Gravitation Notes

Gravitation

Introduction

• We have previously learned about motion, force, and the relationship between them.
• A force is required to change an object's speed or direction.
• Objects fall towards the Earth when dropped from a height.
• Planets revolve around the Sun, and the Moon revolves around the Earth.
• A force, known as the gravitational force, is responsible for these phenomena.
• This chapter will cover gravitation, the universal law of gravitation, the motion of objects under gravitational force, weight variation, and conditions for objects to float in liquids.

Gravitation

• The Moon revolves around the Earth.
• Objects thrown upwards fall downwards after reaching a certain height.
• Newton's observation of an apple falling led him to question if the Earth's attraction could extend to the Moon.
• He hypothesized that the same force is responsible for both the apple falling and the Moon's orbit.
• The Moon continuously falls towards the Earth instead of moving in a straight line due to Earth's attraction.
• Activity 9.1 demonstrates the concept of centripetal force using a thread and stone.

Activity 9.1

• A stone tied to a thread moves in a circular path with constant speed, changing direction at every point, indicating acceleration.
• The force causing this acceleration and maintaining circular motion is directed towards the center, called centripetal force.

Universal Law of Gravitation

• Every object in the universe attracts every other object with a force that is:
  • Proportional to the product of their masses.
  • Inversely proportional to the square of the distance between them.
• The force acts along the line joining the centers of the two objects.
• In the absence of this force, objects would move along a straight line (tangent to the circular path).
• The motion of the Moon around the Earth is due to the centripetal force provided by Earth's gravitational attraction.
• If there were no gravitational force, the Moon would move in a uniform straight line.
• The apple also attracts the Earth, according to the third law of motion. However, the Earth's acceleration towards the apple is negligible due to its large mass (second law of motion).
• Similarly, there exists a force between the Sun and the planets.
• Newton concluded that all objects in the universe attract each other.
• This attractive force between objects is the gravitational force.

Mathematical Representation:

• Let $M$ and $m$ be the masses of two objects A and B, respectively, and let $d$ be the distance between them.

• Let $F$ be the force of attraction between the two objects.

  • $F \propto M \times m$ (9.1)
  • $F \propto \frac{1}{d^2}$ (9.2)

• Combining equations (9.1) and (9.2):

  • $F \propto \frac{M \times m}{d^2}$ (9.3)
  • $F = G \frac{M \times m}{d^2}$ (9.4)

• Where $G$ is the universal gravitation constant.

  • $G = \frac{F \times d^2}{M \times m}$ (9.5)

• The SI unit of $G$ is N m$^2$ kg$^{-2}$.

• The value of $G$ was determined by Henry Cavendish and is approximately $6.673 × 10^{-11} N m^2 kg^{-2}$.

• The law is universal, applicable to all bodies, regardless of size or location.

Inverse-Square Law

• If the distance $d$ increases by a factor of 6, the force $F$ becomes 36 times smaller.

Example 9.1

• Calculate the force exerted by the Earth on the Moon.

  • Mass of Earth, $M = 6 × 10^{24} kg$
  • Mass of Moon, $m = 7.4 × 10^{22} kg$
  • Distance between Earth and Moon, $d = 3.84 × 10^5 km = 3.84 × 10^8 m$
  • $G = 6.7 × 10^{-11} N m^2 kg^{-2}$

• $F = G \frac{M \times m}{d^2} = 6.7 × 10^{-11} \frac{6 × 10^{24}× 7.4 × 10^{22}}{(3.84 × 10^8)^2} = 2.02 × 10^{20} N$

• The force exerted by the Earth on the Moon is $2.02 × 10^{20} N$.

Importance of The Universal Law of Gravitation

• Successfully explains several phenomena:
  • The force that binds us to Earth.
  • The motion of the Moon around the Earth.
  • The motion of planets around the Sun.
  • The tides caused by the Moon and the Sun.

Free Fall

• Objects falling towards the Earth due to gravitational force alone are in free fall.

Activity 9.2

• Demonstrates that objects thrown upwards fall back down due to Earth's gravitational attraction.

• During free fall, objects experience a change in the magnitude of velocity due to Earth's attraction, resulting in acceleration.

• This acceleration is called acceleration due to gravity, denoted by $g$, with units of m s$^{-2}$.

• The gravitational force $F$ is equal to the product of mass $m$ and acceleration due to gravity $g$: $F = m g$ (9.6)

  • $mg = G \frac{M m}{d^2}$
  • $g = G \frac{M}{d^2}$ (9.7)

• For objects on or near the Earth's surface, $d$ is approximately equal to the radius of the Earth, $R$.

  • $mg = G \frac{M m}{R^2}$ (9.8)
  • $g = G \frac{M}{R^2}$ (9.9)

• The Earth is not a perfect sphere, so the value of $g$ varies, being greater at the poles than at the equator.

• For most calculations, $g$ is considered constant near the Earth's surface.

Calculating the Value of g

• Using the values:

  • Universal gravitational constant, $G = 6.7 × 10^{-11} N m^2 kg^{-2}$
  • Mass of the Earth, $M = 6 × 10^{24} kg$
  • Radius of the Earth, $R = 6.4 × 10^6 m$

• $g = \frac{G M}{R^2} = \frac{6.7 × 10^{-11} × 6 × 10^{24}}{(6.4 × 10^6)^2} = 9.8 m s^{-2}$

• The acceleration due to gravity on Earth, $g = 9.8 m s^{-2}$.

Motion of Objects Under the Influence of Gravitational Force of The Earth

• Objects, regardless of size or composition, should fall at the same rate in a vacuum.

Activity 9.3

• Demonstrates that a sheet of paper falls slower than a stone due to air resistance.
• In a vacuum (glass jar with air removed), both paper and stone would fall at the same rate.
• As $g$ is constant near the Earth, equations of uniformly accelerated motion are valid with $a$ replaced by $g$.
  • $v = u + at$ (9.10)
  • $s = ut + \frac{1}{2} a t^2$ (9.11)
  • $v^2 = u^2 + 2 a s$ (9.12)
• $a$ is positive when in the direction of motion and negative when opposing the motion.

Example 9.2

• A car falls off a ledge and drops to the ground in 0.5 s. Assume $g = 10 m s^{-2}$.

  • (i) What is its speed on striking the ground?
  • (ii) What is its average speed during the 0.5 s?
  • (iii) How high is the ledge from the ground?

• Solution:

  • Time, $t = 0.5 s$

  • Initial velocity, $u = 0 m s^{-1}$

  • Acceleration due to gravity, $g = 10 m s^{-2}$

  • Acceleration of the car, $a = +10 m s^{-2}$ (downward)

  • (i) Speed: $v = a t = 10 m s^{-2} × 0.5 s = 5 m s^{-1}$

  • (ii) Average speed: $\frac{u + v}{2} = \frac{0 + 5}{2} = 2.5 m s^{-1}$

  • (iii) Distance travelled: $s = \frac{1}{2} a t^2 = \frac{1}{2} × 10 m s^{-2} × (0.5 s)^2 = 1.25 m$

Example 9.3

• An object is thrown vertically upwards and rises to a height of 10 m. Calculate:

  • (i) The velocity with which the object was thrown upwards.
  • (ii) The time taken by the object to reach the highest point.

• Solution:

  • Distance travelled, $s = 10 m$

  • Final velocity, $v = 0 m s^{-1}$

  • Acceleration due to gravity, $g = 9.8 m s^{-2}$

  • Acceleration of the object, $a = -9.8 m s^{-2}$ (upward motion)

  • (i) $v^2 = u^2 + 2 a s$

    • $0 = u^2 + 2 × (-9.8 m s^{-2}) × 10 m$
    • $-u^2 = -2 × 9.8 × 10 m^2 s^{-2}$
    • $u = \sqrt{196} = 14 m s^{-1}$

  • (ii) $v = u + a t$

    • $0 = 14 m s^{-1} - 9.8 m s^{-2} × t$
    • $t = \frac{14}{9.8} = 1.43 s$

Mass

• Mass is a measure of inertia.
• Greater mass implies greater inertia.
• Mass remains constant regardless of location (Earth, Moon, or outer space).

Weight

• Weight is the force with which an object is attracted towards the Earth.
• $F = m × a$ (9.13)
• $F = m × g$ (9.14)
• $W = m × g$ (9.15), where $W$ is the weight of the object.
• The SI unit of weight is the newton (N), the same as force.
• Weight is a force acting vertically downwards, having both magnitude and direction.
• At a given place, weight is directly proportional to mass: $W \propto m$.
• Mass remains constant, but weight depends on location because $g$ varies.

Weight of An Object on The Moon

• Weight on the Moon is the force with which the Moon attracts an object.

• The Moon's mass is less than Earth's, resulting in a lesser force of attraction.

• Let:

  • $m$ = mass of the object.
  • $W_m$ = weight on the Moon.
  • $M_m$ = mass of the Moon.
  • $R_m$ = radius of the Moon.

• $W<em>m = G \frac{M</em>m m}{R_m^2}$ (9.16)

• Let:

  • $W_e$ = weight of the same object on Earth.
  • $M$ = mass of the Earth.
  • $R$ = radius of the Earth.

• $W_e = G \frac{M m}{R^2}$ (9.17)

• Using values from Table 9.1:

  • $W_m = G \frac{7.36 × 10^{22} kg × m}{(1.74 × 10^6 m)^2} = 2.431 × 10^{10} G m$ (9.18a)

  • $W_e = G \frac{5.98 × 10^{24} kg × m}{(6.37 × 10^6 m)^2} = 1.474 × 10^{11} G m$ (9.18b)

  • $\frac{W<em>m}{W</em>e} = \frac{2.431 × 10^{10}}{1.474 × 10^{11}} = 0.165$ (9.19)

• Weight of the object on the Moon = (1/6) × its weight on the Earth.

Example 9.4

• Mass of an object is 10 kg. What is its weight on Earth?
  • $m = 10 kg$
  • $g = 9.8 m s^{-2}$
  • $W = m × g = 10 kg × 9.8 m s^{-2} = 98 N$

Example 9.5

• An object weighs 10 N on Earth. What would it weigh on the Moon?
• Weight on the Moon = $\frac{1}{6}$ × weight on Earth = $\frac{1}{6}$ × 10 N = 1.67 N.

Thrust and Pressure

• Thrust: Force acting on an object perpendicular to the surface.
• Pressure: Thrust per unit area, given by: $Pressure = \frac{Thrust}{Area}$ (9.20).
• The SI unit of pressure is N/m$^2$ or N m$^{-2}$, also known as pascal (Pa).

Example 9.6

• A wooden block of mass 5 kg with dimensions 40 cm × 20 cm × 10 cm is placed on a tabletop. Find the pressure exerted when it lies on sides:
  • (a) 20 cm × 10 cm
  • (b) 40 cm × 20 cm
  • Solution:
    • Thrust = $F = m × g = 5 kg × 9.8 m s^{-2} = 49 N$
    • (a) Area = 20 cm × 10 cm = 200 cm$^2$ = 0.02 m$^2$
      • Pressure = $\frac{49 N}{0.02 m^2} = 2450 N m^{-2}$
    • (b) Area = 40 cm × 20 cm = 800 cm$^2$ = 0.08 m$^2$
      • Pressure = $\frac{49 N}{0.08 m^2} = 612.5 N m^{-2}$
    • Smaller area exerts larger pressure, and larger area exerts smaller pressure.

Pressure in Fluids

• Liquids and gases are fluids.
• Fluids exert pressure on the base and walls of their container.
• Pressure in a confined fluid is transmitted undiminished in all directions.

Buoyancy

• Buoyancy is the upward force exerted by a fluid on an immersed object.
• The magnitude of buoyant force depends on the density of the fluid.

Why Objects Float or Sink When Placed on The Surface of Water?

• Objects with density less than that of the liquid float.
• Objects with density greater than that of the liquid sink.

Archimedes’ Principle

• When a body is immersed fully or partially in a fluid, it experiences an upward force equal to the weight of the fluid displaced by it.
• Applications: designing ships and submarines, lactometers, and hydrometers.