The speed of the blood in a blood vessel can indicate a person's health. A high speed may indicate a high stress on the walls of the blood vessel. (a) The speed $v$ of the blood in a blood vessel with a diameter $d$ is given by $$v = \frac{kn}{\rho d}$$ where $\eta$ is the viscosity of the blood $\rho$ is the density of the blood $k$ is a constant with no units. Show that the unit for $\eta$ is Pas. (b) As the speed of the blood changes, the wall of the blood vessel expands and contracts: The wall of a blood vessel consists of collagen fibres. The graph shows the stress-strain relationship up to the breaking stress of the collagen fibres.

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The speed of the blood in a blood vessel can indicate a person's health - Edexcel - A-Level Physics - Question 10 - 2023 - Paper 2

Question 10

The speed of the blood in a blood vessel can indicate a person's health. A high speed may indicate a high stress on the walls of the blood vessel. (a) The speed $v$... show full transcript

Worked Solution & Example Answer:The speed of the blood in a blood vessel can indicate a person's health - Edexcel - A-Level Physics - Question 10 - 2023 - Paper 2

Step 1

Show that the unit for η is Pas.

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Answer

To show that the unit for viscosity $\eta$ is Pascals.second (Pas), we start with the given equation:

$v = \frac{kn}{\rho d}$

Rearranging for viscosity:

$\eta = \frac{v \rho d}{k}$

Next, we analyze the units.

Speed $v$ is measured in meters per second (m/s).
Density $\rho$ is measured in kilograms per cubic meter (kg/m³).
Diameter $d$ is measured in meters (m).
The constant $k$ is dimensionless.

Substituting the units in:

$\text{Unit of } \eta = \left( \frac{\text{m}}{\text{s}} \cdot \frac{\text{kg}}{\text{m}^3} \cdot \text{m} \right) = \frac{\text{kg}}{\text{m.s}} = \text{Pas}$

Thus, we have shown that the unit for viscosity $\eta$ is indeed Pascals.second (Pas).

Step 2

Calculate the Young modulus of collagen fibres.

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Answer

To calculate the Young modulus (E) from the stress-strain graph, we can use the formula:

$E = \frac{\text{Stress}}{\text{Strain}}$

Using values from the graph:

For example, take stress at the linear region and strain, e.g., stress = 2.0 x 10⁴ N/m² and strain = 0.06.

Plugging in the values:

$E = \frac{2.0 \times 10^4 \, \text{N/m}^2}{0.06} = 3.33 \times 10^5 \text{N/m}^2$

Thus, the Young modulus of collagen fibres is approximately $3.33 \times 10^5 \, \text{N/m}^2$ .

Step 3

Describe the behaviour of collagen fibres when the stress in the fibres is increased from the elastic limit until the fibres break.

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Answer

Collagen fibres exhibit two distinct behaviors when stress is applied:

Elastic Region: At first, when the stress is increased within the elastic limit, the fibres will stretch proportionally to the applied stress. This region follows Hooke's law where the stress is directly proportional to the strain, and the fibers return to their original length once the load is removed.
Plastic Deformation: As the stress exceeds the elastic limit, the fibres begin to undergo plastic deformation. In this stage, even if the load is removed, the fibres do not return to their original length and permanent changes occur in their structure.

Eventually, if the stress continues to increase past the breaking point, the fibres will reach a failure point where they can no longer withstand the applied stress resulting in rupture. This is marked by the stress-strain curve plateauing before a sudden drop, indicating failure.

The speed of the blood in a blood vessel can indicate a person's health - Edexcel - A-Level Physics - Question 10 - 2023 - Paper 2

Worked Solution & Example Answer:The speed of the blood in a blood vessel can indicate a person's health - Edexcel - A-Level Physics - Question 10 - 2023 - Paper 2

Show that the unit for η is Pas.

Calculate the Young modulus of collagen fibres.

Describe the behaviour of collagen fibres when the stress in the fibres is increased from the elastic limit until the fibres break.

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