Determination of Young’s Modulus by Flextural Method

Tiltle: Determination of Young’s Modulus by Flextural Method.

Theory:

If a bar of breadth b and depth d is placed horizontally on two knife edges separated by a distance L, and a load of mass m, applied at the midpoint of the bar produces a depression x of the bar, then the Young’s modulus of the bar is given by the equation

\[Y=\frac{Wl^3}{4xbd^3}=\frac{mgl^3}{4xbd^3}…………………………….. (i)\]

where g is the acceleration due to gravity. This is the working formula of the experiment and is valid as long as the slope of the bar at any point with respect to the unstrained position is much less than unity. Here Y is determined by measuring the quantities b,d,L and the mean depression x(attainable from the graph) corresponding to a load m. In S.I. units Y is measured in Newton/m2.

Apparatus:

A rectangular bar AB of uniform cross-section and about one meter long is taken. The bar is placed horizontally and symmetrically on two steel knife-edges N1 and N2. The knife-edges in turn are mounted on two steady iron stands. At the mid-point of the bar is placed a light frame F containing a vertical pointer P and a scale pan or hanger S. Weights are placed on the hangers to depress the bar. A microscope, capable of sliding along a vertical scale, is conveniently placed in front of the pointer.

1) beam supported on two knife edge

2) half  kg  loads.

3) traveling microscope.

4) slide calipers.

5) screw gauge.

Procedure:

  • Measure the length of the given bar with a meter scale and mark its mid-point. Draw marks on the scale corresponding to some length L1 L2 / of the bar (say 100 cm).
  • Mount on the bar. Now place the bar with its least dimension vertical, on the knife-edges N1 and N2 such that the L1 L2 / marks coincide with the knife-edges. Mount a spirit level on the bar and adjust the leveling screws until the bar is horizontal.
  • Bring the knife-edge of the frame F on the central transverse mark of the bar. Place the microscope and view the pointer P. Adjust the leveling screws of the microscope until the vertical scale is perfectly vertical and the axis of the microscope is horizontal. Focus the eye-piece on the cross-wires by keeping one of the cross-wires horizontal. Focus the tip 2 of the pointer and adjust the vertical position of the microscope until the image of the tip of the pointer touches that of the horizontal cross-wire. As far as possible, avoid parallax.
  • Determine the vernier constant of the microscope. With zero loads on the hanger, record the position of the microscope on the vertical scale.
  • Place a load of 500 gm on the hanger. This will produce a depression of the bar. Alter the vertical position of the microscope until the image of pointer touches that of the horizontal cross-wire. Note again the vertical scale reading of the microscope. The difference of the two microscope readings gives the depression of the bar for the load of 500gm.
  • Increasing gradually the load in steps of 500 gm and at each observation, and at each step record the vertical scale reading of the microscope. Now decrease the load to zero in the same steps as used for increasing the load, and record the corresponding vertical scale readings of the microscope. Determine the mean of these two readings, and calculate the depression by subtracting the zero-load reading.
  • Remove the bar without disturbing the position of the stands, and measure accurately the distance between the knife-edges (i.e. L1 L2 / ) by placing vertically the marked face of a meter scale across the knife-edges.
  • Determine the vernier constant of the slide callipers and measure with it the breadth b of the bar at three different places. Calculate the mean breadth of the bar. Note the zero error, if any, of the slide callipers and find the correct value of b.
  • Determine the least count of the screw gauge and measure depth d of the bar at a number of places along the length of the bar. Find the mean value. Note the zero error, if any of the screw gauge and obtain the correct value of d.
  • Draw a graph with the load m in gm along the X-axis and the corresponding depression l in cm along the Y-axis and determine the value of Y.

Experimental results:

Determination of the vernier constant of slide calliper:

….. Divisions of the vernier scale = ………… divisions of the main scale.

Vernier Constant = (1-m.s.d/v.s.d.) x value of 1 smallest m.s.d.

Table for the measurment of the breath of the bar by slide calliper:

No of obs        Reading of b in cm Mean b in cm Instrumental error in cm Corrected b in cm
M.S.R V.S.R Total Reading      

Determination of the least count of screw gauge:

Table for the measurment of the depth of the bar by screw gauge:

No of obs        Readind of d in cm Mean d in cm Instrumental error in cm Corrected d in cm
M.S.R C.S.R Total Reading

 

Determination of the vernier constant of the travelling microscope:

….. Divisions of the vernier scale = ………… divisions of the main scale.

Vernier Constant = (1-m.s.d/v.s.d.) x value of 1 smallest m.s.d.

Table for load depression data:

Distance between the knife edge=100 cm

No

Of

obs

Load M (gm) Microscope reading for load increasing(a) Microscope reading for load increasing(b) Mean reading

(a+b)/2 in cm

Depression x in cm
MSR

(cm)

 

VSR

 

Total in cm

 

MSR

(cm)

 

 

VSR

Total in cm

 

1 0
2 500
3 1000

 

Determination of Y from load Depression Graph:

Value of (m) from graph Length(L) in cm Depression (x) from graph Y in dyne/cm2

Percentage Error:

\[\frac{\partial y}{Y}=\frac{\partial m}{m}+\frac{\partial g}{g}+\frac{3\partial l}{l}+\frac{\partial x}{x}+\frac{\partial b}{b}+\frac{3\partial d}{d}\]

\[\partial m=0, \partial g=0,\partial l=2\times0.1cm\]

∂x= 2* v.c  of  travelling mircoscope,

∂b= v.c of the slide calliperse,

∂d= least count of the screw gauge,

So Percentage error = \[\frac{\partial y}{Y}\times100%\]

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