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### Fundamentals Of Physics

Book edition 10th Edition
Author(s) David Halliday
Pages 1328 pages
ISBN 9781118230718

# The Sun has mass ${\mathbf{2}}{\mathbf{.}}{\mathbf{0}}{\mathbit{x}}{{\mathbf{10}}}^{{\mathbf{30}}}{\mathbit{k}}{\mathbit{g}}$and radiates energy at the rate${\mathbf{3}}{\mathbf{.}}{\mathbf{9}}{\mathbf{x}}{{\mathbf{10}}}^{{\mathbf{26}}}{\mathbf{W}}$. (a) At what rate is its mass changing? (b) What fraction of its original mass has it lost in this way since it began to burn hydrogen, about${\mathbf{4}}{\mathbf{.}}{\mathbf{5}}{\mathbf{x}}{{\mathbf{10}}}^{{\mathbf{9}}}{\mathbf{y}}$ ago?

1. The rate at which the mass of the Sun is changing is$4.3×{10}^{9}\frac{\mathrm{kg}}{s}$.
2. The fraction of the original mass lost in this way is$3.1×{10}^{-4}$.
See the step by step solution

## Step 1: The given data

1. Mass of the Sun,${m}_{sun}=2.0x{10}^{30}kg$
2. Rate of energy radiation,$P=3.9x{10}^{26}W$
3. Time of change,$dt=4.5x{10}^{9}y$

## Step 2: Understanding the concept of energy change

In the given problem, the Sun radiates energy in the form of electromagnetic radiation. Thus, this change in energy per given time is given by the change in its mass rate as per the energy relation given by Einstein's energy relation. This results in a change in the mass of the sun which is given by the duration of the change.

Formulae:

The rate of energy in the form of radiationas follows:

$P=\frac{d\mathrm{E}}{dt}$…… (i)

The energy as per the Einstein’s mass-energy relation:

$E=m{c}^{2}$…… (ii)

The mass change of a body is as follows:

$∆m=\frac{dm}{dt}∆t$…… (iii)

## Step 3: a) Calculate the rate of mass changing

Let$m$be the mass of the Sun at time t and$E$be the energy radiated to that time. Then, the power output is given using equation (ii) with the given data in equation (i) as follows:

$\begin{array}{l}P=\frac{d\left(m{c}^{2}\right)}{dt}\\ =\frac{dm}{dt}{c}^{2}\end{array}$

Substitute the values and solve as:

$\frac{dm}{dt}=\frac{P}{{c}^{2}}$

Substitute the values and solve as:

Hence, the rate of mass change is$4.3×{10}^{9}\frac{kg}{s}$.

## Step 4: b) Calculate the original mass lost by the Sun

Using the given data and the above calculated data in equation (iii), the mass change of the body in the given timeis calculated as follows:

Now, the required fractional change of this quantity for the given original mass of the Sun is calculated as follows:

$\frac{∆m}{m+∆m}=\frac{6.15×{10}^{26}kg}{2.0×{10}^{30}kg+6.15×{10}^{26}kg}\phantom{\rule{0ex}{0ex}}=3.1×{10}^{-4}$

Hence, the value of fraction is$3.1×{10}^{-4}$.