Physics 5054/22 — October/November 2011

Walkthrough

The metre rule is balanced on a pivot at the 50 cm mark. According to the principle of moments, for an object to be in rotational equilibrium, the sum of the anticlockwise moments about the pivot must equal the sum of the clockwise moments.

The 0.050 kg mass is placed at the 10 cm mark. Its distance from the pivot is $50 - 10 = 40\text{ cm}$. It creates an anticlockwise moment. The cube is placed at the 75 cm mark. Its distance from the pivot is $75 - 50 = 25\text{ cm}$. It creates a clockwise moment.

Moment = force $\times$ perpendicular distance from the pivot. Since the force is the weight ($mg$), the equation is:

$$0.050 \times g \times 40 = m \times g \times 25$$

The gravitational field strength $g$ cancels out from both sides, leaving:

$$0.050 \times 40 = m \times 25$$

Solving for the mass of the cube $m$:

$$m = \frac{2.0}{25} = 0.080\text{ kg}$$

Key Takeaways

• The principle of moments states that for a balanced object, anticlockwise moments equal clockwise moments.
• Distances in moment calculations must always be measured from the pivot, not from the end of the ruler.
• The gravitational field strength $g$ cancels out when comparing weights on a balance, so it is not needed to find the mass.

Common Mistakes

• Measuring the distance of the mass and cube from the end of the ruler (0 cm or 100 cm) instead of from the pivot (50 cm). Using 10 cm and 75 cm directly in the equation is a common error.
• Forgetting to cancel $g$ or incorrectly including $g$ in the final mass calculation, leading to an answer in Newtons instead of kilograms.

Things to Be Careful About

• Ensure distances are in consistent units. Here, using centimetres for both distances is perfectly fine because the units cancel out in the ratio $\frac{40}{25}$.
• The final answer must include the unit kg. The mark scheme also accepts 80 g, but kg is the standard SI unit and matches the input mass.

Walkthrough

Density $\rho$ is calculated using the formula $\rho = \frac{m}{V}$, where $m$ is the mass and $V$ is the volume.

From part (a), the mass of the wooden cube is $0.080\text{ kg}$. The volume is given as $1.6 \times 10^{-4}\text{ m}^3$.

Substituting these values into the formula:

$$\rho = \frac{0.080}{1.6 \times 10^{-4}} = 500\text{ kg/m}^3$$

Alternatively, if mass is converted to grams ($80\text{ g}$) and volume to cubic centimetres ($0.16\text{ cm}^3$), the density is $\frac{80}{0.16} = 500\text{ g/cm}^3$ wait, $1.6 \times 10^{-4}\text{ m}^3 = 160\text{ cm}^3$. So $\frac{80}{160} = 0.50\text{ g/cm}^3$. Both $500\text{ kg/m}^3$ and $0.50\text{ g/cm}^3$ are correct and equivalent.

Key Takeaways

• Density is mass divided by volume: $\rho = \frac{m}{V}$.
• When mass is in kg and volume in m$^3$, the resulting density is in kg/m$^3$.
• Wood is less dense than water ($1000\text{ kg/m}^3$), which is consistent with the calculated value of $500\text{ kg/m}^3$ and explains why it floats.

Common Mistakes

• Inverting the formula and calculating $\frac{V}{m}$ instead of $\frac{m}{V}$.
• Making errors with powers of ten when dividing by $1.6 \times 10^{-4}$.
• Forgetting to include the correct units in the final answer.

Things to Be Careful About

• Pay attention to the power of ten in the volume. $1.6 \times 10^{-4}$ is a small number, so dividing by it will yield a larger number.
• The mark scheme accepts $0.50\text{ g/cm}^3$ as an equivalent answer, but $500\text{ kg/m}^3$ is the direct result of using SI units (kg and m$^3$). Always ensure your final units match the units used in your calculation.