Effect of Load on Depression of a Metre Scale

Effect of Load on Depression of a Metre Scale

Aim

To study the effect of load on the depression of a suitably clamped metre scale when loaded at:

  1. Its free end
  2. The middle point

Apparatus Required

  • Metre scale
  • Weight hanger
  • Slotted weights (50g, 100g, etc.)
  • Half-metre scale or a reference scale
  • Clamp stand with adjustable clamp
  • Screw gauge (optional, for measuring thickness)
  • Graph paper

Theory

When a beam is fixed at one end and loaded at the other end (or at some point along its length), it bends or sags. This depression or deflection depends on several factors:

For a scale loaded at the free end:

$$y = \frac{Fl^3}{3EI}$$

For a scale loaded at the center:

$$y = \frac{Fl^3}{48EI}$$

Where:

  • $y$ = Depression/deflection of the scale
  • $F$ = Applied force (load)
  • $l$ = Length of the scale from the fixed end to the point of application of load
  • $E$ = Young's modulus of the material of the scale
  • $I$ = Moment of inertia of the cross-section of the scale

For a rectangular cross-section:

$$I = \frac{bd^3}{12}$$

Where:

  • $b$ = Width of the scale
  • $d$ = Thickness of the scale

From these equations, we can deduce that:

$$y \propto F$$

Thus, the depression of the scale is directly proportional to the applied load.

Experimental Setup

Experimental setup showing the metre scale clamped at one end

Procedure

Part I: Load at the Free End

  1. Clamp the metre scale firmly at one end to the table such that about 80 cm of the scale projects horizontally.
  2. Place a half-metre scale vertically near the free end to measure the depression.
  3. Note the initial position (reading) of the free end on the vertical scale.
  4. Hang a weight hanger at the free end of the metre scale.
  5. Add weights in steps (50g, 100g, 150g, etc.) and record the corresponding position of the free end on the vertical scale.
  6. Calculate the depression for each load by finding the difference between the new position and the initial position.
  7. Tabulate your observations.
  8. Remove the weights gradually and check if the scale returns to its original position.

Part II: Load at the Middle

  1. Keep the same setup with the metre scale clamped at one end.
  2. Place the vertical reference scale below the middle point of the horizontal metre scale.
  3. Note the initial position of the middle point on the vertical scale.
  4. Hang the weight hanger at the middle point of the metre scale.
  5. Add weights in steps and record the corresponding position of the middle point on the vertical scale.
  6. Calculate the depression for each load.
  7. Tabulate your observations.
  8. Remove the weights gradually and check if the scale returns to its original position.

Observations

Measurement of Physical Parameters

  • Length of the metre scale projecting from the clamp ($l$) = ______ cm
  • Width of the metre scale ($b$) = ______ cm
  • Thickness of the metre scale ($d$) = ______ cm

Table 1: Load at the Free End

S.No. Mass (m) in g Weight (F=mg) in N Initial Position (cm) Final Position (cm) Depression (y) in cm
1 0 0 0
2 50 0.49
3 100 0.98
4 150 1.47
5 200 1.96

Table 2: Load at the Middle

S.No. Mass (m) in g Weight (F=mg) in N Initial Position (cm) Final Position (cm) Depression (y) in cm
1 0 0 0
2 50 0.49
3 100 0.98
4 150 1.47
5 200 1.96

Calculations

For Load at the Free End

According to the theory of elasticity, for a cantilever beam with a point load at the free end:

$$y = \frac{Fl^3}{3EI}$$

This can be rearranged as:

$$y = K \times F$$

Where $K = \frac{l^3}{3EI}$ is a constant for a given setup.

For Load at the Middle

For a cantilever beam with a point load at the middle:

$$y = \frac{F(l/2)^2(2l - l/2)}{6EI} = \frac{7Fl^3}{48EI}$$

Again, this can be written as:

$$y = K' \times F$$

Where $K' = \frac{7l^3}{48EI}$ is another constant.

To verify the direct proportionality between depression and load, plot a graph of depression (y) vs. load (F) for both cases. If the relationship is linear, it confirms that $y \propto F$.

Graph

Graph 1: Load at Free End

Graph showing depression vs. load for weight at free end

Graph 2: Load at Middle

Graph showing depression vs. load for weight at middle point

Result and Analysis

From the graphs:

  1. For load at the free end:
    • The graph between depression (y) and load (F) is a straight line passing through the origin.
    • This confirms that $y \propto F$.
    • The slope of this line represents the constant $K = \frac{l^3}{3EI}$.
  2. For load at the middle:
    • The graph between depression (y) and load (F) is again a straight line passing through the origin.
    • This confirms that $y \propto F$ in this case as well.
    • The slope of this line represents the constant $K' = \frac{7l^3}{48EI}$.

Theoretically, the ratio of depressions for the same load should be:

$$\frac{y_{\text{free end}}}{y_{\text{middle}}} = \frac{K}{K'} = \frac{l^3/3EI}{7l^3/48EI} = \frac{16}{7}$$

Calculate this ratio from your experimental data and compare it with the theoretical value.

Note: The experimental ratio might differ from the theoretical value due to:
  • Measurement errors
  • Non-uniform cross-section of the metre scale
  • Imperfect clamping
  • Elastic limit effects

Discussion and Conclusion

The experiment verifies the following:

  1. The depression of a clamped metre scale is directly proportional to the applied load, both when the load is applied at the free end and at the middle.
  2. The depression for the same load is greater when the load is applied at the free end compared to when it is applied at the middle.
  3. The theoretical ratio of depressions (free end : middle) is 16:7, which was experimentally verified to be approximately _______ (fill in the value calculated from your data).

Conclusion: The study confirms that the depression of a clamped beam (metre scale) follows the mathematical relationship $y \propto F$ and the distribution of load significantly affects the magnitude of depression.

Applications:

  • Understanding the bending of beams in construction and engineering
  • Designing bridges and cantilever structures
  • Determining the elastic properties of materials

Sources of Error:

  • Improper clamping of the metre scale
  • Non-uniform cross-section of the metre scale
  • Errors in reading the position of the scale
  • Temperature effects on the elasticity of the material
  • The scale might not be perfectly horizontal initially

Precautions:

  • Ensure firm clamping of the metre scale to avoid any movement at the fixed end
  • Add and remove weights gently to avoid any sudden jerks
  • Make sure the scale is elastic and returns to its original position when loads are removed
  • Take readings carefully and avoid parallax errors
  • Do not exceed the elastic limit of the metre scale
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