## Gravitation Science Notes

Gravitation :

• A force can set a body in motion. For example, if a ball at rest on the floor is pushed, it rolls on the floor.
• A force can stop a moving body.
• For example, a moving bicycle can be brought to rest by application of brakes.
• A force acting on a body can change the speed of the body.
• For example, when brakes are applied to a moving bicycle, its speed decreases due to the friction between the brake shoes and the rim of the tyre.
• A force can change the direction of motion of the body.
• For example, in uniform circular motion of a body, the direction of motion of the body keeps on changing due to the applied force.
• A force can change the speed as well as the direction of motion of the body.
• For example, when a ball bowled by a bowler is hit by a batsman, there occurs a change in the speed as well as the direction of motion of the ball.
• A force can change the shape and size of the body force can cha on which it acts. For example, when a rubber ball is pressed, it gets deformed and hence no longer remains spherical.
• Also, there can be a decrease in its volume.
• The gravitational force between the earth and the moon, the electromagnetic force between two charged particles in motion, the nuclear force between a proton and a neutron in the nucleus of an atom.
• The gravitational force is a universal force, i.e., it acts between any two objects in the universe.
• Newton’s first law of motion : An object continues to remain at rest or in a state of uniform motion along a straight line unless an external unbalanced force acts on it.
• Newton’s second law of motion : The rate of change of momentum is proportional to the applied force and the change of momentum occurs in the direction of the force.
• Newton’s third law of motion : Every action force has an equal and opposite reaction force which acts simultaneously.

[Note: Equal in magnitude and opposite in direction)

Circular motion and centripetal force :

→ As long as we are holding the string. Fig. 1.1 : A stone tied to a string, moving we are pulling the stone alons a circular path towards us, i.e., towards and its velocity in the centre of the circle tangential direction and are applying a force towards it.

→ The force stops acting if we release the string. In this case, the stone will fly off along a straight line which is the tangent to the circle at the position of the stone when the string is released, because that is the direction of its velocity at that instant of time (Figure 1.1 (b)].

→ Thus, a force acts on any object moving along a circle and it is directed towards the centre of the circle. This is called the centripetal force. ‘Centripetal’ means centre seeking, i.e., the object tries to go towards the centre of the circle because of this force.

Kepler’s laws:

• “An ellipse is the curve obtained when a cone is cut by an inclined plane.
• It has two focal points. The sum of the distances to the two focal points from every point on the curve is constant.
• F, and F, are two focal points of the ellipse shown in figure 1.2. If A, B and C are three points on the ellipse then, AF1 + AF2 = BF1 + BF2 = CF1 + CF2 Kepler’s laws of planetary motion :
• The orbit of a planet is an ellipse with the Sun at one of the foci.
• The line joining the planet and the Sun sweeps equal areas in equal intervals of time.
• The square of the period of revolution of a planet around the Sun is directly proportional to the cube of the mean distance of the planet from the Sun.

[Note : Strictly speaking. (period of revolution)2 ∝  $$\left(\frac{a b}{2}\right)^{3}$$ (Fig. 12)]

Newton’s universal law of gravitation :

• Every object in the Universe attracts every other object with a definite force.
• This force is directly proportional to the product of the masses of the two objects and inversely proportional to the square of the distance between them.

The earth’s gravitational force :

• The gravitational force on any object due to the earth is always directed towards the centre of the earth.
• If the object is on the earth’s surface, in the usual notation,
$$F=\frac{\mathrm{G} m_{1} m_{2}}{r^{2}}$$
• The value of G was first experimentally measured by Henry Cavendish. In SI units its value is 6.673 × 10-11 Nm2 kg-2

Acceleration due to the gravitational force of the earth :

→ The acceleration produced in a body due to the earth’s gravitational force is called the acceleration due to gravity or the earth’s gravitational acceleration and its magnitude is denoted by g. It is directed towards the earth’s centre.

→ $$g=\frac{G M}{r^{2}}$$ for r = ≥ R (radius of the earth).
It depends on the location of the body.

→ Change in the value of g with height above the earth’s surface

 Place Height (km) g (m /s2) Surface of the earth (average) 0 9.81 Mount Everest 8.8 9.8 Maximum height reached by a man-made balloon 36.6 9.77 Height of a typical weather satellite 400 8.7 Height of a communication satellite 35700 0.225
• Mass : The mass of an object is the amount of matter present in it. Its SI unit is kg.
• Weight : The weight of an object is defined as the force with which the earth attracts the object. Its magnitude is mg and SI unit is the newton (N).

→ Colloquially we use weight for both mass and weight and measure the weight in kilogram which is the unit of mass. But in scientific language when we say that Rajeev’s weight is 75 kg, we are talking about Rajeev’s mass.

→ What we mean is that Rajeev’s weight is equal to the gravitational force on 75 kg mass. As Rajeev’s mass is 75 kg, his weight on the earth is F-mgee 75 × 9.8 = 735 N.

→ The weight of 1 kg mass is 1 × 9.8 = 9.8 N. Our weighing machines tell us the mass. The two-pan scale balance in a shop compares two weights, i.e., two masses.

• The force exerted by the person holding the stone, the force exerted by air and the earth’s gravitational force.
• The stone falls to the ground.
• The forces exerted by air and the earth’s gravitational force.

→ Yes. Two objects kept on a table do not move towards each other because there is a force of friction between each object and the table. Similarly, because there is a force of friction between our body and the floor, we (myself and my friend) do not move towards each other.

• High and low tides occur regularly in the sea.
• The level of sea water at any given location along sea shore increases and decreases twice a day at regular intervals.
• High and low tides occur at different times at different places. The level of water in the sea changes because of the gravitational force exerted by the moon.
• Water directly under the moon gets pulled towards the moon and the level of water there goes up causing high tide at that place.
• At two places on the earth at 90° from the place of high tide, the level of water is minimum and low tides occur there as shown in figure 1.3.

Free fall :

• Whenever an object moves under the influence of gravity alone, it is said to be falling freely.
• For a freely falling object, with v = 0 and a = 8, we have v = $$\frac{1}{2}$$gt, s = gt2 and v2 = 2gs (in the usual notation)
• For an object thrown upward, as the object moves upward, the direction of acceleration is opposite to that of the velocity. Hence, the acceleration is negative, with a = -g.
• The value of g is the same for all objects at a given place on the earth.
• Thus, any two objects, irrespective of their masses or any other properties, when dropped from the same height and falling freely will reach the earth at the same time.
• Galileo is said to have performed an experiment around 1590 in the Italian city of Pisa.
• He dropped two spheres of different masses from the leaning tower of Pisa to demonstrate that both spheres reached the ground at the same time.
• When we drop a feather and a heavy stone at the same time from a height, they do not reach the earth at the same time.
• The feather experiences a buoyant force and a frictional force due to air and therefore floats and reaches the ground slowly, later than the heavy stone.
• The buoyant and frictional forces on the stone are much less than the weight of the stone and do not affect the speed of the stone much.
• Recently, scientists performed this experiment in vacuum and showed that the feather and stone indeed reach the earth at the same time.

• Escape velocity : vese –$$\sqrt{\frac{2 G M}{R}}=\sqrt{2 g R}$$
• For u = vese (from the earth’s surface), the body overcomes the earth’s gravitational attraction.
• It will then move to infinity and come to rest there. The gravitational potential energy of an object at a heighth from the earth’s surface = $$=-\frac{G M m}{R+h}=-\frac{m g R^{2}}{R+h}$$
• The total energy of a body revolving around the earth – kinetic energy + potential energy
= $$\frac{1}{2} m v^{2}+\left(-\frac{G M m}{R+h}\right)$$

→ Uniform circular motion of a planet around the Sun :

• The formula for escape velocity given in the textbook, does not take into account the effect of atmosphere. In practice, the body becomes very hot due to friction with air and may even burn.
• Even when a body is projected obliquely from the earth’s surface, with u – vese, it will overcome the earth’s gravitational influence and move to infinity.

Weightlessness in space :

→ Space travellers as well as objects in the spacecraft appear to be floating. Though the spacecraft is at a height from the surface of the earth, the value of g there is not zero.

→ In the space station the value of g is only 11% less than its value on the surface of the earth. Thus, the height of a spacecraft is not the reason for their weightlessness.

→ Their weightlessness is caused by their being in the state of free fall. Though the spacecraft is not falling on the earth because of its velocity along the orbit, the only force acting on it is the gravitational force of the earth and therefore it is in a free fall.

→ As the velocity of free fall does not depend on the properties of an object, the velocity of free fall is the same for the spacecraft, the travellers and the objects in the craft. Thus, if a traveller releases an object from her hand, it will remain stationary with respect to her and will appear to be floating.

Gravitational waves :

• Waves are created on the surface of water when we drop a stone into it.
• Similarly you must have seen the waves generated on a string when its both ends are held in hand and it is shaken.
• Light is also a wave called the electromagnetic wave.
• Gamma rays, X-rays, ultraviolet rays, infrared rays, microwave and radio waves are all electromagnetic waves with different frequencies.
• Astronomical objects emit these waves and we receive them using our instruments.
• All our knowledge about the universe has been obtained through these waves.
• Gravitational waves are a very different type of waves.
• They have been called the waves on the fabric of space-time.
• Einstein predicted their existence in 1916.
• These waves are very weak and it is very difficult to detect them.
• Scientists have constructed extremely sensitive instruments to detect the gravitational waves emitted by astronomical sources.
• Among these, LIGO (Laser Interferometric Gravitational wave Observatory) is the prominent one.
Exactly after hundred years of their prediction, scientists detected these waves coming from an astronomical source.
• Indian scientists have contributed significantly in this discovery.
• This discovery has opened a new path to obtain information about the Universe.