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
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?
Let the period of revolution of a planet at a distance R from a star be T. Prove that if it was at a distance of 2R from the star, its period of revolution will be \[\sqrt{8}\] T.
Answer the following question.
State Kepler’s law of equal areas.
Answer the following question in detail.
State Kepler’s three laws of planetary motion.
Observe the given figure showing the orbit of a planet moving around the Sun and write the three laws related to it:
The orbit of a planet moving around the Sun
The square of its period of revolution around the sun is directly proportional to the ______ of the mean distance of a planet from the sun.
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.
State Kepler’s laws.
If the distance between the sun and the earth is made three times, then attraction between the two will ______
A planet is revolving around the sun in an elliptical orbit as shown in figure. At which point will its K.E. be maximum?
The earth moves around the sun in an elliptical orbit as shown in the figure. The ratio, `"OA"/"OB"` = x. The ratio of the speed of the earth at Band at A is ______.
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.
Draw areal velocity versus time graph for mars.
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 ______.
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 ______
