Q55P - 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 $\mathbf{8}\mathbf{.}\mathbf{50}\mathbf{\pm }\mathbf{0}\mathbf{.}\mathbf{02}$cm cm and a thickness of$\mathbf{0}\mathbf{.}\mathbf{050}\mathbf{\pm }\mathbf{0}\mathbf{.}\mathbf{005}\mathbf{cm}$ (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

1. The average volume of the cookie is$2.837{\mathrm{cm}}^3$ and the uncertainty in the cookie volume is $\pm 0.296{\mathrm{cm}}^3$.
2. The ratio of the average diameter to the average thickness is $170:1$ and the uncertainty in the ratio is $\pm 20.$

Step 1: Identification of given data

The given data can be listed below,

• The uncertainty in the diameter is, ${\mathrm{D}}_0=\pm 0.02\mathrm{cm}.$
• The uncertainty in the thickness is, ${\mathrm{h}}_0=\pm 0.005 \mathrm{cm}.$
• The maximum radius of the circular disk is, $\mathrm{r}=\frac{8.52\mathrm{cm}}{2}=4.26\mathrm{cm}.$
•  The maximum thickness of the cookie is, ${\mathrm{h}}_{\mathrm{m}}=0.055\mathrm{cm}.$

Step 2: Significance of uncertainty in measurements.

Uncertainty measurements are quantized values of the error that appears in the mathematical parameters due to some unidentifiable flaw in the tools or limit of precision of tools used for the measurements. 

Uncertainty can be of two types: Absolute and Relative. No instrument can be ideal and so it is important to know the percentage error to determine.

Step 3: (a) Determination of the absolute value and average volume of the cookie.

The formula for the average volume of a circular disk is given by,

$V_A=\pi {r_A}^2{h}_A$

Here, r is the radius of the disk, and h is the thickness of the disk.

Substitute the values in the above equation, 

$$\begin{gathered} V=\pi {\left(4.25 \text{cm}\right)}^2\left(0.050 \text{cm}\right) \\ =2.837 {\text{cm}}^{\text{3}} \end{gathered}$$

Thus, the average volume of the circular cookie is $2.837 {\text{cm}}^{\text{3}}$.

The value of thickness could not exceed the maximum value so the maximum volume of the disk can be given by,

$V_m=\pi {r_m}^2{h}_m$

Here, $h_m$is the maximum value of the thickness.

Substitute values in the above,

 

$$\begin{gathered} V_S=\pi {\left(4.26 \text{cm}\right)}^2\left(0.055 \text{cm}\right) \\ =3.1 {\text{cm}}^3 \end{gathered}$$

The volume could be small, so smaller volume can be given by,

$V_s=\pi {r_s}^2{h}_s$

Here, $r_s$is the radius of the disk and $h_s$ is the thickness of the disk

Substitute values in the above,

$$\begin{gathered} V_S=\pi {\left(4.24 \text{cm}\right)}^2\left(0.045 \text{cm}\right) \\ =2.5 {\text{cm}}^3 \end{gathered}$$

The uncertainty in the volume is calculated as,

$V_0=V_A\pm V_S$

Substitute the values in the above equation.

 

$$\begin{gathered} V_0=3.1 {\text{cm}}^{\text{3}}\pm 2.5 {\text{cm}}^3 \\ =\pm 0.03 {\text{cm}}^3 \end{gathered}$$

Thus, the uncertainty in the volume of cookie is $\pm 0.03 {\text{cm}}^3$.

Step 4: (b) Determination of the ratio of the diameter to the thickness and the uncertainty in the ratio.

The ratio of the average diameter to the average thickness is given by,

$$\begin{gathered} \frac{{\mathrm{d}}_{\mathrm{A}}}{{\mathrm{h}}_{\mathrm{A}}}=\frac{8.50 \mathrm{cm}}{0.050 \mathrm{cm}} \\ =170:1 \end{gathered}$$

Thus, the ratio of the average diameter to the average thickness is $170:1.$

By taking the largest possible value of the diameter and the smallest possible thickness largest possible value for this ratio is given by

$$\begin{gathered} \frac{{\mathrm{d}}_{\mathrm{L}}}{{\mathrm{h}}_{\mathrm{L}}}=\frac{8.52 \mathrm{cm}}{0.045 \mathrm{cm}} \\ =190:1 \end{gathered}$$ 

The smallest possible value of the ratio is given by,

$$\begin{gathered} \frac{{\mathrm{d}}_{\mathrm{s}}}{{\mathrm{h}}_{\mathrm{s}}}=\frac{8.48 \mathrm{cm}}{0.055 \mathrm{cm}} \\ =150:1 \end{gathered}$$ 

From the above expressions, it is concluded that uncertainty in the ratio of the diameter and thickness is $\pm 20.$

Thus, the ratio of the diameter and the thickness including the uncertainty is $\left(170\pm 20\right):1$