To determine the relative density of the metallic block, we need to understand how to calculate it based on the weights given in air and water. The relative density (also known as specific gravity) is defined as the ratio of the weight of the object in air to the weight of an equal volume of water. In this case, we have the following weights:
- Weight in air: \( W_{air} = 5.00 \, \text{N} \) with an uncertainty of \( \pm 0.05 \, \text{N} \)
- Weight in water: \( W_{water} = 4.00 \, \text{N} \) with an uncertainty of \( \pm 0.05 \, \text{N} \)
The formula for relative density (RD) is given by:
RD = \frac{W_{air}}{W_{water}}
Substituting the values we have:
RD = \frac{5.00}{4.00} = 1.25
Next, we need to calculate the uncertainty in the relative density. To do this, we can use the formula for the propagation of uncertainty for division:
\( \frac{\Delta RD}{RD} = \sqrt{\left(\frac{\Delta W_{air}}{W_{air}}\right)^2 + \left(\frac{\Delta W_{water}}{W_{water}}\right)^2} \)
Where:
- \( \Delta W_{air} = 0.05 \, \text{N} \)
- \( \Delta W_{water} = 0.05 \, \text{N} \)
Now, we can calculate the relative uncertainties:
\( \frac{\Delta W_{air}}{W_{air}} = \frac{0.05}{5.00} = 0.01 \) (or 1%)
\( \frac{\Delta W_{water}}{W_{water}} = \frac{0.05}{4.00} = 0.0125 \) (or 1.25%)
Now, substituting these values into the uncertainty propagation formula:
\( \frac{\Delta RD}{RD} = \sqrt{(0.01)^2 + (0.0125)^2} \)
\( \frac{\Delta RD}{RD} = \sqrt{0.0001 + 0.00015625} = \sqrt{0.00025625} \approx 0.016 \) (or 1.6%)
Now, we can find the absolute uncertainty in the relative density:
\( \Delta RD = RD \times \frac{\Delta RD}{RD} = 1.25 \times 0.016 \approx 0.02 \)
Thus, the relative density can be expressed as:
RD = 1.25 ± 0.02
Now, let's analyze the options given:
- (a) 5.00 ± 0.05
- (b) 5.00 ± 0.10
- (c) 5.00 ± 6%
- (d) 5.00 ± 11%
None of these options directly match our calculated relative density of 1.25. However, if we consider the relative density as a dimensionless quantity, we can convert our result into a percentage uncertainty:
Percentage uncertainty = \( \frac{0.02}{1.25} \times 100 \approx 1.6\% \)
Since 1.6% is less than 6%, the closest option that reflects the uncertainty in the context of relative density is (c) 5.00 ± 6%. Therefore, the correct answer is:
(c) 5.00 ± 6%
