Assertion (A): The force of attraction between a hollow spherical shell of uniform density and a point mass situated out side is just as if the entire mass of the shell is concentrated at the centre of the shell.
Reason (R): Gravitational forces caused by the various regions of the shell have components along the line joining the point mass to the centre as well as along a direction perpendicular to this line.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true, it's a direct consequence of Newton's Shell Theorem. Reason (R) is false because, by symmetry, the perpendicular components of gravitational forces from various regions of a uniform spherical shell cancel out, resulting in a net force purely along the line joining the point mass to the center. Therefore, (A) is true but (R) is false.
Assertion (A): The gravitational force between two finite bodies is necessarily along the line joining their centre of mass.
Reason (R): The gravitational force between two particles is not central.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is false.
The gravitational force between two finite bodies is generally not directed along the line joining their centers of mass.
Reason (R) is also false. The gravitational force between two point particles is always central, acting along the line connecting them. Therefore, both (A) and (R) are false.
Assertion (A): If the law of gravitation, instead of being inverse square law then planets will still have elliptical orbits.
Reason (R): In that case also, \( T^2 \propto r^3 \) (symbols having usual meanings)
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is false. Only an inverse-square force leads to stable, closed elliptical orbits. An inverse cube law would lead to unstable orbits. Reason (R) is also false. Kepler's third law, \( T^2 propto r^3 \), is a direct consequence of the inverse-square law. If the law changes, this relation no longer holds. Therefore, both (A) and (R) are false.
Assertion (A): Gravitational potential energy of any mass particle may not be zero at earth centre.
Reason (R): Gravitational field intensity at earth centre is zero.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true. Gravitational potential at the center of a uniform solid sphere is finite (e.g., \( -\frac{3}{2} \frac{GM}{R} \)), hence potential energy is non-zero. Reason (R) is true. Due to symmetry, the net gravitational field at the Earth's center is zero. However, zero field does not directly explain non-zero potential energy. Thus, (R) is not the correct explanation of (A).
Assertion (A): If the product of surface area and density is same for two planets, escape velocities at surface will be same for both planets.
Reason (R): For given mass of a planet \( v_e \propto R^{-1/2} \)
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true.
Escape velocity \( v_e = \sqrt{\frac{2GM}{R}} \). Substitute \( M = \rho \frac{4}{3} \pi R^3 \) to get \( v_e = \sqrt{\frac{8G\pi R^2 \rho}{3}} \). If \( R^2 \rho \) is constant (derived from \( A\rho \) being constant), then \( v_e \) is constant. Reason (R) is true, as \( v_e = \sqrt{\frac{2GM}{R}} \) shows \( v_e \propto R^{-1/2} \) for constant \( M \). However, (R) does not explain (A), as the conditions are different.
Assertion (A): If earth stops rotating about its axis, then the value of acceleration due to gravity increases everywhere, except at the poles.
Reason (R): The value of acceleration due to gravity is maximum at the poles.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Assertion (A): The effective gravity is given by \(g' = g - Romega^2cos^2lambda\). If \(omega = 0\), then \(g' = g\). This causes \(g\) to increase everywhere except at poles (where \(coslambda = 0\)). So (A) is true.nReason (R): Due to rotation and equatorial bulge, \(g\) is maximum at poles and minimum at the equator. So (R) is true.n(R) correctly explains (A) as the effect of rotation explains the variation.
Assertion (A): Even when orbit of a satellite is elliptical, its plane of rotation passes through the centre of earth.
Reason (R): According to law of conservation of angular momentum plane of rotation of satellite always remain same.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Assertion (A): For any central force, the orbit is always planar and contains the center of force (Earth's center). So (A) is true.
Reason (R): Gravitational force is a central force, so net torque is zero. This implies conservation of angular momentum, \(\vec{L} = \vec{r} \times \vec{p}\), which means the orbital plane is fixed. So (R) is true. (R) correctly explains (A).
Assertion (A): The radius vector from the sun to a planet sweeps out equal areas in equal times interval.
Reason (R): Transverse (perpendicular to radius vector) acceleration of the planet is zero.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Assertion (A): This statement is Kepler's Second Law, which is a direct consequence of angular momentum conservation. So (A) is true.nReason (R): For a central force, like gravity, the force acts along the radius vector, meaning no transverse force component exists. Thus, transverse acceleration is zero. So (R) is true.n(R) explains (A) because zero transverse acceleration leads to conservation of angular momentum, which implies Kepler's Second Law.
Assertion (A): Earth is continuously pulling moon towards its centre but moon does not fall to earth.
Reason (R): Attraction of sun on moon is greater than that of earth on moon.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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Assertion (A): The Moon has sufficient tangential velocity to orbit the Earth, so it continuously 'falls around' the Earth instead of falling into it. So (A) is true.nReason (R): Calculations show that the gravitational force from the Sun on the Moon is indeed greater than that from the Earth on the Moon. So (R) is true.nHowever, the Sun's attraction does not prevent the Moon from falling to Earth; the Moon's orbital velocity does. Thus, (R) is not the correct explanation for (A).
Given below are two statements: one is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A): Angular momentum conservation can be used to explain Kepler’s second law of planetary motion.
Reason (R): Areal velocity of a planet revolving around the sun is equal to its angular momentum.
In the light of the above statements, choose the correct answer from the options given below:
1. Both (A) and (R) are true and (R) is the correct explanation of (A)
2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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Assertion (A) is true because Kepler's second law represents the conservation of angular momentum. Reason (R) is false because areal velocity is \(\frac{dA}{dt} = \frac{L}{2m}\), which is proportional to angular momentum \(L\) but not equal to it.