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- The Bending of a Cantilever

- Physics
- Lab Report
- Undergraduate
- Pages: 3 (750 words)
- November 11, 2019

The Torsional Pendulum Deflection of cantilever beam has beam measured by varying load and the length of the cantilever beam. The measured values of deflection were used to calculate Young’s Modulus (E) of steel. It was found that deflection of a cantilever beam is directly proportional of the applied load and cube of the length of the cantilever. These measurements provided the value of Young’s modulus of steel to be 203 GPa (Fixed length cantilever) and 174 GPa (Fixed load cantilever). These values are very close to that reported in the literature. The experiment and the calculations are described in this report.

Objective

The experiment was aimed at measuring Young’s Modulus (E) of a material by measuring deflection of a cantilever with varying applied load or length of the cantilever.

Theory and Formulae:

A long and thin bar clamped at one end is termed as cantilever (figure below).

When load is applied at the free end, it deflects. The vertical deflection of a cantilever subjected to a load W is theoretically given by the formula

Here, a and b are width and thickness of the bar respectively; L is length and E is Young’s Modulus of the material. Ideally one should incorporate the correction for deflection of the cantilever under its own weight. It may be ignored if contribution from its weight in the total deflection is small. Thus by measuring deflection of a cantilever, one can measure Young’s Modulus of the material forming the cantilever.

Materials and Procedure:

Metallic strips of different lengths – 50, 60, 70 and 80 cm and 12.8 mm width and 4.8 mm thickness were taken as Cantilever. Three different weights of 0.5, 1 and 2 kg were used to cause deflection in the cantilever.

For foxed length 60 cm of cantilever, vertical deflection was measured for different weights of 0.5, 1 and 2 kg and the same was recorded. The deflection was plotted vs. the weight and from the slope Young’s Modulus (E) was calculated.

For fixed weight of 1 kg, vertical deflection was measured for cantilevers of different lengths – 50, 60, 70 and 80 cm and the same was recorded. The deflection was plotted vs. L3 and from the slope Young’s Modulus (E) was calculated.

Data and Calculations

1) Fixed Length Case

Length of the Cantilever L = 60 cm = 0.60 m

Width of the Cantilever a = 12.8 mm = 0.0128 m

Thickness of the Cantilever b = 4.8 mm = 0.0048 m

Deflection vs. Load Data

Load (kg)

0.5

1

2

Deflection (m)

2*10-2

3.5*10-2

6.5*10-2

Chart Showing Deflection vs. Load Plot

Slope of the deflection vs. load line is 0.03 m/kg

Therefore, Young’s Modulus (E) will be given by

GPa

Error Analysis:

The first calculation was slope of the line (m) - vs W

Error for this (m) will be given as

Finally error for the Young’s Modulus E will be given as

2) Fixed Load Case

Applied Load W = 1 kg

Width of the Cantilever a = 12.8 mm = 0.0128 m

Thickness of the Cantilever b = 4.8 mm = 0.0048 m

Deflection vs. Load Data

Length (m)

0.5

0.6

0.7

0.8

Deflection (m)

2*10-2

3*10-2

6*10-2

8*10-2

Chart Showing Deflection vs. L3 Plot

Slope of the deflection vs. load line is 0.0162 m/m3

Therefore, Young’s Modulus (E) will be given by

GPa

Error Analysis:

The first calculation was slope of the line (m) - vs L3

Error for this (m) will be given as

Finally error for the Young’s Modulus E will be given as

Conclusions:

Based on these experiments it can be concluded that deflection of a cantilever beam is directly proportional to the applied load and cube of the length of the cantilever. This is a very simple method of finding young’s modulus of a material. The results btained are very close to the value reported in literature and different books (~ 210 GPa). This shows how one can use a method as siple as cantilever deflection to measure a very important material property.

Objective

The experiment was aimed at measuring Young’s Modulus (E) of a material by measuring deflection of a cantilever with varying applied load or length of the cantilever.

Theory and Formulae:

A long and thin bar clamped at one end is termed as cantilever (figure below).

When load is applied at the free end, it deflects. The vertical deflection of a cantilever subjected to a load W is theoretically given by the formula

Here, a and b are width and thickness of the bar respectively; L is length and E is Young’s Modulus of the material. Ideally one should incorporate the correction for deflection of the cantilever under its own weight. It may be ignored if contribution from its weight in the total deflection is small. Thus by measuring deflection of a cantilever, one can measure Young’s Modulus of the material forming the cantilever.

Materials and Procedure:

Metallic strips of different lengths – 50, 60, 70 and 80 cm and 12.8 mm width and 4.8 mm thickness were taken as Cantilever. Three different weights of 0.5, 1 and 2 kg were used to cause deflection in the cantilever.

For foxed length 60 cm of cantilever, vertical deflection was measured for different weights of 0.5, 1 and 2 kg and the same was recorded. The deflection was plotted vs. the weight and from the slope Young’s Modulus (E) was calculated.

For fixed weight of 1 kg, vertical deflection was measured for cantilevers of different lengths – 50, 60, 70 and 80 cm and the same was recorded. The deflection was plotted vs. L3 and from the slope Young’s Modulus (E) was calculated.

Data and Calculations

1) Fixed Length Case

Length of the Cantilever L = 60 cm = 0.60 m

Width of the Cantilever a = 12.8 mm = 0.0128 m

Thickness of the Cantilever b = 4.8 mm = 0.0048 m

Deflection vs. Load Data

Load (kg)

0.5

1

2

Deflection (m)

2*10-2

3.5*10-2

6.5*10-2

Chart Showing Deflection vs. Load Plot

Slope of the deflection vs. load line is 0.03 m/kg

Therefore, Young’s Modulus (E) will be given by

GPa

Error Analysis:

The first calculation was slope of the line (m) - vs W

Error for this (m) will be given as

Finally error for the Young’s Modulus E will be given as

2) Fixed Load Case

Applied Load W = 1 kg

Width of the Cantilever a = 12.8 mm = 0.0128 m

Thickness of the Cantilever b = 4.8 mm = 0.0048 m

Deflection vs. Load Data

Length (m)

0.5

0.6

0.7

0.8

Deflection (m)

2*10-2

3*10-2

6*10-2

8*10-2

Chart Showing Deflection vs. L3 Plot

Slope of the deflection vs. load line is 0.0162 m/m3

Therefore, Young’s Modulus (E) will be given by

GPa

Error Analysis:

The first calculation was slope of the line (m) - vs L3

Error for this (m) will be given as

Finally error for the Young’s Modulus E will be given as

Conclusions:

Based on these experiments it can be concluded that deflection of a cantilever beam is directly proportional to the applied load and cube of the length of the cantilever. This is a very simple method of finding young’s modulus of a material. The results btained are very close to the value reported in literature and different books (~ 210 GPa). This shows how one can use a method as siple as cantilever deflection to measure a very important material property.