Chapter 1: Physical Quantities and MeasurementsNumericals Solved Step by Step

Given Data

• 1 hour = 3600 s
• 1 day = 24 hours
• 1 month = 30 days (taken)

To Find

• Seconds in (a) a day (b) a week (c) a month, written with SI prefixes

Formula

Calculation

1. One day: 24 × 3600 s = 86 400 s. Since 1 ks = 1000 s, this is 86.4 ks.
2. One week: 7 × 86 400 s = 604 800 s = 604.8 ks.
3. One month: 30 × 86 400 s = 2 592 000 s. Since 1 Ms = 10⁶ s, this is 2.592 Ms.

Given Data

• 1 day = 86 400 s
• 1 week = 604 800 s
• 1 month = 2 592 000 s (from Numerical 1.1)

To Find

• Each answer of Numerical 1.1 in scientific notation

Formula

Calculation

1. 86 400 s: move the point 4 places left, so 8.64 × 10⁴ s.
2. 604 800 s: move the point 5 places left, so 6.048 × 10⁵ s.
3. 2 592 000 s: move the point 6 places left, so 2.592 × 10⁶ s.

Given Data

• (a) 4 × 10⁻⁴ kg and 3 × 10⁻⁵ kg
• (b) 5.4 × 10⁻⁶ m and 3.2 × 10⁻⁵ m

To Find

• (a) the sum (b) the difference, in scientific notation

Formula

Calculation

1. To add or subtract, first make the powers the same.
2. (a) Write 3 × 10⁻⁵ as 0.3 × 10⁻⁴. Then (4 + 0.3) × 10⁻⁴ = 4.3 × 10⁻⁴ kg.
3. (b) Write 5.4 × 10⁻⁶ as 0.54 × 10⁻⁵. Then (0.54 − 3.2) × 10⁻⁵ = −2.66 × 10⁻⁵ m.

Given Data

• (a) 5 × 10⁴ m and 3 × 10⁻² m
• (b) 6 × 10⁸ kg and 3 × 10⁴ m³

To Find

• (a) the product (b) the quotient, in scientific notation

Formulas

Calculation

1. (a) Multiply the numbers: 5 × 3 = 15. Add the powers: 4 + (−2) = 2. So 15 × 10² m² = 1.5 × 10³ m².
2. (b) Divide the numbers: 6 ÷ 3 = 2. Subtract the powers: 8 − 4 = 4. So 2 × 10⁴ kg m⁻³.

Given Data

• 1 km = 10³ m

To Find

• The value in base units, in scientific notation

Formula

Calculation

1. Multiply the top: 3 × 4.0 = 12, so the top is 12 × 10² kg km.
2. Divide: 12 × 10² ÷ (5 × 10²) = 2.4, since 10² ÷ 10² = 1. The result is 2.4 kg km s⁻².
3. Change km to m: 2.4 × 10³ kg m s⁻².

Given Data

• (a) 0.0045 m (b) 2.047 m (c) 3.40 m (d) 3.420 × 10⁴ m

To Find

• The number of significant figures in each

Calculation

1. (a) Zeros at the start do not count; 4 and 5 remain, so 2.
2. (b) The zero between 2 and 4 counts, so 4.
3. (c) The end zero after the decimal counts, so 3.
4. (d) Count only the digits in 3.420, including the end zero, so 4.

Given Data

• (a) 0.0035 m (b) 206.4 × 10² m

To Find

• Each in scientific notation

Formula

Calculation

1. (a) Move the point 3 places right to get 3.5. The power is −3, so 3.5 × 10⁻³ m.
2. (b) 206.4 = 2.064 × 10², so 206.4 × 10² = 2.064 × 10² × 10².
3. Add the powers: 2 + 2 = 4, so 2.064 × 10⁴ m.

Given Data

• (a) 5.0 × 10⁴ cm (b) 580 × 10² g (c) 45 × 10⁻⁴ s

To Find

• Each with a suitable prefix

Formula

Calculation

1. (a) 5.0 × 10⁴ cm = 50 000 cm. Divide by 100 to get 500 m, then by 1000 to get 0.50 km.
2. (b) 580 × 10² g = 58 000 g. Divide by 1000 to get 58 kg.
3. (c) 45 × 10⁻⁴ s = 4.5 × 10⁻³ s. Since 10⁻³ s = 1 ms, this is 4.5 ms.

Given Data

• v = 3.0 × 10⁸ m s⁻¹
• t = 1 year = 365 days

To Find

• Distance light travels in one year (1 light year)

Formula

Calculation

1. Time in seconds: t = 365 × 24 × 3600 s = 31 536 000 s.
2. Distance = speed × time = v × t.
3. Distance = (3.0 × 10⁸) × 31 536 000 = 9.4608 × 10¹⁵ m.

Given Data

• 1 g = 10⁻³ kg
• 1 cm = 10⁻² m, so 1 cm³ = 10⁻⁶ m³

To Find

• Density of mercury in kg m⁻³

Formulas

Calculation

1. Put the conversions in: 13.6 g cm⁻³ = 13.6 × 10⁻³ kg ÷ 10⁻⁶ m³.
2. Subtract the powers: −3 − (−6) = +3, so 13.6 × 10³ kg m⁻³.
3. In scientific notation: 1.36 × 10⁴ kg m⁻³.