Comprehensive Notes on Deforming Solids, Hooke's Law, and Spring Extension Experiments

Deforming Solid Materials

• Deforming Forces: Forces can change the size and shape of solid objects by stretching, squashing, bending, or twisting them.

• Categorization of Deforming Forces (demonstrated using a foam rubber cylinder):

  • Tensile forces: Applied forces that stretch an object, increasing its overall length.

  • Compressive forces: Applied forces that compress or squash an object, decreasing its length.

  • Bending forces: Applied forces that arch or curve an object.

  • Torsional forces: Applied forces that twist an object around its central axis.

• Elastic Deformation: the

  • Definition: Deformation in which an object returns to its original shape and dimensions when all deforming forces are removed. Foam rubber is ideal for investigating deformation because it springs back to its original shape upon unloading.

  • Kicked Football Example: When a football is kicked, it undergoes temporary compression during impact. It then springs back to its original spherical shape as it pushes off the player's boot. The player's boot also compresses slightly, but because it is stiffer than the ball, the effect is less noticeable.

  • Tennis Ball Example: A tennis ball compresses temporarily when struck by a tennis racket before returning to its original shape.

  • Bungee Jumping Example: Bungee jumping relies on the springiness of a heavy rubber rope to break a jumper's fall. As the bungee cord reaches its maximum extension, it exerts a restoring force that pulls the jumper upward. This springiness causes the jumper to bounce up and down, coming to a halt gradually. If the cord stretched permanently, the jumper would stop abruptly at the bottom of the fall instead of bouncing.

• Inelastic (Permanent) Deformation:

  • Definition: Deformation in which an object undergoes a permanent change in shape and fails to return to its original length or shape when applied forces are removed.

  • Car Collision Example: When two cars collide, the metal bodywork panels bend permanently.

  • Precious Metals Hammering Example: Precious metals such as gold and silver are relatively soft at room temperature and can be hammered into desired shapes (such as ornaments, rings, and wrist bands) without heating. Tibetan silversmiths have utilized this property for thousands of years.

Stretching Springs and Hooke's Law

• Spring Characteristics: Springs are engineered to stretch significantly when small forces are applied, allowing precise measurements of length changes.

• Fundamental Terminology and Equation:

  • Load: The force (typically weight) applied to stretch an object such as a spring.

  • Extension: The increased length of an object beyond its original length when a load is applied.

  • Equation for Stretched Length:     $\text{length of stretched spring} = \text{original length} + \text{extension}$

• Elastic versus Inelastic Behavior in Springs:

  • Doubling the load stretching a spring does not double the total length of the spring; it doubles the extension.

  • Within the elastic region, increasing the load in regular steps increases the length in regular steps. Removing the load allows the spring to return completely to its original length.

  • If the load is increased too far, the spring becomes permanently stretched and will not return to its original length. It has undergone inelastic deformation and permanent damage.

• Hooke's Law:

  • First described mathematically by British scientist Robert Hooke.

  • Hooke established that the extension of a spring is directly proportional to the stretching load: doubling the load doubles the extension, and tripling the load triples the extension ($3 \times \text{load} = 3 \times \text{extension}$).

  • Graph of Load against Extension:

  • Linear Region: At first, the graph is a straight line passing through the origin $(0,0)$, showing that extension is directly proportional to load.

  • Curved Region: At higher loads, the line curves and slopes upward less steeply, indicating that the spring is permanently damaged.

  • Unloading Behavior: When an overstretched spring is unloaded, the load-extension line does not return to zero extension, indicating permanent structural elongation.

• Limit of Proportionality:

  • Definition: The maximum point on a load-extension graph up to which the extension of a spring remains directly proportional to the load.

  • Beyond this point, the spring suffers permanent damage and will remain permanently elongated even after the load is completely removed.

Experimental Setup and Methods for Stretching Springs

• Engineering Context and Significance:

  • Understanding material behavior under applied forces is essential in engineering and structural design.

  • RMS Titanic Example: The hull of the RMS Titanic was constructed of mild steel plates held together by $3 \times 10^6$ ($3\,\text{million}$) rivets. When the ship struck an iceberg in $1912$, the impact forces sheared several of these rivets, contributing directly to the ship sinking.

• Objective: To investigate how the extension of a spring changes as load is increased and determine whether extension is directly proportional to load.

• Required Apparatus:

  • Eye protection

  • Clamp stand, boss, and clamp

  • Steel spring

  • Mass hanger with slotted masses ($100\,g$ each)

  • G-clamp

  • Measuring ruler

  • Plumb line (a thread with a lump of metal on the end)

• Laboratory Safety Precautions:

  • Eye protection must be worn because a spring under tension can snap and fly into someone's eye.

  • Place a mat on the floor directly beneath the masses so that if the spring snaps, falling masses will not damage the floor.

  • Avoid standing on the mat because falling masses could land on feet if the spring breaks.

• Step-by-Step Experimental Procedure:

  1. Secure the clamp stand to the laboratory bench using a G-clamp. Fix the boss and clamp to the stand.

  2. Suspend the spring vertically from the clamp.

  3. Mount the ruler vertically beside the spring with the $0\,cm$ mark at the top, so ruler readings increase as the spring stretches downward. Align the ruler vertically using a plumb line.

  4. Attach the empty mass hanger to the bottom of the spring.

  5. Record the load as $0\,N$ and record the baseline ruler reading where it lines up with the bottom of the hanger.

  6. Add a $100\,g$ slotted mass (representing an applied load of $1.0\,N$) to the hanger and record the new ruler reading at the bottom marker.

  7. Remove the mass and record whether the steel spring returns to its original baseline length.

  8. Repeat steps 6 and 7, adding another $100\,g$ mass ($1.0\,N$ load) each time, recording ruler readings and verifying return to original length until the data table is filled or the spring breaks.

  9. Calculate spring extension by subtracting the baseline ruler reading for a load of $0\,N$ from all subsequent ruler readings. The calculated extension should be $0\,cm$ at $0\,N$ load.

  10. Plot a load-extension graph with load on the vertical axis ($N$) and extension on the horizontal axis ($cm$), including axis labels, title, and line of best fit.

Analysis of Glass Fibers and Solid Deformation

• Elasticity in Glass Materials:

  • Glass is fundamentally a brittle material that shatters once it reaches its limit of proportionality.

  • Thin glass fibers (such as those used in optical fibers and loft insulation) can bend easily into circles without shattering due to their small cross-sectional thickness.

• Mechanics of Bending Solids:

  • When a solid object (such as a pencil eraser or textbook) is bent into an arch shape:

  • The convex outer surface stretches (tensile strain).

  • The concave inner surface compresses and gets shorter (compressive strain).

• Comparison: Textbook versus Solid Block of Paper:

  • Textbook Experiment: A textbook with a bottom width of approximately $21\,cm$ bent into an arch shape keeps its left-hand spine at right angles to both front and back covers. The individual pages slide past each other, ensuring that the length of all pages and covers remains unchanged ($21\,cm$).

  • Solid Block Behavior: If a solid, bound block of paper (such as a stack in its wrapper) is bent into an arch, the pages cannot slide past one another. The top surface stretches and the bottom surface compresses, while the right-hand edge stays at right angles to the top and bottom surfaces, removing a small triangular section of material at the end relative to sliding pages.

Experimental Data Tables

• Table 5.1: Results from Spring Stretching Experiment:

  • Load $0.0\,N$: Length $24.0\,cm$, Extension $0.0\,cm$

  • Load $1.0\,N$: Length $24.6\,cm$, Extension $0.6\,cm$

  • Load $2.0\,N$: Length $25.2\,cm$, Extension $1.2\,cm$

  • Load $3.0\,N$: Length $25.8\,cm$, Extension $1.8\,cm$

  • Load $4.0\,N$: Length $26.4\,cm$, Extension $2.4\,cm$

  • Load $5.0\,N$: Length $27.0\,cm$, Extension $3.0\,cm$

  • Load $6.0\,N$: Length $27.6\,cm$, Extension $3.6\,cm$

  • Load $7.0\,N$: Length $28.6\,cm$, Extension $4.6\,cm$

  • Load $8.0\,N$: Length $29.5\,cm$, Extension $5.6\,cm$

  • Data Analysis: From $0.0\,N$ to $6.0\,N$, extension increases in equal steps of $0.6\,cm$ per $1.0\,N$ load. Beyond $6.0\,N$, the steps become larger ($1.0\,cm$ and $0.9\,cm$), showing that the spring has passed its limit of proportionality and suffered permanent inelastic deformation.

• Table 5.2: Results from Elastic Cord Stretching Experiment:

  • Load $0\,N$: Length $75\,cm$, Extension $0\,cm$

  • Load $2\,N$: Length $81\,cm$, Extension $6\,cm$

  • Load $4\,N$: Length $87\,cm$, Extension $12\,cm$

  • Load $6\,N$: Length $93\,cm$, Extension $18\,cm$

  • Load $8\,N$: Length $99\,cm$, Extension $24\,cm$

  • Load $10\,N$: Length $105\,cm$, Extension $30\,cm$

  • Load $12\,N$: Length $118\,cm$, Extension $43\,cm$

  • Load $14\,N$: Length $135\,cm$, Extension $60\,cm$

  • Load $16\,N$: Length $156\,cm$, Extension $81\,cm$

  • Data Analysis: Extension increases linearly by $6\,cm$ per $2\,N$ load up to $10\,N$. Beyond $10\,N$, increments increase rapidly ($13\,cm$, $17\,cm$, $21\,cm$), showing the limit of proportionality is reached at $10\,N$.

Questions and Numerical Problems

• Elastic Cord Extension Calculation Problem:

  • Question: A piece of elastic cord is $75\,cm$ long. When it is stretched, its length increases to $97\,cm$. What is its extension?

  • Calculation:     $\text{Extension} = \text{Stretched Length} - \text{Original Length}$     $\text{Extension} = 97\,cm - 75\,cm = 22\,cm$

• Question 1a (Experimental Skills 5.1): Explain the purpose of the G-clamp.

  • Answer: The G-clamp secures the clamp stand firmly to the laboratory bench, preventing the apparatus from tipping over when heavy loads are suspended from the spring.

• Question 1b (Experimental Skills 5.1): Explain the purpose of the plumb line.

  • Answer: The plumb line provides a true vertical baseline to ensure that the ruler and suspended spring are oriented completely vertical.

• Question 2 (Experimental Skills 5.1): Each slotted weight is $100\,g$. Calculate what load this represents.

  • Calculation:     $\text{Mass } m = 100\,g = 0.100\,kg$     $\text{Load Force } F = m \times g = 0.100\,kg \times 10\,N/kg = 1.0\,N$

• Question 3 (Experimental Skills 5.1): Identify the independent (input) and dependent (output) variables.

  • Independent (Input) Variable: Applied load ($N$).

  • Dependent (Output) Variable: Ruler reading / Spring extension ($cm$).

• Question 4 (Experimental Skills 5.1): Graph Features and Proportionality Identification:

  • Origin Check: The line of best fit passes through the origin $(0,0)$ after correcting for the original length of the spring by subtracting the baseline ruler reading at $0\,N$ load.

  • Linear Region Identification: The straight-line section extending from the origin identifies the range where extension is directly proportional to applied load.

  • Limit of Proportionality Identification: The point where the graph bends and ceases to be a straight line is the limit of proportionality. For Table 5.1, this limit occurs at a load of $6.0\,N$ and an extension of $$3.6\,cm$.

• Question 5 (Experimental Skills 5.1): Spring Behavior Beyond Limit of Proportionality:

  • Answer: Beyond the limit of proportionality, the spring does not return to its original length when unloaded, demonstrating permanent inelastic deformation.