Explain whether or not each material obeys Hooke's law. (a sketch graph in each case may be necessary) (4 marks) Define elastic limit and state its value for the wire. (2 marks)

Here are the solutions to the problems:

Problem a) i):
This part involves calculations based on the provided graph, which plots pressure (in units of $10^5 \text{ Nm}^{-2}$) against depth (in meters).

• Step 1: Calculate the gradient of the graph. We use the two points indicated on the graph for the gradient calculation: $(1.4 \text{ m}, 1.2 \times 10^5 \text{ Nm}^{-2})$ and $(6.6 \text{ m}, 1.55 \times 10^5 \text{ Nm}^{-2})$.

  $$\text{Gradient} = \frac{\Delta P}{\Delta h} = \frac{(1.55 \times 10^5 \text{ Nm}^{-2}) - (1.2 \times 10^5 \text{ Nm}^{-2})}{6.6 \text{ m} - 1.4 \text{ m}}$$ $$\text{Gradient} = \frac{(1.55 - 1.2) \times 10^5 \text{ Nm}^{-2}}{5.2 \text{ m}}$$ $$\text{Gradient} = \frac{0.35 \times 10^5 \text{ Nm}^{-2}}{5.2 \text{ m}}$$ $$\text{Gradient} = \frac{35000 \text{ Nm}^{-2}}{5.2 \text{ m}}$$ $$\text{Gradient} \approx 6730.77 \text{ Nm}^{-3}$$

  The gradient is $\boxed{\text{6730.77 Nm}^{-3}}$.

• Step 2: Determine the pressure when depth is 0m. From the graph, when the depth is $0 \text{ m}$, the line intercepts the pressure axis (y-axis) at a value of 1.0. Since the y-axis is scaled by $10^5 \text{ Nm}^{-2}$, the pressure at $0 \text{ m}$ depth is $1.0 \times 10^5 \text{ Nm}^{-2}$. This value represents the atmospheric pressure at the surface. The pressure when depth is $0 \text{ m}$ is $\boxed{1.0 \times 10^5 \text{ Nm}^{-2}}$.

• Step 3: Calculate the density of the liquid. The gradient of a pressure-depth graph ($\Delta P / \Delta h$) represents $\rho g$, where $\rho$ is the density of the liquid and $g$ is the acceleration due to gravity. We have $\rho g = 6730.77 \text{ Nm}^{-3}$. Assuming $g = 10 \text{ N kg}^{-1}$ (or $10 \text{ m s}^{-2}$):

  $$\rho = \frac{6730.77 \text{ Nm}^{-3}}{10 \text{ N kg}^{-1}}$$ $$\rho = 673.077 \text{ kg m}^{-3}$$

  Rounding to two decimal places:

  $$\rho \approx 673.08 \text{ kg m}^{-3}$$

  The density of the liquid is $\boxed{\text{673.08 kg m}^{-3}}$.

Problem a) ii): Explain whether or not each material obeys Hooke's law.

• Hooke's Law states that the extension of a spring or wire is directly proportional to the force applied, provided the elastic limit is not exceeded. Mathematically, this is expressed as $F = kx$, where $F$ is the applied force, $x$ is the extension, and $k$ is the spring constant.
• Materials that obey Hooke's Law: Materials like metals (e.g., steel, copper wire) obey Hooke's Law within their elastic limit. Their force-extension graph is a straight line passing through the origin.

  Force (F)
  ^
  |   /
  |  /
  | /
  |/
  +----------> Extension (x)
  

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• Materials that do not obey Hooke's Law: Some materials, such as rubber, do not obey Hooke's Law. Their force-extension graph is not a straight line, indicating that the extension is not directly proportional to the applied force. The relationship is often non-linear from the start.

  Force (F)
  ^
  |   /
  |  /
  | /
  |/
  +----------> Extension (x)
  

  Copy
  (Note: The graph for rubber would typically curve, showing a non-linear relationship between force and extension.)

Problem a) iii): Define elastic limit and state its value for the wire.

• Elastic limit is the maximum force or stress that a material can withstand without undergoing permanent deformation. If the applied force is removed when it is below the elastic limit, the material will return to its original dimensions. If the force exceeds the elastic limit, the material will be permanently stretched or deformed.
• The value of the elastic limit for "the wire" cannot be determined from the information provided in the problem, as there is no force-extension graph or data for a wire.

Problem b) i): Elasticity

• Elasticity is the property of a material that enables it to return to its original shape and size after the deforming force or stress has been removed.

Problem b) ii): The wire obeys Hooke's law because the molecules of the wire recoil in the same way as they uncoil.

• A wire obeys Hooke's Law within its elastic limit due to the nature of the interatomic forces within the material. When a small force is applied, the atoms are slightly displaced from their equilibrium positions. The interatomic forces act like tiny springs, exerting a restoring force that is directly proportional to the displacement of the atoms. This microscopic proportionality leads to the macroscopic observation that the applied force is directly proportional to the extension of the wire. The "recoil" refers to these restoring forces pulling the atoms back to their original positions.