472x+171.03;x=-87.833
Using
m=4π2k=1.9472:k=20.27
c= 4π2m3k;m=171.03*3*20.274π2=263.443g
Discussion
The measured mass of the spring was found to be 196 g. from the calculations the mass was obtained as 263.433g. Both graphs tend to have an almost linear relationship based on the line of best fit drawn. The spring obeys Hooke’s law, which makes it effective to conduct the experiment. According to Hanna and Picciotto (2002), for the spring to exhibit characteristics of a simple harmonic motion, it has to have an equilibrium position from which displacement can occur. Further, the spring has a tendency to return to its equilibrium position. Further, according to Myers (2006) for a system to be in simple harmonic motion is that the force has to be proportional to the displacement, which in this case is as shown in the graph.
Conclusion
Accuracy is very vital in a simple harmonic motion test. The spring has to obey Hooke’s law if the desired results are to be achieved. In case the masses exceed the elastic limit of the spring, the spring does not adhere to certain requirements of simple harmonic motion. Thus, it becomes useful to ensure accuracy is achieved by not stretching the spring too far.
ACCELERATION DUE TO GRAVITY LABORATORY
Objective
Apparatus
- Fly wheel and attached axle
- Variable incline track
- Meter rule
- Stopwatch
- Spirit level
Method
- The track was set in the horizontal in it lowest position.
- The track was checked to be level using the spirit level and the adjustable screw feet.
- Any greases and matter was removed from the track. The distance to the white mark was checked to be 1m.
- The pin of the horizontal position was removed and the rack moved to the first incline position. The pin was then secured at this position
- The flywheel was placed at the white line. It was then released and the time taken for the flywheel to roll down the track to the end recorded. This was repeated to obtain average time.
- The same procedure was done for all other increments.
Theory
For each run of the track in the experiment, the acceleration of the flywheel is constant. The force on the flywheel due to gravity is given by Newton’s second law: Force=mass*acceleration
The flywheel starts from rest and thus:-
s=ut+12at2
Where
s=distance travelled (m)
u=initial velocity (m/s)
T=Duration of travel (s)
a=acceleration (m/s2)
At the top of the incline, the flywheel has potential energy, which is converted into kinetic energy as it accelerates during its travel down the slope.
For the flywheel used in this experiment, it is known that the transfer of potential to kinetic energy leads to the following expression:
g=2sk2r2+1∆t-2∆sinθ
For this experiment, s=1m and k2r2=50
∆t-2∆sinθcan be found by plotting a graph of 1average time squaredagainst Sinθ
Results
The equation for the line of best fit is given as
y=0.1179x-0.005, which means the gradient is 0.01179
The value of g is obtained by
g=2sk2r2+1*slope of the graph
g=2*150+1*0.1179=12.0258(m/s2)
Discussion
There possible sources of errors that might have affected the accuracy of the experiment. According to (Wagh and Deshpande, 2012) errors may be introduced when using the stopwatch. According to Sathyaseelan (2002), a simple pendulum experiment can also be used to determine the acceleration due to gravity.
Conclusion
The value of g obtained from the experiment was12.0258(m/s2), which is higher than the normal 9.81(m/s2). This has been attributed to errors that were experienced during the experiment.
SINGLE PLANE BALANCING LABORATORY
Objective
Apparatus
- Norwood dynamic balancing rig (NDBR)
- Weighing scales
- Masses
- Fixtures
- Rule
Method
- Three mass systems were produced 600g at a radius of 127mm; 860g at a radius of 103mm and 555g at a radius of 127mm
- A vector diagram was constructed on an A3 paper to the largest scale possible to determine the angles of vectors.
- The system was made using the masses fixed at the correct radii and angles. A 600g mass was placed at an angle of 400 to the 00 reference plane.
- The NDBR rig was then operated to indicate the system was in balance.
Theory
For a single plane rotating system to be in balance, the sum of the algebraic centripetal forces is zero. Since the angular velocity is the same, it becomes dependent on the forces based on the mass multiplied by the radius of each component.
mω2r=0
This reduces to
mr=0
Discussion
Balancing is important in rotating systems as it reduces vibrations during the operation of these systems (IMechE, 2004). Furthermore, balanced rotating systems have a higher precision in executing their functions. Genta (2007) notes that a system is balanced when all its components are balanced. A system may become unbalanced because of lack of frequent maintenance. For instance, centrifugal pumps may become unbalanced if not well maintained and serviced.
Conclusion
References
Gambhir, M. L. 2009. Fundamentals of solid mechanics: a treatise on strength of materials. New Delhi: PHI Learning.
Genta, G. 2007. Dynamics of rotating systems. New York: Springer.
Hanna, N., & Picciotto, R. 2002. Making Development Work: Development Learning in a World of Poverty and Wealth. Philadephia: Transaction Publishers.
IMechE, I. 2004. Eighth International Conference on Vibrations in Rotating Machinery, 7-9 September 2004, University of Wales, Swansea, UK. Bury St. Edmunds: Published by Professional Engineering Pub. for the Institution of Mechanical Engineers.
Myers, R. L. 2006. The basics of physics. Westport, Conn.: Greenwood Press.
Rajput, R. 2005. Basic mechanical engineering: (for B.E. 1st year - as per prescribed syllabus of Rajiv Gandhi Proudyogiki Vishwavidyalaya, Bhopal, M.P.) (2nd rev. ed.). Daryanganj, New Delhi: Laxmi Publications.
Sathyaseelan, H. 2002. Laboratory Manual in Applied Physics. New Delhi: New Age International.
Wagh, S. M., & Deshpande, D. A. 2012. Essentials of Physics, Volume 1. New Delhi: PHI Learning Pvt. Ltd.
