Question
Scientists are investigating the variation in air pressure at different locations on a mountain.
(a) The scientists make measurements of several physical quantities at each location.
Complete Table 1.1 by stating the SI base unit for each quantity and identifying with a tick (\(\checkmark\)) whether each quantity is a scalar or a vector. Use the space for any working.
Table 1.1
| quantity measured | SI base unit | scalar | vector |
|---|---|---|---|
| air temperature | |||
| air pressure |
(b) (i) At one location, the density of the air is \(1.1\,\mathrm{kg\,m^{-3}}\). A spherical weather balloon is filled with a gas and released from rest. The balloon has radius \(0.90\,\mathrm{m}\).
Calculate the upthrust acting on the balloon. (2 marks)
upthrust = __________________________________________ \( \mathrm{N} \)
(ii) Explain why an upthrust acts on the balloon. (2 marks)
________________________________
________________________________
________________________________
(iii) The balloon has weight \(19\,\mathrm{N}\).
Calculate the magnitude of the initial acceleration of the balloon. (3 marks)
acceleration = __________________________________________ \( \mathrm{m\,s^{-2}} \)
(c) A quantity \(c\) relating to the motion of the balloon is calculated from three measured quantities \(k\), \(F\) and \(v\) using the formula
\(c=\dfrac{2kF}{v^2}\)
The percentage uncertainties in the measured quantities are given in Table 1.2.
Table 1.2
| measured quantity | percentage uncertainty |
|---|---|
| \(k\) | 5% |
| \(F\) | 3% |
| \(v\) | 4% |
The calculated value of \(c\) is \(1.8\).
Determine the absolute uncertainty in \(c\). (2 marks)
absolute uncertainty = __________________________________________
Syllabus Topic Codes (Cambridge International AS & A Level Physics 9702):
• 1.2: SI units — part (a)
• 1.3: Errors and uncertainties — part (c)
• 1.4: Scalars and vectors — part (a)
• 3.1: Momentum and Newton’s laws of motion — part (b)(iii)
• 4.3: Density and pressure — parts (b)(i) and (b)(ii)
▶️ Answer/Explanation
(a) SI base units and scalar/vector quantities [2 marks]
Air temperature has the SI base unit kelvin, \( \mathrm{K} \).
Air pressure has the SI base units \( \mathrm{kg\,m^{-1}\,s^{-2}} \).
Both air temperature and air pressure are scalars, because they have magnitude but no direction.
Answer:
| quantity | SI base unit | scalar | vector |
|---|---|---|---|
| air temperature | \(\mathrm{K}\) | \(\checkmark\) | |
| air pressure | \(\mathrm{kg\,m^{-1}\,s^{-2}}\) | \(\checkmark\) |
(b)(i) Upthrust on the balloon [2 marks]
The upthrust is equal to the weight of air displaced by the balloon:
\(F_{\mathrm{up}}=\rho Vg\)
For a sphere, \(V=\dfrac{4}{3}\pi r^3\).
Therefore,
\(F_{\mathrm{up}}=1.1\times9.81\times\dfrac{4}{3}\pi(0.90)^3\)
\(F_{\mathrm{up}}=33\,\mathrm{N}\)
Answer: \( \boxed{33\,\mathrm{N}} \)
(b)(ii) Explanation of upthrust [2 marks]
There is a difference in height or depth between the top and bottom of the balloon, so there is a difference in pressure between the top and bottom.
The pressure is greater at the bottom, so the upward force on the bottom of the balloon is greater than the downward force on the top.
Therefore, the resultant force is upwards and this is the upthrust.
(b)(iii) Initial acceleration of the balloon [3 marks]
The upthrust is \(33\,\mathrm{N}\) and the weight is \(19\,\mathrm{N}\), so the resultant force is
\(\Sigma F=33-19\)
\(\Sigma F=14\,\mathrm{N}\)
The mass of the balloon is obtained from \(W=mg\):
\(m=\dfrac{19}{9.81}\)
\(m=1.94\,\mathrm{kg}\)
Using \(F=ma\),
\(a=\dfrac{33-19}{19/9.81}\)
\(a=7.2\,\mathrm{m\,s^{-2}}\)
Answer: \( \boxed{7.2\,\mathrm{m\,s^{-2}}} \)
(c) Absolute uncertainty in \(c\) [2 marks]
The quantity is
\(c=\dfrac{2kF}{v^2}\)
For multiplication and division, percentage uncertainties are added. Since \(v\) is squared, its percentage uncertainty is doubled.
Percentage uncertainty in \(c=5+3+(2\times4)\)
\(=16\%\)
The absolute uncertainty is
\(\text{absolute uncertainty}=1.8\times0.16\)
\(=0.288\)
Answer: \( \boxed{\pm0.3} \)