The student attempt to measure the diameter of a steel ball by using meter rule to measure four similar ball in a row

The student attempt to measure the diameter of a steel ball by using meter rule to measure four similar ball in a row. – cm 0 2 3 4 6 5 The student estimate the positions on the scale to be as follows. X (1.0 20.2) cm Y(5.0 +0.2) cm What is the diameter of a steel ball together with its associated uncertainty? A) (1.0 20.05) cm C) (1.0 +0.1) cm B) (1.0 +0.2) cm D) (1.0 +0.24) cm

The Correct Answer and Explanation is:

To find the diameter of a single steel ball and its associated uncertainty, we follow a simple physics measurement method using the meter rule.

Given:

• Four steel balls are measured in a row.
• The student recorded:
  • X = (1.0 ± 0.2) cm
  • Y = (5.0 ± 0.2) cm

These are the positions of the ends of the four balls on the meter rule.

Step 1: Calculate the total length occupied by the four balls

Length=Y−X=5.0 cm−1.0 cm=4.0 cm\text{Length} = Y – X = 5.0\ \text{cm} – 1.0\ \text{cm} = 4.0\ \text{cm}Length=Y−X=5.0 cm−1.0 cm=4.0 cm

Step 2: Find the diameter of one steel ball

Since the total length is for 4 balls, the diameter of one ball is:Diameter=4.0 cm4=1.0 cm\text{Diameter} = \frac{4.0\ \text{cm}}{4} = 1.0\ \text{cm}Diameter=44.0 cm​=1.0 cm

Step 3: Calculate uncertainty

Each position (X and Y) has an uncertainty of ±0.2 cm. When subtracting measurements with uncertainties, you add the uncertainties:Uncertainty in total length=0.2 cm+0.2 cm=0.4 cm\text{Uncertainty in total length} = 0.2\ \text{cm} + 0.2\ \text{cm} = 0.4\ \text{cm}Uncertainty in total length=0.2 cm+0.2 cm=0.4 cm

Now, divide this by 4 to get the uncertainty of the diameter of one ball:Uncertainty in diameter=0.4 cm4=0.1 cm\text{Uncertainty in diameter} = \frac{0.4\ \text{cm}}{4} = 0.1\ \text{cm}Uncertainty in diameter=40.4 cm​=0.1 cm

Final Answer:

(1.0±0.1) cm\boxed{(1.0 \pm 0.1)\ \text{cm}}(1.0±0.1) cm​

Explanation

To find the diameter of one steel ball, the student used a common technique in experimental physics: measure multiple similar items together, then divide the total length by the number of items. This method reduces relative error and increases accuracy, especially when each individual item is small compared to the least count of the measuring instrument.

Here, four steel balls were aligned in a straight line. The student read the position of one end as (1.0 ± 0.2) cm and the other end as (5.0 ± 0.2) cm. These are X and Y, and the difference between Y and X gives the total length of the four balls: 4.0 cm. Dividing this by 4, each ball’s diameter is 1.0 cm.

Uncertainty is also crucial. Since both measurements X and Y have a ±0.2 cm uncertainty, the uncertainty in the total length is 0.4 cm. We then divide this total uncertainty by 4 (the number of balls) to find the uncertainty in one diameter measurement, giving ±0.1 cm.

Therefore, the diameter of a single steel ball is best expressed as (1.0 ± 0.1) cm.

✅ Correct Choice:

C) (1.0 ± 0.1) cm