• An understanding of the fundamental concepts, laws and analytic methods for the

solution of applied mechanics problems

Skills

• Proficiency in the use of mechanical technology, and the ability to evaluate and

appraise the results of practical experiments.

1 Abstract

• A self-contained summary of what you did and your main findings

and conclusions.

• Maximum length: 200 words. You should include a Word count

at the end of the abstract.

10

2 Results

• Full and complete table of experimental readings.

• Well-presented and fully labelled graphs.

10

10

3 Analysis of results

Discuss briefly:

Experiment 1

• Show the relationship between deflection and beam material.

• Comparison of experimental and theoretical values of Young’s

modulus (E) for brass, aluminium and mild steel

Experiment 2

• Estimate the inertia of the flywheel by treating it as a solid steel disc.

Full working should be shown.

• Critically analyse the tabulated data and comment on your graph and

results and compare the value with the textbook value for the all three

beam materials.

10

4 Discussion

• A critical discussion of the accuracy of the results, which includes a

description of the main sources of error/inaccuracy.

5

5 Quality of presentation of the report

The use of an appropriate technical style for the presentation of the

report.

5

Total marks for each report 50

Page 4 of 13

Part A: MOMENT OF INERTIA OF A FLYWHEEL

During the course of the experiment a set of weights were hung in turn on a piece of cord

from a pulley attached to the flywheel. The objects were allowed to fall under gravity from a

height of 1 metre, as described in the laboratory sheet. This was performed three times and

the data recorded is shown in the table below. The diameter of the spindle d (= 2r) was found

to be 38 mm (0.038 m).

Total weight

(including hanger)

(N)

1

st descent

time

(s)

2

nd descent

time

(s)

3

rd descent

time

(s)

3.2 29.69 34.60 34.60

6.0 24.25 24.28 25.22

8.5 19.60 19.70 19.95

11.0 17.60 17.06 17.29

13.5 15.12 15.65 15.12

Complete the following tasks:

• The flywheel dimensions are as follows: diameter = 390mm, mean thickness

= 30mm. Also, it is made of steel, with density 7850 kg/m3

. Use this

information to estimate the inertia, by considering the formula for the inertia

of a solid disc. This value should be used as an approximate validation of the

inertia derived.

• Critically analyse the tabulated data and use it to determine the inertia of the

flywheel and the frictional torque Tf. You should use Excel to calculate and

tabulate relevant values, and to plot Pxr versus α, as per the laboratory sheet.

NB: do not confuse weight with mass (a common mistake!)

Part B: STIFFNESS OF DIFFERENT MATERIALS

During the experiment, three beams of different materials (Aluminium, Brass and Steel), but

of the same dimensions (9.5 mm x 3 mm), were tested to find their material stiffness values.

All steps to perform the complete experiment are given in the laboratory instruction sheet.

Tables 1 to 3 were obtained after completing the experiment.

Complete the following tasks:

• Complete tables 1 to 3 and obtain the values for equation 48δI / L3

.

Page 5 of 13

• Plot your load and equation 48δI / L3

results on the chart paper (as shown in

Figure 1).

• Find the gradient of the chart.

• Critically analyse the tabulated data and comment on your graph and results

and compare the value with the textbook value for the all three beam

materials.

Beam Modulus

Load (m)

(g)

Load (W)

(N)

Deflection

(mm)

Deflection (δ)

(m)

48δI / L3

0 0 0

100 0.98 0.65

200 1.96 1.16

300 2.94 1.73

400 3.92 2.23

500 4.9 2.74

Material: Aluminium

Beam Size: b= 9.5 mm d= 3 mm I = 2.1375 x 10-11 m4

Table 1: Results table for Aluminium.

Beam Modulus

Load (m)

(g)

Load (W)

(N)

Deflection

(mm)

Deflection (δ)

(m)

48δI / L3

0 0 0

100 0.98 0.36

200 1.96 0.80

300 2.94 1.06

400 3.92 1.41

500 4.9 1.75

Material: Brass

Beam Size: b= 9.5 mm d= 3 mm I = 2.1375 x 10-11 m4

Table 2: Results table for Brass.

Beam Modulus

Page 6 of 13

Load (m)

(g)

Load (W)

(N)

Deflection

(mm)

Deflection (δ)

(m)

48δI / L3

0 0 0

100 0.98 0.19

200 1.96 0.42

300 2.94 0.58

400 3.92 0.78

500 4.9 0.98

Material: Steel

Beam Size: b= 9.5 mm d= 3 mm I = 2.1375 x 10-11 m4

Table 3: Results table for Steel.

Figure 1: Blank results chart.

Laboratory instructions

Page 7 of 13

TITLE

EAT103: MOMENT OF INERTIA OF A FLYWHEEL

OBJECT

To determine the moment of inertia of a flywheel by accelerating it with a falling mass.

THEORY

APPARATUS

m

d

a

P

P

mg

,

Page 8 of 13

Flywheel, variety of masses m attached to length of cord.

m = falling mass, kg

h = distance fallen by mass, m

t = time of fall of mass, s

d = diameter of spindle, m

r = radius of spindle, m (= d/2)

a = acceleration of mass, m/s2

= angular velocity of flywheel when mass reaches ground, rad/s

= angular acceleration of flywheel with mass falling, rad/s2

P = tension in cord, N

g = gravitational acceleration = 9.81 m/s2

METHOD

1. For a particular mass, time the rate of descent and measure the distance fallen.

2. Use the formula given below in conjunction with Excel to determine the moment of inertia of

the flywheel, by plotting the straight line described.

3. Repeat steps 1-3 for a number of different masses (try at least five values of m).

4. Estimate the inertia using the formula for the flywheel of a solid disc, to enable you to check

that your experimental results are of the correct order of magnitude.

Considering acceleration of the flywheel:

T I

d

T = I P − f =

2

————————————- (1)

Where, Tf = friction torque due to bearings etc.

I = moment of inertia of flywheel and spindle

Newton’s 2nd law applied to falling mass: mg − P = ma ——- (2)

Page 9 of 13

From (s = ut + 0.5at2

), acceleration:

2

2

t

h

a = ———————— (3)

Thus: (3) into (2)

2

2

t

mh P = mg − ———————————- (4)

From (v = u + at), and using the expression for acceleration a given by equation (3), the speed of

mass m when it hits ground is:

t

h

v

2

= —————————- (5)

Since = v/r, from (5) the angular velocity of the flywheel when the mass hits ground is:

rt

2h

== ————————— (6)

Also, the angular acceleration of the flywheel as mass falls is:

t

= —————————- (7)

Thus, (6) into (7)

t r

h

2

2

= ————————— (8)

Equation (1) can be re-written as follows:

y mx c

P r I Tf

= +

= +

——————— (9)

In other words, if we plot P r (y-axis) versus (x-axis), we should get a straight line, where the

gradient is I, the inertia of the flywheel, and the intercept with the y-axis is Tf, the friction torque.

P is calculated using equation (4), and is calculated using equation (8).

TITLE

Page 10 of 13

EAT103: Stiffness of materials

OBJECT

To achieve an understanding of the stiffness of different materials under load.

THEORY

Stiffness is an important property for many applications of materials. The stiffness of a

structure depends on both the properties of the material from which it is made and on the

geometry of the structure.

Theory: Deflection of a simply supported beam.

This formula applies to beams which are subjected to a central load acting at a right

angle to their length.

The equation for the deflection of a beam supported between two points is given by:

Δ =

3

48

Where:

= Load (N)

Δ = deflection (m)

L = Length (m)

K = Flexural rigidity constant

K may be calculated due to the second moment of area of the material I, and the value of

Young’s modulus E

Page 11 of 13

= .

Some standard values of (E) for the following materials:

Brass 105 Gpa

Aluminium 69 GPa

Mild Steel 200 GPa

Second moment of area (I) for solid and hollow beams:

For a solid beam of rectangular cross section:

=

3

12

The units of I are m4

APPARATUS

The ES4 apparatus uses an example of a very simple structure, a beam in three point

bending. The apparatus explores that beams of different material but identical dimensions

(and I value) deflect by different amounts for the same load, providing that their material

property (Young’s Modulus) affects deflection.

The apparatus is very simple to use, the operator places the specimen into the apparatus

and applies a load. A dial indicator measures the deflection of the specimen at the point

of loading.

The apparatus is simple and demonstrated in Figure 1.

Page 12 of 13

Figure 1: Experiment Setup

METHOD

Page 13 of 13

Experimental Procedure – Beam Modulus

1. Find the three beams of different materials (Aluminium, Brass and Steel) but of

the same dimensions (9.5 mm x 3 mm). The aluminium beam weight less than

the other two and the brass beam has a golden colour.

2. For each beam, use the Dial Calliper Gauge (supplied with the kid) to accurately

measure the beam dimensions (b and d). Calculate the I values for the beam and

note this and the other beam properties into your results table 1-3.

3. Fit the aluminium beam to the supports.

4. Adjust the Dial indicator and Wire ‘stirrup’ along to the middle of the beam (180

mm).

5. Set and zero the Dial indicator.

6. In increments of 100 g, add weights up to a maximum of 500 g. Each time you

add a weight. Tap the work panel to reduce the effects of friction. Record the

deflection at each weight increment.

7. Calculate the load in Newton and calculate the results of the equation shown in

the table 1.

8. Rearranging the equation in the form y = mx+c means that the gradient of a chart

of W against

48 δI

3

will give the value of E.

9. Plot your load and equation results on the chart paper.

10.Find the gradient of the chart.

11.Compare the value with the textbook value for the beam material.

12. Repeat for the other two beams.

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