Problem 55 - 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 cm and a thickness of 0.050 \(\pm\) 0.005 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) Volume: 2.837±0.283 cm³; (b) Ratio: 170±1.72.

Step 1: Calculate the volume of a cookie

The volume of a cylindrical shape, like our cookie, is calculated using the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the thickness (or height) of the cookie. Since the diameter is given, the radius \( r = \frac{d}{2} = \frac{8.50}{2} = 4.25 \) cm. The thickness \( h = 0.050 \) cm. Therefore, the volume of the cookie can be calculated as follows:\[ V = \pi \times (4.25)^2 \times 0.050 \approx 2.837 \text{ cm}^3 \].

Step 2: Determine the uncertainty in volume calculation

To find the uncertainty in the volume, we use the formula for propagation of uncertainty for products, which in this case involves a constant \( \pi \), radius \( r \), and thickness \( h \):\[ \left( \frac{\Delta V}{V} \right)^2 = \left( 2 \times \frac{\Delta r}{r} \right)^2 + \left( \frac{\Delta h}{h} \right)^2 \]Given \( \Delta d = 0.02 \) cm, \( \Delta r = \frac{0.02}{2} = 0.01 \) cm, and \( \Delta h = 0.005 \) cm, we substitute:\[ \left( \frac{\Delta V}{2.837} \right)^2 = \left( 2 \times \frac{0.01}{4.25} \right)^2 + \left( \frac{0.005}{0.050} \right)^2 \]Evaluating this gives:\[ \left( \frac{\Delta V}{2.837} \right)^2 = 0.0000221 + 0.01 = 0.0100221 \]Thus, \( \Delta V \approx 0.283 \text{ cm}^3 \).

Step 3: Calculate the diameter to thickness ratio

The ratio \( R \) of the diameter \( d \) to the thickness \( h \) is given by \( R = \frac{d}{h} = \frac{8.50}{0.050} = 170 \).

Step 4: Determine the uncertainty in the ratio

To find the uncertainty in the ratio, use the formula for uncertainty in division:\[ \frac{\Delta R}{R} = \sqrt{\left( \frac{\Delta d}{d} \right)^2 + \left( \frac{\Delta h}{h} \right)^2} \]Substitute \( \Delta d = 0.02 \) cm, \( \Delta h = 0.005 \) cm:\[ \frac{\Delta R}{170} = \sqrt{\left( \frac{0.02}{8.50} \right)^2 + \left( \frac{0.005}{0.050} \right)^2} \]Calculating, we find:\[ \frac{\Delta R}{170} = 0.0011765 + 0.01 \approx 0.01011765 \]Therefore, \( \Delta R \approx 1.72 \).

Key Concepts

Volume of a Cylinder

In mathematics, the volume of a cylinder is a fundamental concept, especially when dealing with physical shapes such as cookies, cans, or any cylindrical object. To determine the volume, we use the formula \( V = \pi r^2 h \). Here, \( r \) is the radius of the base of the cylinder, and \( h \) is the height or thickness. This formula essentially says that the volume is equal to the area of the base circle multiplied by the height. In our cookie example, we converted diameter to radius by dividing by two since the diameter \( d = 8.50 \) cm was given, which gives \( r = \frac{8.50}{2} \) cm. The thickness or height \( h \) is directly provided as 0.050 cm. Using these values, the calculation gives us \( V = \pi \times (4.25)^2 \times 0.050 \approx 2.837 \text{ cm}^3 \). This means each cookie occupies approximately 2.837 cubic centimeters of space.

Error Propagation

Error propagation is a critical concept in scientific measurements, allowing for estimation of how uncertainties in measurements affect calculated results. When we deal with physical equations involving products or ratios, such as the volume of a cylinder, errors in each measurement propagate through the calculation. The formula for the propagation of uncertainty in a product is given by:

• \( \left( \frac{\Delta V}{V} \right)^2 = \left( 2 \times \frac{\Delta r}{r} \right)^2 + \left( \frac{\Delta h}{h} \right)^2 \)

Here, \( \Delta r \) and \( \Delta h \) are the uncertainties in radius and thickness, respectively. For our example, \( \Delta r = 0.01 \) cm and \( \Delta h = 0.005 \) cm. Substituting these into the formula helps us find the uncertainty in the volume \( \Delta V \), which comes out to approximately 0.283 cm³. This tells us how accurate our volume measurement is, considering the original uncertainties in diameter and thickness.

Ratio Calculation

Ratio calculation is an essential mathematical operation used to compare the sizes of two quantities while considering their uncertainties. For example, finding the ratio of the diameter to the thickness of our cookie is useful in quantifying how wide it is in relation to its height. The formula for the ratio \( R \) is \( R = \frac{d}{h} \). Inserting the values from our cookie, \( d = 8.50 \) cm and \( h = 0.050 \) cm, we find that the ratio \( R = 170 \). To calculate the uncertainty in this ratio, we use the rule for division errors, given by:

• \( \frac{\Delta R}{R} = \sqrt{\left( \frac{\Delta d}{d} \right)^2 + \left( \frac{\Delta h}{h} \right)^2} \)

Plugging in the uncertainties \( \Delta d = 0.02 \) cm and \( \Delta h = 0.005 \) cm yields an uncertainty in the ratio of approximately 1.72. Thus, the diameter to thickness ratio of the cookie, considering uncertainties, is 170 with an uncertainty of 1.72.