# IB DP Physics HL Unit 1. Measurements and uncertainties Question Bank paper 1

### Question

What are the units of specific energy and energy density?

 Â Specific energy Energy density A. J mâ€“1 J mâ€“2 B. J kgâ€“1 J mâ€“3 C. J kgâ€“1 J mâ€“2 D. J mâ€“1 J mâ€“3

Ans: B

Specific energyÂ orÂ massic energyÂ isÂ energyÂ per unitÂ mass. TheÂ SIÂ unit for specific energy is theÂ jouleÂ perÂ kilogramÂ (J/kg).

Energy densityÂ is the amount ofÂ energyÂ stored in a given system or region of space per unitÂ volume. SIÂ unit J/m3

Question

Two lengths, $$a$$ and $$b$$, are measured to be $$51 \pm 1{\text{ cm}}$$ and $$49 \pm 1{\text{ cm}}$$ respectively. In which of the following quantities is the percentage uncertainty the largest?

A. Â  Â  $$a + b$$

B. Â  Â  $$a – b$$

C. Â  Â  $$a \times b$$

D. Â  Â  $$\frac{a}{b}$$

### Ans.

B

$$a+b=100\pm 2 \;cm$$ so the percentage error is 2/100= 2%
$$a-b=2\pm 2 \; cm$$ so the percentage error is 2/2= 100%
The percentage errors for a and b separately are 1/51= 0.0196 or about 2% and 1/49= 0.02041 or about 2%

so the percentage errors of a x b and a/b are 2+ 2= 4%

Â

### Question

Which of the following expresses the units of capacitance in terms of fundamental units?

A. $${{\rm{s}}^{\rm{4}}}{{\rm{A}}^{\rm{2}}}{{\rm{m}}^{{\rm{ – 2}}}}{\rm{k}}{{\rm{g}}^{{\rm{ – 1}}}}$$

B. $${{\rm{s}}^{\rm{2}}}{\rm{A}}{{\rm{m}}^{{\rm{ – 2}}}}{\rm{k}}{{\rm{g}}^{{\rm{ – 1}}}}$$

C. $${{\rm{s}}^{\rm{4}}}{{\rm{A}}^{\rm{2}}}{{\rm{m}}^{{\rm{ – 2}}}}$$

D. $${{\rm{s}}^{\rm{2}}}{\rm{A}}{{\rm{m}}^{{\rm{ – 2}}}}$$

### Ans:

A

A farad is aÂ derived unitÂ based on four of the seven base units of theÂ International System of Units:Â kilogramÂ (kg),Â metreÂ (m),Â secondÂ (s), andÂ ampereÂ (A).

$$C=\frac{Q}{V} =\frac{Q}{\frac{W}{Q}}=\frac{Q^2}{W}$$

$$=\frac{(It)^2}{F\times d} =\frac{I^2t^2}{m\times a\times d}$$

$$=\frac{A^2s^2}{kg\times m\times s^{-2}\times m}$$

$$=s^4A^2 m^{-2}kg^{-1}$$

### Question

The volume V of a cylinder of radius R and height H is given by V = $$\pi$$R2H. The volume of the cylinder was measured with an uncertainty of 10% and the height was measured with an uncertainty of 6%. What is the uncertainty in the radius of the cylinder?

A. 1%
B. 2%
C. 4%
D. 8%

Ans.

D

$$V=\pi R^2H$$
$$R^2 =\frac{V}{\pi H}$$

The rule of dealing with uncertainties in mathematical calculations;

â€¢ you add percentage uncertainties when you multiply or divide quantities.

â€¢ if a quantity is squared you double the uncertainty, if cubed it is 3 times etc. i.e. multiply by the index. Square root is index half so divide by 2.

Uncertainty in RÂ  will be $$\frac{10+6}{2}=8%$$