
Two bodies of mass $10\,kg$ and $5\,kg$ moving in concentric orbits of radii R and r such that their periods are the same. Then the ratio between their centripetal acceleration is
A. $\dfrac{R}{r} \\ $
B. $\dfrac{r}{R} \\ $
C. $\dfrac{{{R^2}}}{{{r^2}}} \\ $
D. $\dfrac{{{r^2}}}{{{R^2}}}$
Answer
164.4k+ views
Hint:In order to solve this question, we will use the general formula of centripetal acceleration for the circular motion and using this we will solve for the ratio of the centripetal acceleration of two bodies.
Formula used:
For a circular motion, centripetal acceleration is given by,
$a = \dfrac{{{v^2}}}{r}$
where v is velocity and r is the radius of a circular path.
Time period is given by,
$T = \dfrac{{2\pi r}}{v}$
Complete step by step solution:
According to the question, let us suppose velocities of given two masses $10kg$ and $5kg$ are ${v_1},{v_2}$ and their radius of circular path given to us R and r, given that time period are equal so,
$T = \dfrac{{2\pi R}}{{{v_1}}}$ and $T = \dfrac{{2\pi r}}{{{v_2}}}$
So, we have
$\dfrac{{2\pi R}}{{{v_1}}} = \dfrac{{2\pi r}}{{{v_2}}} \\
\Rightarrow \dfrac{{{v_1}}}{{{v_2}}} = \dfrac{R}{r} \to (i) \\ $
Now, the centripetal acceleration of both bodies are,
${a_1} = \dfrac{{{v_1}^2}}{R} \\
\Rightarrow {a_2} = \dfrac{{{v_2}^2}}{r} \\ $
On dividing both values of centripetal acceleration, we get
$\dfrac{{{a_1}}}{{{a_2}}} = {\left( {\dfrac{{{v_1}}}{{{v_2}}}} \right)^2}\dfrac{r}{R}$
Using value from equation (i) we get
$\dfrac{{{a_1}}}{{{a_2}}} = {\left( {\dfrac{R}{r}} \right)^2}\dfrac{r}{R} \\
\therefore \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{R}{r} \\ $
So, the ratio of the centripetal acceleration of both the bodies is $\dfrac{R}{r}$.
Hence, the correct answer is option A.
Note: It should be remembered that, centripetal acceleration and time period of a body moving in circular motion is independent of the mass of the body as centripetal acceleration and time period depends only upon the velocity of the body and radius of the circular path. and concentric orbits are those when two bodies move in circular orbits having different radii but their centers are the same.
Formula used:
For a circular motion, centripetal acceleration is given by,
$a = \dfrac{{{v^2}}}{r}$
where v is velocity and r is the radius of a circular path.
Time period is given by,
$T = \dfrac{{2\pi r}}{v}$
Complete step by step solution:
According to the question, let us suppose velocities of given two masses $10kg$ and $5kg$ are ${v_1},{v_2}$ and their radius of circular path given to us R and r, given that time period are equal so,
$T = \dfrac{{2\pi R}}{{{v_1}}}$ and $T = \dfrac{{2\pi r}}{{{v_2}}}$
So, we have
$\dfrac{{2\pi R}}{{{v_1}}} = \dfrac{{2\pi r}}{{{v_2}}} \\
\Rightarrow \dfrac{{{v_1}}}{{{v_2}}} = \dfrac{R}{r} \to (i) \\ $
Now, the centripetal acceleration of both bodies are,
${a_1} = \dfrac{{{v_1}^2}}{R} \\
\Rightarrow {a_2} = \dfrac{{{v_2}^2}}{r} \\ $
On dividing both values of centripetal acceleration, we get
$\dfrac{{{a_1}}}{{{a_2}}} = {\left( {\dfrac{{{v_1}}}{{{v_2}}}} \right)^2}\dfrac{r}{R}$
Using value from equation (i) we get
$\dfrac{{{a_1}}}{{{a_2}}} = {\left( {\dfrac{R}{r}} \right)^2}\dfrac{r}{R} \\
\therefore \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{R}{r} \\ $
So, the ratio of the centripetal acceleration of both the bodies is $\dfrac{R}{r}$.
Hence, the correct answer is option A.
Note: It should be remembered that, centripetal acceleration and time period of a body moving in circular motion is independent of the mass of the body as centripetal acceleration and time period depends only upon the velocity of the body and radius of the circular path. and concentric orbits are those when two bodies move in circular orbits having different radii but their centers are the same.
Recently Updated Pages
How to Calculate Moment of Inertia: Step-by-Step Guide & Formulas

Dimensions of Charge: Dimensional Formula, Derivation, SI Units & Examples

Dimensions of Pressure in Physics: Formula, Derivation & SI Unit

Environmental Chemistry Chapter for JEE Main Chemistry

Uniform Acceleration - Definition, Equation, Examples, and FAQs

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

NCERT Solutions for Class 11 Physics Chapter 1 Units and Measurements

Units and Measurements Class 11 Notes: CBSE Physics Chapter 1

NCERT Solutions for Class 11 Physics Chapter 2 Motion In A Straight Line

Motion in a Straight Line Class 11 Notes: CBSE Physics Chapter 2

Important Questions for CBSE Class 11 Physics Chapter 1 - Units and Measurement
