Calculate the percentage error in the volume of a cylinder of diameter 2.4 cm and height 10.5 cm, both measured using a metre scale of least count 0.1 cm. What will be the percentage error if dimensions are measured using a vernier calipers of least count 0.01 cm?

Explanation

To calculate the percentage error in the volume of a cylinder, we need to follow these steps:

### Step 1: Understand the formula for the volume of a cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height.

### Step 2: Identify the measurements and their uncertainties
Given:
- Diameter \( d = 2.4 \) cm, so the radius \( r = \frac{d}{2} = \frac{2.4}{2} = 1.2 \) cm.
- Height \( h = 10.5 \) cm.
- The least count of the measuring instrument (meter scale) is \( 0.1 \) cm.

The absolute error in diameter and height when using the meter scale is:
\[ \Delta d = 0.1 \text{ cm}, \quad \Delta h = 0.1 \text{ cm} \]

### Step 3: Calculate the percentage error in volume
The formula for percentage error in volume \( V \) is:
\[ \frac{\Delta V}{V} \times 100 = \left( \frac{2 \Delta d}{d} + \frac{\Delta h}{h} \right) \times 100 \]
Substituting the values:
\[ \frac{\Delta V}{V} \times 100 = \left( \frac{2 \times 0.1}{2.4} + \frac{0.1}{10.5} \right) \times 100 \]

### Step 4: Calculate each term
Calculating the first term:
\[ \frac{2 \times 0.1}{2.4} = \frac{0.2}{2.4} \approx 0.0833 \]

Calculating the second term:
\[ \frac{0.1}{10.5} \approx 0.00952 \]

### Step 5: Add the percentage errors
Now, adding the two percentage errors:
\[ 0.0833 + 0.00952 \approx 0.09282 \]

### Step 6: Convert to percentage
Converting to percentage:
\[ 0.09282 \times 100 \approx 9.282\% \]

### Step 7: Round off
Rounding off to two decimal places, we get:
\[ \text{Percentage error} \approx 9.28\% \]

### Step 8: Repeat for Vernier calipers
Now, if the dimensions are measured using Vernier calipers with a least count of \( 0.01 \) cm:
\[ \Delta d = 0.01 \text{ cm}, \quad \Delta h = 0.01 \text{ cm} \]

Using the same formula for percentage error:
\[ \frac{\Delta V}{V} \times 100 = \left( \frac{2 \times 0.01}{2.4} + \frac{0.01}{10.5} \right) \times 100 \]

Calculating each term:
\[ \frac{2 \times 0.01}{2.4} = \frac{0.02}{2.4} \approx 0.00833 \]
\[ \frac{0.01}{10.5} \approx 0.000952 \]

Adding these:
\[ 0.00833 + 0.000952 \approx 0.009282 \]

Converting to percentage:
\[ 0.009282 \times 100 \approx 0.9282\% \]

Rounding off:
\[ \text{Percentage error} \approx 0.93\% \]

### Final Answer
- Percentage error using meter scale: **9.28%**
- Percentage error using Vernier calipers: **0.93%**

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