Question

The earth moves round the sun once in a year with a speed of 30 kilometre per second. What is the centripetal acceleration of earth towards the sun ?
(A) $6 \times {10^{ - 7}}m/{s^2}$
(B) $6 \times {10^{ - 6}}m/{s^2}$
(C) $6 \times {10^{ - 5}}m/{s^2}$
(D) $6 \times {10^{ - 3}}m/{s^2}$

Answer 1

Hint: In order to solve this problem, first we calculate the distance between earth and sun by following expression :
$d = ct$
Where
d $ = $ distance between earth and sun
c $ = $ speed of light
t $ = $ time taken by light from sun to reach earth
After then apply this value in the formula of centripetal acceleration i.e., ${a_c} = \dfrac{{{v^2}}}{R}$
Here $R = d = $ distance between sun and earth
v $ = $ velocity of earth


Step by step answer:
Given that the velocity of earth when it revolves around the sun is 30 kilometre per second.
${v_e} = 3 \times {10^4}m/s$ …..(1)
We know that light takes 8 minutes from the sun to reach earth.
\[t = 8minute\]
$t = 8 \times 60 second$
$t = 480 seconds$
So, the distance between earth and sun is
$d = c \times t$
Here $c = 3 \times {10^8}m/s$ $($speed of light in vacuum$)$
$d = 3 \times {10^8} \times 480$
$d = 1440 \times {10^8}m$
$d = 144 \times {10^9}m$ …..(2)
Earth revolves around the sun in a circular orbit. So centripetal acceleration is given as
${a_c} = \dfrac{{v_e^2}}{d}$
On putting the values from equation 1 and 2
${a_c} = \dfrac{{{{(3 \times {{10}^4})}^2}}}{{144 \times {{10}^9}}}$
${a_c} = \dfrac{{3 \times 3 \times {{10}^4} \times {{10}^4}}}{{144 \times {{10}^9}}}$
${a_c} = \dfrac{9}{{144}} \times {10^8} \times {10^{ - 9}}$
${a_c} = \dfrac{9}{{144}} \times {10^{ - 1}}$
${a_c} = \dfrac{{900}}{{144}} \times {10^{ - 3}}$
${a_c} = 6.25 \times {10^{ - 3}}m/{s^2}$
Hence option D is correct answer i.e., $6 \times {10^{ - 3}}m/{s^2}$


Note: in circular motion, the main reason for circular direction of a particle is centripetal acceleration. i.e., centripetal acceleration is the property of the motion of a body traversing a circular path and the direction is radially towards the centre of the circle.