Question

Gravity causes objects to be attracted to one another. This attraction keeps our feet firmly planted on the ground and causes the moon to orbit the earth. The force of gravitational attraction is represented by the equationF=Gm1m2/r2where F is the magnitude of the gravitational attraction on either body,m1 and m2 are the masses of the bodies, r is the distance between them, and G is the gravitational constant. In SI units, the units or force are kg m/s2, the units of mass are kg, and the units of distance are m. For this equation to have consistent units, the units of Gmust be which of the following?One consequence of Einstein's theory of special relativity is that mass is a form of energy. This mass-energy relationship is perhaps the most famous of all physics equations:,where m is mass, c is the speed of the light, and E is the energy. In SI units, the units of speed are m/s. For the preceding equation to have consistent units (the same units on both sides of the equation), the units of E must be which of the following?ANSWER:

Answer 1

1) $[m^3\cdot kg^{-1}\cdot s^{-2}]$

2) $[kg\cdot m^2 \cdot s^{-2}]$

Explanation:

1)

The magnitude of the gravitational force is given by

$F=\frac{G m_1 m_2}{r^2}$

where

F is the force

G is the gravitational constant

m1, m2 are the two masses

r is the distance between the two masses

The SI units used for the various quantities are:

Force: Newton [N], equivalent to $[kg \cdot m\cdot s^{-2}]$

Mass: kilogram [kg]

Distance: meter [m]

If we re-arrange the equation making G the subject,

$G=\frac{Fr^2}{m_1m_2}$

And substituting the units, we find the units of G:

$[G]=\frac{[kg\cdot m \cdot s^{-2}][m]^2}{[kg][kg]}=\frac{[m^3]}{[kg][s^2]}=[m^3\cdot kg^{-1}\cdot s^{-2}]$

2)

The mass-energy relationship from Einstein's equivalence is

$E=mc^2$

where

E is the energy

m is the mass

c is the speed of light

The SI units for the various quantities are:

mass: kilogram [kg]

speed: meters per second, $[m\cdot s^{-1}]$

Therefore, the SI units for the energy must be:

$[E]=[kg]\cdot [m\cdot s^{-1}]^2 =[kg]\cdot [m]^2 \cdot [s^{-2}]=[kg\cdot m^2 \cdot s^{-2}]$