Question

A student attempts to measure the radius of a steel ball by using a ruler to measure three identical balls in a row. The student estimates the positions on the scale to be as follows X: (2.0 ±0.3) cm Y: (7.0±0.3) cm What is the radius of the ball together with its associated uncertainity? [3]

• A. (0.9±0.1) cm
• B. (0.8±0.1) cm
• C. (0.7±0.2) cm
• D. (0.70±0.15) cm

Answer 1

Answer: (B)

(B) (0.8 ± 0.1) cm

Answer 2

1. The total distance measured between X and Y is
     $\Delta = (7.0 \pm 0.3) - (2.0 \pm 0.3) = (5.0 \pm 0.424)$ cm,
     where the uncertainty is obtained by
     $\delta\Delta = \sqrt{(0.3)^2 + (0.3)^2} \approx 0.424$ cm.

2. Since three identical balls are placed in a row, the total length covers three diameters, i.e.,
     $\Delta = 6r$.
     Thus,
     $r = \frac{\Delta}{6} = \frac{5.0}{6} \approx 0.83$ cm.

3. The uncertainty in $r$ is given by
     $\delta r = \frac{\delta\Delta}{6} \approx \frac{0.424}{6} \approx 0.07$ cm.
     Rounded suitably, we have:
     $r \approx 0.8 \pm 0.1$ cm.