PH8151EPUnit1

1.11. UNIFORM BENDING AND NONUNIFORM BENDING

l

Load

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2

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y

W

Figure 1.33: Non-uniform bending - Equivalence to a pair of inverted cantilevers

of each inverted cantilever is equal to the depressionyat the midpoint of the bar. The
Young’s ModulusY is can be written from equation (1.34)as,

Y =

1

W

2

21

l

2

(^23)
3 yIg

(1.42)

For a beam of rectangular cross-section with breadthband thicknessd,

Y=

Mgl^3

48 y

1

bd^3

12

2

Therefore, the Young’s Modulus of a non-uniformly bent bar is given by,

Y =

Mgl^3

4 bd^3 y

(1.43)

Experimental Procedure

As shown in Figure1.32, support the given beam symmetrically on two knife edges with
lengthlbetween knife edges and suspend a loadLfrom the middle of the beam. A pin
is fixed vertically at the midpoint of the beam. Focus a travelling microscope to the tip
of the pin such that the horizontal cross-wire coincides with the image of pin-tip. Note
down the reading on the vertical scale of the microscope. Increase the load by adding 50
g. Lower the microscope such that the horizontal cross-wire now coincides with the new
position of pin tip image and note down the reading from vertical scale. This is repeated
by increasing the load in steps of 50 g tillL+250 gm and also for decreasing loads. The
readings are tabulated as shown (Figure1.34) and mean depression (y) for M= 0.05 kg
is calculated. Using a vernier calipers, the breadth (b) of the beam is determined. The
thickness (d) of the beam is measured using a screw gauge. Young’s Modulus of the beam
is then calculated from the expression:

Y =

Mgl^3

4 bd^3 y

PH8151 37 LICET