PH8151EPUnit1

1.10. CANTILEVER: THEORY AND EXPERIMENT

whereY is the Young’s Modulus of the material of the cantilever and the geometric
moment of inertiaIg=(bd^3 )/ 12 (for a beam of rectangular cross-section).

Moment of the weightWabout the point P =W(l≠x) (1.32)

At equilibrium, since the moment of the external bending couple (equation1.32) is bal-
anced by the internal bending moment (equation1.31),

W(l≠x)=

YIg

(l≠x)

dy

dx

Rearranging,

dy=

W

YIg

(l≠x)^2 dx (1.33)

The total depressionyat the end of the cantilever is obtained by integrating the expression
(1.33) for the entire length (from 0 tol) of the cantilever.

y=

⁄y

0

dy=

⁄l

0

W

YIg

(l≠x)^2 dx=

W

YIg

⁄l

0

(l^2 +x^2 ≠ 2 lx)dx=

W

YIg

C

l^2 x+

x^3

3

≠lx^2

Dl

0

Hence,

y=

Wl^3

3 YIg

(1.34)

IfW=mgandIg=(bd^3 )/ 12 ,

y=

(mg)l^3

3 Y

(^1) (bd (^3) )
12
(^2) =
4 Mgl^3
bd^3 Y
The Young’s Modulus of the cantilever is then given by:

Y =

4 Mgl^3

bd^3 y

(1.35)

Experimental Procedure

Fix the given beam horizontally at one end and suspend a loadLfrom the free end as
shown in Figure1.26. Length of the cantileverlis measured. A pin is fixed vertically at
the free end 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 procedure is carried out repeated by
increasing the loads in steps of 50 g tillL+250g. From this fully loaded position, this
procedure is carried out also for decreasing loads and readings noted.

The readings are tabulated as shown in Figure 1.27 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 =

4 Mgl^3

bd^3 y

PH8151 31 LICET