Class 11

The modulus of elasticity refers to the material's capacity to withstand or resist deformation when stress is applied. There are three types of the modulus of elasticity including:

Young's Modulus (E): It is used to measure a solid's elasticity when longitudinal stress (tensile or compressive) is applied to it. Mathematically, it is represented by the following formula;

SI Unit is N/m² (Pascal, Pa).

Bulk Modulus (K): It is used to measure the ability of the material to resist volume change when uniform pressure is applied to it. Mathematically, it is:

SI Unit is N/m² (Pa)

Shear Modulus (G) or Modulus of Rigidity: It is used when a shear force is applied to measure the resistance to shape change. Mathematically, it is represented by:

SI Unit: N/m² (Pa)

The value of each modulus varies for different materials, and it describes different types of deformation.

Hooke's Law states that within the elastic limit, the stress applied to a material is directly proportional to the strain produced. Mathematically,  

E stands for the modulus of elasticity also called Young's modulus in case of linear deformation.
The Hooke's Law defines the elastic behaviour of materials, determines its elastic limit and helps in material selection for construction and manufacturing. It is used to design springs and elastic components in machines. Hook's law is applicable for the solids within their elastic limit. 
Hooke's law for a spring and other elastic material is shown as:

F is the force applied on the material, k is the spring constant i.e. the stiffness of the material and it depends on the material's geometry and properties, and x is the extension or compression produced.

While studying the mechanical properties of solids, stress and strain are fundamental concepts.

Stress: It refers to the condition when the external force is applied to a solid body and it experiences an internal restoring force per unit area. The resisting deformation due to the internal force is called stress. The formula is:

The SI unit of stress is N/m² (Pascal, Pa).

There are three types of stress -

1. When the applied force increases the length of the body, it is called tensile stress.
2. When the applied force decreases the length of the body, it is termed as the compressive stress.
3. When the force is applied tangentially and creates deformation in shape, it is called shear stress.

Strain: It is a dimension less quantity. It refers to the relative deformation produced in a body due to applied stress. The formula is:

There are three types of strain:

1. Change in length per unit original length is called the Longitudinal Strain.
2. The change in volume per unit original volume is called the Volumetric Strain.
3. Angular deformation due to shear stress is called Shear Strain.

Hooke's Law relates stress and strain by a formula. It states that within the elastic limit, stress is directly proportional to strain:

E stands for the modulus of elasticity.

In Newton's Law of Gravitation, G (Universal Gravitational Constant) has a fixed value as it is a fundamental constant which is:

G determines the strength of gravitational interactions and it remains the same everywhere in the universe. It plays a significant role in cosmology and astrophysics and it is important in calculating the forces between celestial bodies. Due to the gravitational pull of a massive body like Earth, 'g' (Acceleration due to Gravity) is the acceleration which the object experiences. The value of g changes based on the planetary composition, latitude, and altitude. G is universal and g is dependent on the location. This relation can be understood by the following formula:

M represents the mass of the planet, and R is the radius. The concept and its differences help in fields such as space exploration where astronauts experience microgravity in orbit even though the gravitational forces are present.

It is due to a concept taken from Newton's Law of Gravitation. It says that each mass element of the shell puts an attractive force on a particle inside but due to the fact that the shape of the shell is symmetrical, these forces cancel out in every direction.
To prove it mathematically, one can use Gauss's Law for Gravitation. It states that inside a uniformly distributed spherical shell, the net gravitational field is zero. It differs from electrostatics, which states that a conducting shell blocks or stops the external electric fields. The shielding concept is not part of gravitation where the external bodies exert a gravitational force on an object inside the shell. It is the fundamental property of astrophysics and helps in understanding why the gravitational field inside a star or planet depends only on the mass enclosed within a given radius.

Gravitational Potential Energy (U) is the energy an object has due to its position in a gravitational field. It is defined as:

Where M represents the mass of the larger body, m is the mass of the smaller body, r is the distance between them, and G stands for the gravitational constant.
G is written with a negative sign to show that the gravitational potential energy is always lower than zero and the values even decrease as the objects move apart from each other. It implies that to separate two masses, work should be done against the gravity. At infinity, U = 0, implying that no external force is required to keep them apart. When two objects are closer, potential energy is more negative which implies a stronger gravitational attraction. The concept is very important in fields such as orbital mechanics and astrophysics.

Kepler's laws refer to the motion of planets around the Sun and it offers various insights into the celestial mechanics. It includes the following:

• The First Law (Law of Orbits) states that all planets move in elliptical orbits, and the Sun is at one focus. It is against the previous belief that planetary orbits were perfect circles.
• The Second Law (Law of Areas) states the line joining the Sun and a planet sweeps out equal areas in equal time intervals. It means that the planets near to Sun move faster and the ones at more distance, move slower.
• The third Law (Law of Periods) says that the square of a planet's orbital period is directly proportional to the cube of the semi-major axis of its orbit. The mathematical formula is:

According to Newton's Law of Universal Gravitation, each object in the universe attracts every other object with a force that depends on the distance between them and their masses. This force is mathematically given by the following formula:

Here, F is the gravitational force, G is the gravitational constant and m? and m? refers to the masses of the objects, and r represents the distance between them.

Newton's Law of Universal Gravitation explains the behaviour of tides due to the Moon's gravity, planetary motion, and even how satellites orbit the Earth. This law is widely used in space research and astrophysics to find out the masses of the celestial bodies and how they interact with each other. It is accurate in most cases but for extreme gravitational fields, instead of Newton's law, Einstein's General Theory of Relativity is more useful.