Advertisements
Advertisements
प्रश्न
A star like the sun has several bodies moving around it at different distances. Consider that all of them are moving in circular orbits. Let r be the distance of the body from the centre of the star and let its linear velocity be v, angular velocity ω, kinetic energy K, gravitational potential energy U, total energy E and angular momentum l. As the radius r of the orbit increases, determine which of the above quantities increase and which ones decrease.
Advertisements
उत्तर
The situation is shown in the diagram, where a body of mass m is revolving around a star of mass M.
Linear velocity of the body `v = sqrt((GM)/r)`
⇒ `v ∝ 1/sqrt(r)`
Therefore, when r increases, v decreases.
Angular velocity of the body `ω = (2π)/T`
According to Kepler's law of the period,
`T^2 ∝ r^3` ⇒ `T = kr^(3/2)`
Where k is a constant
∴ `ω = (2π)/(kr^(3/2))` ⇒ `ω ∝ 1/r^(3/2)` .....`(∵ ω = (2π)/T)`
Therefore, when r increases, ω decreases.
The kinetic energy of the body,
K = `1/2 mv^2`
= `1/2 m xx (GM)/r`
= `(GMm)/(2r)`
∴ `K ∝ 1/r`
Therefore, when r increases, KE decreases.
The gravitational potential energy of the body,
`U = - (GMm)/r` ⇒ `U ∝ 1/r`
Therefore, when r increases, PE becomes less negative i.e., increases.
Total energy of the body,
`E = KE + PE`
= `(GMm)/(2r) + (- (GMm)/r)`
= `- (GMm)/(2r)`
Therefore, when r increases, total energy becomes less negative, i.e., increases.
Angular momentum of the body,
`L = mvr`
= `mr sqrt((GM)/r)`
= `msqrt(GMr)`
∴ `L ∝ sqrt(r)`
Therefore, when r increases, angular momentum L increases.
APPEARS IN
संबंधित प्रश्न
Let us assume that our galaxy consists of 2.5 × 1011 stars each of one solar mass. How long will a star at a distance of 50,000 ly from the galactic centre take to complete one revolution? Take the diameter of the Milky Way to be 105 ly
A Saturn year is 29.5 times the earth year. How far is the Saturn from the sun if the earth is 1.50 ×108 km away from the sun?
State Kepler's laws of planetary motion.
In the Following figure shows the elliptical path of a planet about the sun. The two shaded parts have equal area. If t1 and t2 be the time taken by the planet to go from a to b and from c to d respectively,
Answer the following question in detail.
State Kepler’s three laws of planetary motion.
Observe the given figure and answer these following questions.
- What is the conclusion about the orbit of a planet?
- What is the relation between velocity of planet and distance from sun?
- ASB, CSD and ESF relation between areas explain.
Write the Kepler’s laws.
In our solar system, the inter-planetary region has chunks of matter (much smaller in size compared to planets) called asteroids. They ______.
If the sun and the planets carried huge amounts of opposite charges ______.
- all three of Kepler’s laws would still be valid.
- only the third law will be valid.
- the second law will not change.
- the first law will still be valid.
If the sun and the planets carried huge amounts of opposite charges ______.
- all three of Kepler’s laws would still be valid.
- only the third law will be valid.
- the second law will not change.
- the first law will still be valid.
Supposing Newton’s law of gravitation for gravitation forces F1 and F2 between two masses m1 and m2 at positions r1 and r2 read F1 = – F2 = `- r_12/r_12^3 GM_0^2 ((m_1m_2)/M_0^2)^n` where M0 is a constant of dimension of mass r12 = r1 – r2 and n is a number. in such a case.
- the acceleration due to gravity on earth will be different for different objects.
- none of the three laws of Kepler will be valid.
- only the third law will become invalid.
- for n negative, an object lighter than water will sink in water.
The centre of mass of an extended body on the surface of the earth and its centre of gravity ______.
- are always at the same point for any size of the body.
- are always at the same point only for spherical bodies.
- can never be at the same point.
- is close to each other for objects, say of sizes less than 100 m.
- both can change if the object is taken deep inside the earth.
Out of aphelion and perihelion, where is the speed of the earth more and why?
The maximum and minimum distances of a comet from the Sun are 1.6 × 1012 m and 8.0 × 1010 m respectively. If the speed of the comet at the nearest point is 6 × 104 ms-1, the speed at the farthest point is ______.
lf the angular momentum of a planet of mass m, moving around the Sun in a circular orbit is L, about the center of the Sun, and its areal velocity is ______.
Halley's Comet revolves around the sun for a time period of 76 years. The aphelion distance if perihelion is given by 8.9 × 1010 m, will be ______.
(Take, the mass of sun = 2 × 1030 kg and G = 6.67 × 10-11 Nm3/kg2)
How can an ellipse be drawn using pins and thread?
The time taken by a planet to orbit the Sun depends on ______.
Two identical particles each of mass ‘m’ go round a circle of radius a under the action of their mutual gravitational attraction. The angular speed of each particle will be ______
