Problem 17 - University Physics with Modern Physics

Question

As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of \(8.50 \pm 0.02 \mathrm{cm}\) and a thickness of \(0.050 \pm 0.005 \mathrm{cm} .\) (a) Find the average volume of a cookie and the uncertainty in the volume. (b) Find the ratio of the diameter to the thickness and the uncertainty in this ratio.

Step-by-Step Solution

Answer

(a) Average volume of a cookie: \(2.834 \text{ cm}^3\) with uncertainty \(0.411 \text{ cm}^3\). (b) Diameter to thickness ratio: 170 with uncertainty 17.47.

Step 1: Determine the Formula for Volume

To find the volume of a cookie, we assume the shape of a cylindrical disk. Hence, the volume \(V\) is given by the formula: \[ V = \pi \left( \frac{D}{2} \right)^2 h \] where \(D\) is the diameter, and \(h\) is the thickness.

Step 2: Calculate the Average Volume

Substitute the given values into the volume formula to calculate the average volume. The diameter \(D = 8.50\, \text{cm}\) and thickness \(h = 0.050\, \text{cm}\). So, \[V = \pi \left( \frac{8.50}{2} \right)^2 (0.050) \approx 2.834 \text{ cm}^3\] rounded to three decimal places.

Step 3: Calculate the Uncertainty in Volume

To find the uncertainty in volume \((\Delta V)\), use the formula for uncertainty propagation: \[ \frac{\Delta V}{V} = 2 \frac{\Delta D}{D} + \frac{\Delta h}{h} \]Substituting \(\Delta D = 0.02\, \text{cm}\), \(D = 8.50\, \text{cm}\), \(\Delta h = 0.005\, \text{cm}\), and \(h = 0.050\, \text{cm}\), we find:- \( \Delta V \approx 2.834 \left( 2 \cdot \frac{0.02}{8.50} + \frac{0.005}{0.050} \right) \approx 0.411 \text{ cm}^3\).

Step 4: Determine the Formula for Diameter to Thickness Ratio

The ratio of diameter to thickness \( R \) is given by: \[ R = \frac{D}{h} \] This ratio is used to assess the relationship between the disk's lateral dimensions.

Step 5: Calculate the Average Ratio

Calculate \( R \) using the average values for diameter and thickness: \[ R = \frac{8.50}{0.050} = 170 \]

Step 6: Calculate the Uncertainty of the Ratio

To find the uncertainty \((\Delta R)\) in the ratio, use the uncertainty propagation formula for division: \[ \frac{\Delta R}{R} = \frac{\Delta D}{D} + \frac{\Delta h}{h} \]Substitute \(\Delta D = 0.02\, \text{cm}\), \(D = 8.50\, \text{cm}\), \(\Delta h = 0.005\, \text{cm}\), and \(h = 0.050\, \text{cm}\):- \( \Delta R \approx 170 \left( \frac{0.02}{8.50} + \frac{0.005}{0.050} \right) \approx 17.47 \) rounded to two decimal places.

Key Concepts

Uncertainty Propagation

In the world of measurements, achieving absolute certainty is often impossible. This is where uncertainty propagation comes into play. When you calculate the volume of a cylinder, such as a cookie shaped like a disk, you use specific measurements for diameter and thickness. However, these measurements come with uncertainties. Uncertainty propagation helps us understand how these small measurement errors influence the final calculated quantity—like volume in our case.

To calculate the uncertainty in volume \(\Delta V\), start by determining the relative uncertainties for each measurement. The formula used here is:

• \(\frac{\Delta V}{V} = 2 \frac{\Delta D}{D} + \frac{\Delta h}{h}\)

The relative uncertainty of diameter is \(\frac{0.02}{8.50}\) and for thickness is \(\frac{0.005}{0.050}\). Multiply the volume by the total relative uncertainty to find \(\Delta V\). This careful calculation ensures that any variability in diameter and thickness is accounted for in the volume's uncertainty. It's crucial in scientific studies and engineering where precision matters.

Diameter to Thickness Ratio

The diameter to thickness ratio gives insight into the proportion and shape of a cylindrical object like a cookie. This ratio, represented as \(R = \frac{D}{h}\), essentially compares how 'wide' a disk is to how 'tall' it is. For the cookies, with a diameter \(D\) of 8.50 cm and a thickness \(h\) of 0.050 cm, this ratio calculates to:

• \(R = \frac{8.50}{0.050} = 170\)

This high ratio indicates that the cookies are much broader than they are thick—a common characteristic of baked goods.

When assessing the precision of this ratio, knowing its uncertainty is vital. Using the uncertainty propagation principles again, the formula employed is:

• \(\frac{\Delta R}{R} = \frac{\Delta D}{D} + \frac{\Delta h}{h}\)

This propagation formula helps in understanding how small errors in diameter and thickness measurements contribute to the overall uncertainty in the ratio \(\Delta R\).

It's a useful calculation when designing or evaluating materials where shape and dimension consistency are critical.

Measurement Error

Measurement error is an unavoidable aspect of taking physical measurements. It's the difference between the measured value and the actual value. When you measure the diameter and thickness of a cookie, the tool or method used introduces tiny discrepancies—these are your measurement errors.

These errors could arise from several sources:

• Imperfections in the measuring instrument.
• Environmental conditions affecting measurements.
• Human error in reading or recording measurements.

Understanding measurement error is essential for interpreting results. By quantifying measurement errors, such as \(\pm 0.02\, \text{cm}\) for diameter and \(\pm 0.005\, \text{cm}\) for thickness, you can better evaluate the reliability of calculated properties like volume and the diameter to thickness ratio.

Correctly accounting for measurement errors enables scientists, engineers, or even bakers to make more informed decisions based on data that comes with known limitations.