Some college physics students measured the dimensions of a soda can. They reported the following results: i) Diameter (d) of the can: 4.35 cm ii) Circumference (C) of the can: 13.48 cm Their professor has asked them to use their results to calculate the value of Ï€ (pi) to the correct number of significant figures. They have come to you for help. You remember that the equation relating circumference to diameter is: Circumference = Ï€ * Diameter. Show how you would calculate and impress your friends. (T= [13.48/4.35] = 3.10 Experimental value of Ï€: 3.10 You know that the value obtained above is an experimental value because it was determined from measurements made during an experiment. You also know that the accepted value (also called the actual value or “textbook” value) of Ï€ is 3.1416.
The Correct Answer and Explanation is:
To calculate the experimental value of π (pi) from the given measurements, we can use the formula that relates the circumference of a circle to its diameter:Circumference=π×Diameter\text{Circumference} = \pi \times \text{Diameter}Circumference=π×Diameter
Given:
- Diameter (d) = 4.35 cm
- Circumference (C) = 13.48 cm
We need to rearrange the formula to solve for π:π=CircumferenceDiameter\pi = \frac{\text{Circumference}}{\text{Diameter}}π=DiameterCircumference
Substitute the given values:π=13.484.35=3.10\pi = \frac{13.48}{4.35} = 3.10π=4.3513.48=3.10
The experimental value of π is approximately 3.10.
Explanation:
- Using the Formula:
The relationship between the circumference and the diameter of a circle is fundamental in geometry, given by the equation C=π×dC = \pi \times dC=π×d. To calculate π, we rearranged this to π=Cd\pi = \frac{C}{d}π=dC, where CCC is the circumference, and ddd is the diameter. - Performing the Calculation:
We divided the given circumference (13.48 cm) by the given diameter (4.35 cm). This gave us an experimental value of 3.10 for π. - Significant Figures:
We must consider significant figures when reporting the result. The given diameter (4.35 cm) has 3 significant figures, and the circumference (13.48 cm) also has 4 significant figures. The result should be reported with the fewest significant figures from the measurements. Therefore, the experimental value of π should be reported with 3 significant figures, which is 3.10. - Comparison to Accepted Value:
The accepted value for π is 3.1416, which is more precise. The experimental value (3.10) is an approximation based on the measurements, and it is slightly less accurate. This small difference could be due to measurement errors, rounding, or limitations in the precision of the instruments used.
Conclusion:
The experimental value of π calculated from the given data is 3.10, which is reasonably close to the actual value of 3.1416. The difference in values highlights how experimental measurements may vary due to limitations such as instrument precision and rounding.
