Need Help?

Get in touch with us

bannerAd

Gravitational Acceleration – Experiments

Aug 24, 2022
link

Key Concepts:

  • Gravitational acceleration
  • Variation in gravitational acceleration
  • Weight 

Introduction: 

Suppose an object of mass m is kept on the surface of the Earth, then according to the universal law of gravitation, the force of gravity acting on the body FF can be given as follows. 

Mass of the earth = mE 

The radius of the earth = rE 

Mass of the object = m 

According to the universal law of gravitation the gravitational force on the mass m due to the gravitational pull of the Earth = FG 

parallel

Gravitational force on the object = FG = G mmE / r2E

Acceleration due to gravity on the surface of the earth

We know from Newton’s second law = Force = mass x acceleration 

We can write 

Gravitational force = mass x gravitational acceleration 

We denote gravitational acceleration = g  

parallel

FG = mg 

mg = G mmE / r2E

g = GmE / r2E

Gravitational acceleration = g =  GmE / r2E

Mass of the earth = mE  = 6 x 1024 kg 

Radius of the earth = rE = 6.4 x 103 km 

Gravitational constant = G = 6.7 x 10-11 Nm2/kg 

g = GmE / r2E

g = 6.7 ×10−11× 6 × 1024 = 9.8 m/s2 

Gravitational acceleration on the earth = g = 9.8 m/s2 

Explanation: 

  • The below image is taken from a multi-flash photograph showing the pattern of movement when an object falls on the surface of the earth. 
The increasing speed of a falling ball is captured in this multi-flash image. 
  • There are seven images of the ball, taken at equal intervals of time. The ball falls further in each successive time interval. This shows that its speed is increasing. 
  • This ball falling under gravity is executing uniformly accelerated motion. 

Example – An apple falling under gravity 

 An apple falling under gravity 
Velocity-time graph for a body falling under gravity

Acceleration due to gravity does not depend upon mass. 

The acceleration due to gravity is independent of the mass of the object falling towards the center of the earth. If air resistance and friction are negligible (or in vacuum) both the object fell at the same rate. 

Object falling under gravity 

We are so accustomed to the effect of air resistance and friction that we think that the light object (feather) falls slower than the heavy (hammer) one. If light and a heavy object dropped from the same height simultaneously, they will reach the ground at the same time. 

The earth applies gravitational force on all objects and objects also apply the same gravitational force on the earth then why do we not see the Earth be moving towards the mango or the moon? 

Gravitational forces  

From Newton’s 2nd law for a given force, the acceleration is proportional to the mass of the object. 

I.e., a α 1/m 

mmango <<< mearth 

Thus,  amango >>> aearth 

The Earth also moves towards the mango as a result of the gravitational pull of the mango on the Earth. However, the acceleration of the earth towards the mango is negligibly small and hence, not apparent. 

We do not see the Earth moving towards the mango or the ball similarly, the mass of the Moon is very small as compared to that of the Earth.  

mmoon << mearth 

Thus,  amoon >> aearth 

Weight of a body: 

The earth attracts everybody towards its centre with a certain force that depends on the mass of the body and the gravitational acceleration at that place. Weight is determined by the force it is attracted towards the centre of the earth.  

Measuring weight using spring balance 

Weight of a body on earth = Gravitational force exerted by the earth on the body  

W = mass of the body x gravitational acceleration 

W = mg 

Weight is measured using a spring balance. 

Variation in g due to shape of the earth: 

Variation in g due to shape of the Earth

Earth is elliptical in shape. It is flattened at the poles and bulged out at the equator. The equatorial radius is about 21 km longer than the polar radius. 

Mass of the earth = M 

Gravitational acceleration = g = GM/R2

Gravitational acceleration at equator = ge = GM/R2 e

Gravitational acceleration at pole = gp = GM/R2p

Since Re > Rp therefore gequator < gpole and approximately ge + 0.018 m/s2 = gp 

Therefore, the weight of a body increases as it is taken from equator to the pole.  

It has been experienced when astronauts land on the surface of the moon they feel less weight. 

Weight on the surface of the Moon: 

If we take an object to the moon, it will be weightless, because the Moon’s gravity is weaker than the Earth’s as comparatively its mass is small. However, the mass of the object will remain the same. 

Size of Earth = 4 x Size of Moon 

Gravitational Acceleration on Earth = ge

Gravitational Acceleration on Moon = gm=1/6ge

gm = 1/6ge= 9.8/6 = 1.66 m/s2 

 Weight on Earth = mge

Weight on Moon = mgm

Weight on Moon = (1/6) × Weight on Earth 

Question: 

What would be the weight of a person on the Moon and Earth whose mass is 12 kg? 

Answer: 

Weight of the person on earth  =  mgE  = 12 kg × 9.8 m/ s2  

     = approximately 120 N 

Weight of the person on moon = mgm = 120 N/6 = 20 N 

Summary:

  • According to the universal law of gravitation the gravitational force on the mass m due to the gravitational pull of the Earth =
  • According to Newton’s second law-: Force = mass x acceleration
  • Gravitational force = mass x gravitational acceleration
  • We denote gravitational acceleration = g

Fg = mg

mg = G mmE / r2E

Gravitational acceleration = g = GmE/r2E

  • Gravitational acceleration on the earth = g = 9.8 m/s?
  • The acceleration due to gravity is independent of the mass of the object falling towards
    the canter of the earth. If air resistance and friction are negligible (or in a vacuum) all
    objects fall down at the same rate.
  • Weight of a body on earth = Gravitational force exerted by the earth on the body
  • W= mass of the body x gravitational acceleration
  • W=mg
  • Earth is elliptical in shape. It is flattened at the poles and bulged out at the equator. The
    equatorial radius is about 21 km longer than the polar radius; therefore, the weight of a
    body increases as it is taken from the equator to the pole.
  • If we take an object to the moon, it will weightless, because the Moon’s gravity is weaker
    than the Earth’s as comparatively its mass is small. However, the mass of the object will
    remain the same.
  • Weight on Moon = (1/6) X Weight on Earth

Comments:

Related topics

Define Position Time Graph and its Types

Key Concepts • Slope of a graph • Position time graph • Slope of s-t graph = Velocity • Types of position time graphs Introduction An object in a uniform motion covers equal distances in equal intervals of time. This also indicates that it moves at a constant velocity. When its position at different instants […]

Read More >>

Magnetic Field Lines: Definition, Explanation and Q&A

Key Concepts Magnetic Field Magnetic Field Lines properties of magnetic field lines Uniform and non uniform magnetic lines Introduction Two magnets when placed close to each other attract and stick to each other. However, if we go on increasing the distance between them, the attraction between them reduces gradually to such an extent that they […]

Read More >>

The Life Cycles of Stars: Meaning and Example

Key Concepts Stars Analysis of starlight Composition of stars Stars’ temperature Size and mass of stars Stages of life cycle of a star Introduction Stars are huge, shining balls of extremely hot gas (known as plasma) in space. The Sun is our nearest star. During the nighttime, many other stars are visible to the naked […]

Read More >>

Mirror Formula

Key Concepts New cartesian sign convention Mirror formula Solving problems using the mirror formula Introduction When dealing with the reflection of light by spherical mirrors mathematically, a set of sign conventions is followed, called the New Cartesian Sign Convention. According to this convention, the pole of a spherical mirror is taken as the origin and […]

Read More >>

Other topics