THP

Gravitation Notes

Notes by Mohit Sir(MSc)

Gravitational Force

All objects in this universe attract each other; this force of attraction is called Gravitational Force.

Newton's Observations

  • Why does an apple fall on Earth from a tree?
  • Because Earth attracts it towards itself.

Centripetal Force

When a body moves in a circular path, it changes its direction at every point. The force which keeps the body in the circular path acts towards the centre of the circle. This force is called centripetal force.

Key Point

If there is no centripetal force, the body will move in a straight line tangent to the circular path.

Universal Law of Gravitation

Newton's law of gravitation (1687)-

The universal law of gravitation states that, 'Every object in the universe attracts every other object with a force which is directly proportional to product of the masses and inversely proportional to the square of the distance between them.'

Mathematical Representation

Let \( m_1 \) and \( m_2 \) be the masses of two objects and distance \( d \), then force of attraction between them:

\[ F \propto m_1 \times m_2 \quad \text{(i)} \] \[ F \propto \frac{1}{d^2} \quad \text{(ii)} \]

From above equations, we can write:

\[ F \propto \frac{m_1 m_2}{d^2} \] \[ F = \frac{G m_1 m_2}{d^2} \]

where \( G \) is the gravitational constant.

  • SI Unit: \( Nm^2 kg^{-2} \)
  • Value of \( G \): \( 6.673 \times 10^{-11} Nm^2 kg^{-2} \) (found out by Henry Cavendish)

Importance of Universal Law of Gravitation

  1. Binds us to the Earth.
  2. Controls Moon's motion around Earth.
  3. Explains Earth's motion around the Sun.
  4. Causes tides in seas due to the Moon.

Free Fall

Free Fall: When an object falls under the influence of Earth's gravitational force alone, it is called free fall.

Acceleration due to Gravity: The acceleration experienced by an object during free fall due to Earth's gravity is called acceleration due to gravity.

  • Denoted by 'g'
  • SI unit = \( ms^{-2} \) (same as acceleration)

    • Value of 'g' on the Surface of Earth

    We know that force acting on an object:

    \[ F = \frac{G M_e m}{R^2} \quad \text{(i)} \]

    where:
    \( M_e \) = mass of Earth
    \( m \) = mass of object
    \( R \) = radius of Earth

    We also know from 2nd law of motion:

    \[ F = mg \quad \text{(ii)} \]

    From equations (i) and (ii), we have:

    \[ m \times g = \frac{G M_e m}{R^2} \] \[ g = \frac{G M_e}{R^2} \]

    When a body is at distance 'r' from the center of Earth then:

    \[ g = \frac{G M}{r^2} \]

    Key Point

    Value of 'g' may vary at different parts of Earth:

    • From the equation \( g = \frac{G M}{r^2} \), it is clear that the value of g depends on \( r \) (distance of object from Earth's center).
    • Because the shape of Earth is not a perfect sphere - it is flat at poles and bulged out at the equator.
    • Therefore, the value of 'g' is greater at the poles and lesser at the equator. But for convenience, we take 'g' as constant.
    Calculation of g \[ g = \frac{G M}{R^2} \] \[ = \frac{6.673 \times 10^{-11} \times 6 \times 10^{24} kg}{(6.4 \times 10^6 m)^2} \] \[ g = 9.8 ms^{-2} \]

    Mass of Earth = \( 6 \times 10^{24} kg \)

    Radius of Earth = \( 6.4 \times 10^6 m \)

    Standard value: \( g = 9.8 ms^{-2} \)

    Difference between G and g

    Gravitational Constant (G) Gravitational Acceleration (g)
    1. Its value is \( 6.673 \times 10^{-11} Nm^2/kg^2 \). 1. Its value is \( 9.8 m/s^2 \).
    2. Its value remains constant everywhere. 2. Its value varies at different places.
    3. Its unit is \( Nm^2/kg^2 \). 3. Its unit is \( m/s^2 \).
    4. It is a scalar quantity. 4. It is a vector quantity.

    Equations of Motion for Free Fall

    Case 1: Object falling towards Earth with initial velocity (u)

    \[ v = u + gt \] \[ s = ut + \frac{1}{2}gt^2 \] \[ v^2 = u^2 + 2gh \]

    Case 2: Object falling from rest (initial velocity u=0)

    \[ v = gt \] \[ s = \frac{1}{2}gt^2 \] \[ v^2 = 2gh \]

    Case 3: Object thrown vertically upwards with initial velocity (u)

    Gravitational acceleration (g) will be negative:

    \[ v = u - gt \] \[ s = ut - \frac{1}{2}gt^2 \] \[ v^2 = u^2 - 2gh \]

    Mass and Weight

    Mass

    Mass of an object is the measure of its inertia.

    • SI unit = Kilogram (kg)
    • Mass is a scalar quantity.
    • Mass of a body remains constant.
    • Mass of a body cannot be zero.

    Weight

    The force by which an object is attracted towards the center of Earth is called Weight of that object.

    We know, force = mass × acceleration:

    \[ F = m \times g \quad (\text{as } a = g) \]

    From above definition, this force is weight so:

    \[ W = m \times g \]
    • SI unit = Newton (N)
    • Vector quantity.

    Weight of an Object on the Moon

    The Moon, having less mass than Earth, pulls objects with less force.

    Important Conclusion

    An object’s weight on the Moon is one-sixth of its weight on Earth.

    Thrust and Pressure

    a) Thrust

    Thrust is the force acting on an object perpendicular to the surface.

    Example

    When you stand on loose sand the force (weight) of your body is acting on an area equal to the area of your feet. When you lie down, the same force acts on an area equal to the contact area of the whole body. In both cases the force acting on the sand (thrust) is the same.

    b) Pressure

    Pressure is the force acting on unit area of a surface.

    \[ \text{Pressure} = \frac{\text{Thrust}}{\text{Area}} \]

    Example

    The effect of thrust on loose sand is larger while standing than while lying down.

    • The SI unit of thrust is \( N/m^2 \) or \( N m^{-2} \).
    • It is called Pascal (Pa).

    Pressure in Fluids

    a) Pressure in fluids (Liquids and gases)

    • Fluids exert pressure on the base and walls of the container.
    • Fluids exert pressure in all directions.
    • Pressure exerted on fluids is transmitted equally in all directions.

    b) Buoyancy (Upthrust)

    When an object is immersed in a fluid it experiences an upward force called buoyant force. This property is called buoyancy or upthrust.

    The force of gravity pulls the object downward and the buoyant force pushes it upwards.

    The magnitude of the buoyant force depends upon the density of the fluid.

    c) Why objects float or sink in water?

    If the density of an object is less than the density of a liquid, it will float on the liquid and if the density of an object is more than the density of a liquid, it will sink in the liquid.

    Activity

    Take some water in a beaker. Take a piece of cork and an iron nail of the same mass. Place them on the water. The cork floats and the nail sinks.

    • The cork floats because the density of cork is less than the density of water and the upthrust of water is more than the weight of the cork.
    • The nail sinks because the density of the iron nail is more than the density of water and the upthrust of water is less than the weight of the nail.

    Archimedes' Principle

    Archimedes' principle states that, 'When a body is partially or fully immersed in a fluid it experiences an upward force that is equal to the weight of the fluid displaced by it.'

    Applications

    Archimedes principle has many uses:

    • It is used in designing ships and submarines.
    • Hydrometers used to determine the density of liquids.
    • Lactometers used to determine purity of milk.
    Weight on Moon vs Earth

    Density and Relative Density

    i) Density

    The density of a substance is the mass of a unit volume of the substance.

    \[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
    • The unit of density is kilogram per metre cube \( (kg m^{-3}) \).

    ii) Relative Density

    The relative density of a substance is the ratio of the density of a substance to the density of water.

    \[ \text{Relative density} = \frac{\text{Density of a substance}}{\text{Density of water}} \]

    Key Point

    Since relative density is a ratio of similar quantities, it has no unit.

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