Given below are two statements : one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant.
Reason (R) : For a central force field the angular momentum is a constant.
In the light of the above statements, choose the most appropriate answer from the options given below :
Assertion (A) : The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant.
Reason (R) : For a central force field the angular momentum is a constant. In the light of the above statements, choose the most appropriate answer from the options given below :
Show Hint
- Both (A) and (R) are correct and (R) is the correct explanation of (A)
- Both (A) and (R) are correct but (R) is not the correct explanation of (A)
- (A) is correct but (R) is not correct
- (A) is not correct but (R) is correct
The Correct Option is A
Approach Solution - 1
To solve the given assertion and reason problem, we first need to analyze the statements individually and then collectively:
- Understanding the Assertion (A): "The radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time and thus areal velocity of planet is constant."
- This statement is based on Kepler's Second Law of Planetary Motion, which states that a line joining a planet to the sun sweeps out equal areas in equal times. This implies that the areal velocity, the area swept per unit time, is constant for a planet in orbit. Thus, Assertion (A) is correct.
- Understanding the Reason (R): "For a central force field the angular momentum is a constant."
- A central force field is a force field in which the force acts along the line joining the bodies and depends only on the distance between them, such as the gravitational force between a planet and the Sun.
- For a central force, the angular momentum is conserved because there is no external torque acting on the system, which supports the reason (R) being correct.
- Connecting Assertion and Reason:
- The constancy of areal velocity as mentioned in Assertion (A) is due to the conservation of angular momentum, as stated in Reason (R). Hence, (R) is indeed the correct explanation of (A).
Therefore, the correct answer is: Both (A) and (R) are correct and (R) is the correct explanation of (A).
Approach Solution -2
Step 1: Analyze Assertion (A).
Assertion (A) states Kepler's second law of planetary motion: the radius vector from the Sun to a planet sweeps out equal areas in equal intervals of time, implying constant areal velocity. This is a fundamental law of planetary motion and is correct.
Step 2: Analyze Reason (R).
Reason (R) states that for a central force field, the angular momentum is a constant. Gravitational force, which governs planetary motion around the Sun, is a central force. Under a central force, the torque on the planet with respect to the Sun is zero, leading to the conservation of the planet's angular momentum.
Thus, Reason (R) is also correct.
Step 3: Determine if Reason (R) is the correct explanation of Assertion (A).
The areal velocity \( \frac{dA}{dt} \) of a planet is mathematically related to its angular momentum \( L \) by \( \frac{dA}{dt} = \frac{L}{2m} \), where \( m \) is the mass of the planet. Since the gravitational force is central, the angular momentum \( L \) is conserved. As the mass \( m \) is also constant, the areal velocity \( \frac{dA}{dt} \) remains constant.
Therefore, the conservation of angular momentum (Reason (R)) directly explains the constant areal velocity (Assertion (A)).
Top JEE Main Physics Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
- $I _{ CM }$ is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc $I_{A B}$ is it's moment of inertia about an axis $AB$ perpendicular to plane and parallel to axis $CM$ at a distance $\frac{2}{3} R$ from center Where $R$ is the radius of the dise The ratio of $I _{ AB }$ and $I _{ CM }$ is $x: 9$ The value of $x$ is______
- A nucleus disintegrates into two smaller parts, which have their velocities in the ratio $3: 2$ The ratio of their nuclear sizes will be $\left(\frac{x}{3}\right)^{\frac{1}{3}}$ The value of ' $x$ ' is:-
Top JEE Main Gravitation Questions
- A planet takes 200 days to complete one revolution around the Sun. If the distance of the planet from Sun is reduced to one fourth of the original distance, how many days will it take to complete one revolution ?
- If \( R \) is the radius of the earth and the acceleration due to gravity on the surface of the earth is \( g = \pi^2 \, \text{m/s}^2 \), then the length of the second's pendulum at a height \( h = 2R \) from the surface of the earth will be:
- Four identical particles of mass \( m \) are kept at the four corners of a square. If the gravitational force exerted on one of the masses by the other masses is \[ \left( \frac{2 \sqrt{2} + 1}{32} \right) \frac{G m^2}{L^2}, \] the length of the sides of the square is:
- Assertion: The angular velocity of the moon revolving around the earth is more than that of the earth revolving around the Sun.
Reason: The time taken by the moon to revolve around the earth is less than the time taken by the earth to revolve around the sun. - A spherical body of mass 100 g is dropped from a height of 10 m from the ground. After hitting the ground, the body rebounds to a height of 5 m. The impulse of force imparted by the ground to the body is given by: (given \( g = 9.8 \, \text{m/s}^2 \))
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.