We need to convert physical quantities between different systems of units by applying appropriate conversion factors for mass, length, and time.
Concept: Unit conversion
Unit conversion involves multiplying by ratios of equivalent units to change the scale while maintaining physical consistency. The dimensional analysis method ensures that the units on both sides of the conversion remain balanced.
Given:
- (a) 1 kg⋅m2⋅s−2
- (b) 1 m
- (c) 3.0 m⋅s−2
- (d) G=6.67×10−11 N⋅m2⋅kg−2
Solving:
-
Conversion of Energy Units
- We know that 1 kg=103 g and 1 m=102 cm.
- Substituting these into the expression:
1 kg⋅m2⋅s−2=(103 g)⋅(102 cm)2⋅s−2=103⋅104 g⋅cm2⋅s−2=107 g⋅cm2⋅s−2
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Conversion of Length to Light Years
- A light year (ly) is the distance light travels in a vacuum in one year.
- 1 ly=c⋅t=(2.9979×108 m⋅s−1)⋅(365.25×24×3600 s)≈9.461×1015 m.
- Therefore:
1 m=9.461×10151 ly≈1×10−16 ly
-
Conversion of Acceleration Units
- We convert meters to kilometers (1 m=10−3 km) and seconds to hours (1 s=36001 h).
3.0 m⋅s−2=3.0⋅(1/3600 h)210−3 km=3.0⋅10−3⋅(3600)2 km⋅h−2=3.0⋅10−3⋅12960000 km⋅h−2=38880 km⋅h−2
- Rounding to two significant figures:
3.9×104 km⋅h−2
-
Conversion of the Gravitational Constant
- First, express G in base SI units using 1 N=1 kg⋅m⋅s−2:
G=6.67×10−11 m3⋅kg−1⋅s−2
- Convert m3 to cm3 (106) and kg−1 to g−1 (10−3):
G=6.67×10−11⋅(102 cm)3⋅(103 g)−1⋅s−2=6.67×10−11⋅106⋅10−3 cm3⋅g−1⋅s−2=6.67×10−8 cm3⋅g−1⋅s−2
Answer:
(a)
107
(b)
1.0×10−16
(c)
3.9×104
(d)
6.67×10−8