Young's Modulus Determination by Cantilever Method

DETERMINATION OF YOUNG'S MODULUS BY CANTILEVER METHOD

1. AIM

To determine the Young's modulus of elasticity of the material of a given beam using the cantilever method.

2. APPARATUS USED

Rectangular beam of the test material (typically steel, aluminum, or brass)
Rigid support with clamp for fixing one end of the beam
Micrometer screw gauge
Vernier caliper
Scale/ruler (meter scale)
Traveling microscope or dial gauge
Set of standard weights with hanger
Knife edge to support the weight
Spirit level

3. DIAGRAM

Oscillations of Mass with combination of two speings

Fig 1: Experimental setup for determination of Young's modulus by cantilever method

4. THEORY

A cantilever is a beam that is fixed at one end and free at the other. When a load is applied at the free end, the beam deflects. The amount of deflection depends on the material properties (specifically Young's modulus), the dimensions of the beam, and the applied load.

When a load (W) is applied at the free end of a cantilever beam, the beam deflects. According to the beam theory, for small deflections, the relationship between the deflection (y) at the free end and the applied load (W) is given by:

\[y = \frac{W \cdot L^3}{3 \cdot E \cdot I}\]

Where:

\(y\) is the deflection at the free end
\(W\) is the applied load
\(L\) is the effective length of the cantilever (distance from fixed end to point of load application)
\(E\) is Young's modulus of elasticity
\(I\) is the moment of inertia of the cross-section of the beam

For a rectangular cross-section, the moment of inertia is given by:

\[I = \frac{b \cdot h^3}{12}\]

Where:

\(b\) is the width of the beam
\(h\) is the thickness (height) of the beam in the direction of bending

5. FORMULA

Rearranging the equation to find Young's modulus:

\[E = \frac{W \cdot L^3}{3 \cdot y \cdot I}\]

Substituting the expression for \(I\):

\[E = \frac{4 \cdot W \cdot L^3}{b \cdot h^3 \cdot y}\]

Where:

\(E\) is Young's modulus in N/m² or Pa
\(W\) is the load in N
\(L\) is the effective length of the cantilever in m
\(b\) is the width of the beam in m
\(h\) is the thickness of the beam in m
\(y\) is the deflection at the free end in m

6. PROCEDURE

Measure the dimensions of the beam (length, width, and thickness) using vernier caliper.
Fix one end of the beam firmly in the support clamp to create a cantilever setup.
Use a spirit level to ensure the beam is horizontal.
Measure the effective length (L) of the cantilever from the fixed end to the point where the load will be applied.
Set up the measuring device (traveling microscope or dial gauge) to measure the deflection at the free end.
Record the initial reading of the measuring device without any load.
Place the hanger at the free end of the cantilever.
Add weights to the hanger in increments (e.g., 50g, 100g, 150g, etc.).
For each weight, record the new reading on the measuring device.
Calculate the deflection for each load by finding the difference between the loaded and unloaded readings.
Repeat steps 8-10 for at least 5 different weights.
After taking all readings with increasing weights, repeat the process with decreasing weights to check for hysteresis.
Calculate Young's modulus using the formula for each set of readings and find the average value.

7. OBSERVATION TABLE

Beam Dimensions:

Length of the beam (L): _____ m
Width of the beam (b): _____ m
Thickness of the beam (h): _____ m

Deflection Measurements:

S.No.	Mass (m) kg	Weight (W = m×g) N	Scale Reading (mm)	Deflection (y) m	W·L³/(3·I·y) (N/m²)
1
2
3
4
5

8. CALCULATIONS

1. Calculate the moment of inertia (I) of the beam:

\[I = \frac{b \cdot h^3}{12}\]

2. For each observation, calculate Young's modulus using the formula:

\[E = \frac{W \cdot L^3}{3 \cdot I \cdot y}\]

\[E = \frac{4 \cdot W \cdot L^3}{b \cdot h^3 \cdot y}\]

3. Find the average value of Young's modulus from all observations:

\[E_{avg} = \frac{E_1 + E_2 + E_3 + ... + E_n}{n}\]

9. RESULT

The Young's modulus of elasticity of the material of the given beam is _____ × 10^__ N/m² or Pa.

10. PRECAUTIONS

Ensure that the beam is properly clamped at one end to prevent any movement.

Verify that the beam is perfectly horizontal using a spirit level before starting the experiment.

Apply loads gradually to avoid impact or sudden loading.

Ensure that the deflection measuring device is placed at the exact point where the load is applied.

Keep the deflections small (preferably less than 1/50 of length) to maintain the validity of the beam theory.

Avoid exceeding the elastic limit of the material to prevent permanent deformation.

Make sure the knife edge supporting the weight is sharp and placed correctly.

Take readings for both increasing and decreasing loads to check for hysteresis.

Measure the dimensions of the beam accurately, as small errors in thickness can lead to significant errors in results.

Avoid vibrations during measurements.

11. VIVA VOCE QUESTIONS

Q1: What is Young's modulus? What are its units?

Q2: What is the difference between stress and strain?

Q3: Why is the cantilever method suitable for determining Young's modulus?

Q4: What changes would you observe if the load exceeds the elastic limit of the material?

Q5: How does the thickness of the beam affect its deflection?

Q6: What are the sources of error in this experiment?

Q7: What is the significance of Young's modulus in engineering applications?

Q8: Can this method be applied to all materials? Why or why not?

Q9: How would you modify this experiment to measure Young's modulus of a wire?

Q10: Why does the formula contain the cube of the length (L³)?

Q11: What happens to the deflection if the width of the beam is doubled?

Q12: What happens to the deflection if the thickness of the beam is doubled?

Q13: What is the relationship between Young's modulus and the stiffness of a material?

Q14: Explain the concept of elastic deformation in terms of atomic bonding.

Q15: How would temperature changes affect the Young's modulus measurement?

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