Newton's Law of Gravitation

Net gravitational force at the center of a square is found to be $F_1$ when four particles having mass $M, 2M, 3M$ and $4M$ are placed at the four corners of the square as shown in figure and it is $F_2$ when the positions of $3M$ and $4M$ are interchanged. The ratio $\frac{F_1}{F_2}$ is $\frac{\alpha}{\sqrt{5}}$. The value of $\alpha$ is ______.

Three identical spheres of mass m, are placed at the vertices of an equilateral triangle of length $a$. When released, they interact only through gravitational force and collide after a time $T = 4$ seconds. If the sides of the triangle are increased to length $2a$ and also the masses of the spheres are made 2 m, then they will collide after__________seconds.

A small point of mass $m$ is placed at a distance $2R$ from the centre ' $O$ ' of a big uniform solid sphere of mass M and radius R. The gravitational force on ' m ' due to M is $F_1$. A spherical part of radius $R/3$ is removed from the big sphere as shown in the figure and the gravitational force on m due to remaining part of $M$ is found to be $F_2$. The value of ratio $F_1 : F_2$ is

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 $\left(\frac{2\sqrt{2}+1}{32}\right)\frac{Gm^{2}}{L^{2}}$, the length of the sides of the square is

Two identical metal spheres are kept in contact with each other, each having radius ' $R$ ' cm and ' $\rho$ ' is the density of material of metal spheres. The gravitational force ' $F$ ' of attraction between them is proportional to

A metal wire of uniform mass density having length $L$ and mass $M$ is bent to form a semicircular arc and a particle of mass $m$ is placed at the centre of the arc. The gravitational force on the particle by the wire is:

A thin rod of length '$L$' is bent in the form of a circle. Its mass is '$M$'. What force will act on mass '$m$' placed at the centre of this circle?

( $G =$ constant of gravitation)

A system consists of three particles each of mass '$m_1$ ' placed at the corners of an equilateral triangle of side ' $\frac{L}{3}$ ', A particle of mass '$m_2$ ' is placed at the mid point of any one side of the triangle. Due to the system of particles, the force acting on $m_2$ is

If two identical spherical bodies of same material and dimensions are kept in contact, the gravitational force between them is proportional to $R^{x}$, where $x$ is non zero integer [Given: $R$ is radius of each spherical body]

Match List - I with List - II

Choose the correct answer from the options given below

Two objects of equal masses placed at certain distance from each other attracts each other with a force of F. If one-third mass of one object is transferred to the other object, then the new force will be:

Given below are two statements:

Statement I: The law of gravitation holds good for any pair of bodies in the universe. Statement II: The weight of any person becomes zero when the person is at the centre of the earth. In the light of the above statements, choose the correct answer from the options given below.

Three identical particles $A, B$ and $C$ of mass $100\text{ kg}$ each are placed in a straight line with $AB = BC = 13\text{ m}$. The gravitational force on a fourth particle $P$ of the same mass is $F$, when placed at a distance $13\text{ m}$ from the particle $B$ on the perpendicular bisector of the line $AC$. The value of $F$ will be approximately:

Find the gravitational force of attraction between the ring and sphere as shown in the diagram, where the plane of the ring is perpendicular to the line joining the centres. If $\sqrt{8}$R is the distance between the centres of a ring (of mass 'm') and a sphere (mass 'M') where both have equal radius 'R'.

A solid sphere of radius R gravitationally attracts a particle placed at 3R from its centre with a force F1. Now a spherical cavity of radius $\left(\frac{R}{2}\right)$ is made in the sphere (as shown in figure) and the force becomes F2. The value of F1: F2 is

A straight rod of length L extends from x = a to x = L + a. The gravitational force it exerts on a point mass 'm' at x = 0, if the mass per unit length of the rod is A + Bx2, is given by:

Take the mean distance of the moon and the sun from the earth to be $0.4 \times 10^{6}$ km and $150 \times 10^{6}$ km respectively. Their masses are $8 \times 10^{22}$ kg and $2 \times 10^{30}$ kg respectively. The radius of the earth is $6400$ km. Let $\Delta F_1$ be the difference in the forces exerted by the moon at the nearest and farthest points on the earth and $\Delta F_2$ be the difference in the force exerted by the sun at the nearest and farthest points on the earth. Then, the number closest to $\frac{\Delta F_1}{\Delta F_2}$ is:

Two astronauts are floating in gravitational free space after having lost contact with their spaceship. The two will

From a sphere of mass M and radius R, a smaller sphere of radius $\frac{R}{2}$ is carved out such that the cavity made in the original sphere is between its centre and the periphery. (See figure). For the configuration in the figure where the distance between the center of the original sphere and the removed sphere is 3R, the gravitational force between the two spheres is:

The change in the value of acceleration of earth towards sun, when the moon comes from the position of solar eclipse to the position on the other side of earth in line with sun is: (mass of the moon = $7.36 \times 10^{22}$ kg, radius of the moon's orbit = $3.8 \times 10^{8}$ m).

(1) Centre of gravity (C.G) of a body is the point at which the weight of the body acts (2) Centre of mass coincides with the centre of gravity if the earth is assumed to have infinitely large radius.

(3) To evaluate the gravitational field intensity due to any body at an external point, the entire mass of the body can be considered to be concentrated at its C.G.

(4) The radius of gyration of any body rotating about an axis is the length of the perpendicular dropped from the C.G. of the body to the axis.

Which one of the following pairs of statements is correct?

A man of 50 kg mass is standing in a gravity free space at a height of 10 m above the floor. He throws a stone of 0.5 kg mass downwards with a speed 2 m/s. When the stone reaches the floor, the distance of the man above the floor will be

Two spheres of masses m and M are situated in air and the gravitational force between them is F. The space around the masses is now filled with a liquid of specific gravity 3. The gravitational force will now be

Two spherical bodies of mass $M$ and $5M$ & radii $R$ & $2R$ respectively are released in free space with initial separation between their centers equal to $12R$. If they attract each other due to gravitational force only, then the distance covered by the smaller body just before collision is