Paper 2 Measurement Drills

Topic-local structured practice calibrated to recurring Paper 2 patterns. Show working, retain suitable precision during intermediate steps, and give units with final numerical answers.

Question 1: Units, prefixes and homogeneity [10]

A student proposes the relation

$$T=2\pi \sqrt{\frac{L}{g}},$$

where $T$ is time, $L$ is length and $g$ is acceleration.

(a) State the SI base units of $L$ and $g$. [2]

(b) Show that $L/g$ has unit ${\mathrm{s}}^2$. [2]

(c) Hence show that the equation for $T$ is homogeneous. [2]

(d) Explain why homogeneity does not prove that the equation is physically correct. [2]

(e) Convert $75.0\mathrm{cm}$ to metres and express it in standard form. [2]

Question 2: Uncertainty in a derived quantity [12]

A student measures the mass and dimensions of a small rectangular block.

quantity	measured value
mass, $m$	$84.0\pm 0.1\mathrm{g}$
length, $l$	$5.20\pm 0.05\mathrm{cm}$
width, $w$	$3.10\pm 0.05\mathrm{cm}$
height, $h$	$1.60\pm 0.05\mathrm{cm}$

The density is calculated using

$$\rho =\frac{m}{lwh}.$$

(a) Calculate the volume of the block. [2]

(b) Calculate the density in $\mathrm{g}\mathrm{c}{\mathrm{m}}^{-3}$. [2]

(c) Calculate the percentage uncertainty in the volume. [3]

(d) Calculate the absolute uncertainty in the density. [3]

(e) State the final density with uncertainty to a sensible precision. [2]

Question 3: Vector addition and representation [11]

A small boat travels $120\mathrm{m}$ due east and then $90\mathrm{m}$ due north.

(a) State why displacement is a vector quantity. [1]

(b) Calculate the magnitude of the resultant displacement. [2]

(c) Calculate the direction of the resultant displacement, measured north of east. [2]

(d) The boat then travels $50\mathrm{m}$ due west. Find the components of the new resultant displacement from the starting point. [2]

(e) Explain why the total distance travelled is not the same as the magnitude of the final displacement. [2]

(f) State two features of a clear vector diagram for this journey. [2]