Experimental determination of Young's Modulus by Cantilever depression

EXPERIMENTAL DETERMINATION OF YOUNG'S MODULUS BY CANTILEVER DEPRESSION

STATICAL METHOD

Description:

It consists of a beam clamped rigidly at one end on the table by the use of a G‒clamp. The weight hanger (H) is suspended at the other end of the beam, through a small groove on the beam as shown in Fig 1.11. A pin is fixed at the free end of the beam by means of a wax. A microscope (M) is placed in front of this arrangement for measuring the variation of height of the pin.

Procedure:

The weight hanger is kept hanged in a dead load position (W), (i.e) without any slotted weights.

The microscope is adjusted and the tip of the pin is made to coincide with the horizontal cross wire. The reading in the vertical scale of the microscope is noted.

Then the weights m, 2m, 3m, 4m, etc., are increased in steps to the weight hanger. Each time the microscope is adjusted to coincide the tip of the pin to the horizontal cross wire and the readings are noted from the vertical scale of the microscope.

The experiment is repeated by decreasing the loads and the respective readings are noted from the vertical scale of the microscope and are tabulated in the tabular column as shown. The mean depression 'y' for a load M kg is found.

Theoretically, we know that the depression produced for a load of M kg is

          .....(1)

Where ‘l’ is the length of the beam (i.e) the distance between the clamped end and the loaded end.

If 'b' is the breadth of the beam and 'd' is the thickness of the beam, then

The geometrical moment of inertia for a rectangular cross sectional bar I_g = bd³ / 12  ……….(2)

also the weight W = Mg           ……………(3)

Substituting eqn.(2) and eqn. (3) in eqn.(1), we have

…………….. (4)

Rearranging eqn.(4) we get

Youngs modulus

Substituting the mean value of M/y from the tabular column the youngs modulus of the given beam can be calculated.

GRAPHICAL METHOD (OR) DYNAMICAL METHOD

A graph is drawn by taking load (M) along x‒axis and depression (y) along y axis. It will be a straight line as shown in Fig. 1.12. The slope tweeted of the graph gives the value of y/M.

Eqn.(5) can also be written as

 Y= [ 4gl³ / bd³ ].  [ 1/slope ]

Substituting the value of the slope in the given formula, the youngs modulus of the beam can be calculated.