Force ( Solution of Modern Graded Science, Class 10)

2. Answer the following questions.

a. What is gravitation? What are the factors that affect gravitation? Write the SI unit of gravitation.

The mutual force of attraction between any two bodies in the universe is called gravitation. It is also called gravitational force. 

The factors that affect gravitation are:

i.                Masses of two bodies

ii.               Distance between their centres.             

The SI unit of gravitation is Newton (N)

    

 

b. State Newton's universal law of gravitation.

Newton's universal law of gravitation states that "every object attracts every other object in the universe with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centres."

 

 

c. What will happen to the gravitational force between two bodies if the distance between them is halved keeping their masses constant?

Do yourself

 

d. What will be the gravitational force between two heavenly bodies if the masses of both are tripled keeping the distance between them constant?

Ans:

Let us consider, two bodies, A and B have the masses 'm₁' and 'm₂'. The distance between them from their centres is 'd'. Then the force of gravitation 'F' between them is given by,

       F = Gm₁m₂  …….eq(i)

                 d²

According to the question, the mass of both bodies are tripled i. e. mass of body A is 3m₁ and mass of body B is 3m₂ keeping the distance constant, then,

 

       F₁ = G x 3m₁ x 3m₂  

                         d²

Or, F₁ = 9Gm₁m₂  

                   d²

Or, F₁ = 9F      (from eq i)

 

Hence, the force of gravitation between the bodies becomes nine times of the initial force if the mass of both the bodies are tripled keeping the distance between them constant.

 

e. What will be the gravitational force between two bodies if the mass of each is doubled and the distance between them is halved?

Do yourself

 

f. Define G. Also write its value and SI unit.

The universal gravitational constant 'G' is defined as the force of gravitation produced between two bodies of unit mass each, separated by unit distance between their centres. Its value is 6.67 X 10^-11 and SI unit is Nm²Kg^-2.

g. In this formula, ,what do G, M, and R stand for? Prove that:  

G stands for Universal gravitational constant, M stands for mass of the planet and R stands for radius of planet.

h. What will be the effect on the acceleration due to gravity of the earth if it is compressed to a the size the moon?

i. The acceleration due to gravity of Jupiter is 25 m/s². What does it mean?

The acceleration due to gravity of jupiter is 25 m/s² , it means that the velocity of a falling body increases by 25 m/s in each second if there is no air resistance.

j. A coin and a feather are dropped in vacuum. Which one will reach the ground first? Why?

A coin and feather when dropped in vacuum falls simultaneously due to free fall as there is no external resistance in vacuum.

k. Where does a body have more weight: at the pole or at the equator of the earth? Explain with reason.

The weight of an object depends on the magnitude of acceleration due to gravity (g) and the magnitude of acceleration due to gravity (g) depends on the radius of the earth. The value of 'g' is more at poles (i.e. 9.83 m/s²) due to less radius and less at equator (i.e. 9.78 m/s²) due to more radius. Therefore, the weight of an object is more at the pole than that in the equator of the earth.

l. When an apple falls towards the earth, the earth moves up to meet the apple. Is this true? If yes, why is the earth's motion not noticeable?

The earth and the apple pull each other with the same force.The acceleration due to the force is given by a=F/m. Where, F is the force and m is the mass of the object.

Since the apple has a much smaller mass, the acceleration it experiences is significantly higher when compared to the earth. Also, the mass of Earth is significantly larger. Hence, the acceleration it experiences is very small in magnitude. Thus, Earth doesn't move towards the apple.

m. Show with the help of an experiment that the acceleration of a freely falling body does not depend on the mass of that body.

n. An astronaut is said to be weightless when he/she travels in a satellite. Does it mean that the earth does not attract him/her?

An astronaut is said to be weightlessness when he travels in a satelite but it doesn't mean that earth doesn't attract him. When an astronaut is in satellite they both are in freefall towards the earth. The astronaut does not exert any force on the side of the satellite. As a result the astronaut feels to be floating weightlessly though the force of gravity at the distance may not be zero. As a result the earth attracts the astronaut but negligibly.

o. What is the importance of gravitational force? Write any two points. 

The importance of Gravitational force is: 

i. Existence of atmosphere on the surface of the earth.

ii. Flowing of river.

p. What is free fall? What happens to the weight of a body when it is falling freely under the action of gravity?

When a body is falling freely towards the centre of the earth under the influence of the gravitational pull of the earth, the motion of the body is called free fall.

The weight of a freely falling body under the action of gravity seems to be zero because the body fells zero reaction force.

q. What is the effect of gravity on a falling object? Write the conclusion obtained from the coin and feather experiment.

The effect of gravity on a falling object is: Gravity causes falling object to accelerate towards the surface of the earth. While falling, the velocity of the falling body goes on increasing.

Conclusion of feather and coin experiment:

Acceleration due to gravity of freely falling bodies, whatever their masses, is the same in the absence of air resistance.

r. Write the difference between the fall of a parachute on the earth and its fall on the moon? On what condition does a body have free fall?

On earth, due to the large surface area of the parachute, the air resistance will be more. As a result, the acceleration of the parachute decreases. The decreased acceleration due to gravity causes the paratrooper to fall slowly. As a result of slow motion, the paratrooper can balance their body and land safely without any hurt.

But, on the moon, the use of a parachute is not safe because of the absence of the atmosphere, no air pressure is felt. Thus, the parachute gets attracted by the gravitational force of the moon and the parachute falls forcibly onto the surface of the moon which may hurt paratroopers.

The condition necessary for free fall is: 

An object should fall only under the influence of gravity without external resistance.

s. Write any two examples of weightlessness.

The examples of weightlessness are:

i. Astronauts in the spaceship will experience weightlessness.

ii. People using elevators will experience weightlessness when the elevator is in free fall motion.

t.Define:

i. acceleration due to gravity

Acceleration due to gravity is defined as the acceleration produced in a freely falling body due to the force of the gravity of the earth. It is denoted by g and its SI unit is ms⁻² .

ii. free fall

When a body is falling freely towards the centre of the earth under the influence of the gravitational pull of the earth, the motion of the body is called free fall.

iii. weightlessness

Weightlessness is the condition of a body when its weight seems to be zero. In other words, weightlessness is the state of the body in which it feels that the body is not influenced by any force.

iv. null point

The space between any two heavenly bodies at which the resultant gravity is zero, is known as the null point.

3. Distinguish between:

a. Gravity and gravitation

b. Gravity and acceleration due to gravity

c. Mass and weight

d. g and G

e. Free fall and weightlessness

f. Gravitational force and acceleration due to gravity 

Gravitational force	Acceleration due to gravity
1.   It is the mutual force of attraction between any two bodies in the universe due to their masses.	1.  It is the acceleration produced in a freely falling body due to the force of gravity of the earth.
2.   It depends on the masses of two bodies and distance between their centres.	2.   It depends on the mass and radius of   planet, but doesn’t depend on the mass of falling body.
3. SI unit of gravitational force is N.	3. SI unit of acceleration due to gravity is  ms-2

g.

4. Give reasons.

a. Newton's law of gravitation is called a universal law.

Newton's law of gravitation is called universal law because this law holds truth for all the bodies, whether microscopic or massive placed everywhere in the universe.

b. The weight of a body decreases with increase in distance from the earth's surface.

We have relation: Weight= mass x acceleration due to gravity

The value of  acceleration due to gravity decreases when the object is far from the surface of the earth so weight of the object also decreases when distance is increased.

C.The weight of a stone is less at the top of a mountain than that at its bottom.

The weight of an object depends on the magnitude of acceleration due to gravity (g) and the magnitude of acceleration due to gravity (g) is inversely proportional to the square of the radius of the earth. The  value of 'g' is more at the bottom of mountain due to less radius and less at the top of mountain due to more radius. Therefore, the weight of a stone is less at the top of mountain than at its bottom.

d. The weight of a body is less at the moon than its weight on the earth.

The weight of body depends upon the mass and acceleration due to gravity (g). As we know that 'g' of the earth (9.8 m/s²) is more than that of the moon (1.6 m/s²), hence the weight of the body is less at the moon than its weight on the earth.

e. The moon has no atmosphere.

As there is less gravitational force on the moon, it does not attract the atmosphere on its surface as a result there is no atmosphere on the surface of the moon.

f. The value of g is greater at the poles than at the equator.

The shape of the earth is not perfectly spherical. It is flattened at the poles and bulges out at the equator. Therefore, the radius of the equator is more than the radius of the pole. As we know that, 'g' is inversely proportional to the square of the radius of the earth, i.e. g∝1/R² so, the value of 'g' is more at the poles than at the equator.

g. There is more role of the moon than the sun to cause tides in oceans.

The ocean tides on earth are caused by both the moon's gravity and the sun's gravity. Even though the sun is much more massive and therefore has stronger gravity than the moon, the moon is closer to the earth so that its gravitational gradient(gravitational force from one location to another) is stronger than that of the sun. Thus, the moon plays a more role in creating tides than does the sun. 

h. Tides occur in oceans but not in lakes.

Tide is an effect of gravity on ocean water that causes the sea level to rise. As gravity is directly proportional to the product of two masses, it shows its effect on the ocean. Due to less mass of the water of lake, the gravity of the moon is not much active there and hence tides occurs in oceans but not in lakes.

i. It is difficult to lift a large stone on the surface of the earth but it is easy to lift a small one.

At a particular place on the earth, the value of acceleration due to gravity is constant. The weight of an object is directly proportional to its mass, i.e. W ∝ m. It shows that a lighter object is attracted with less force and the heavier object is attracted with more force. So, it is difficult to lift a large stone than to lift a small one.

j. Mass of the Jupiter is 319 times more than that of the earth but its acceleration due to gravity is only 2.5 times more than the earth.

The acceleration due to gravity of a planet having its mass 'M' and radius 'R' is given by g= GM/R². It means  g∝1/R². Eventhough the mass of the Jupiter is so much heavier than that of the earth, the square of its radius reduces the value of acceleration due to gravity. Thus, the mass of the Jupiter is about 319 times heavier than that of the earth, but its acceleration due to gravity is only about 2.5 times greater than that of the earth.

k. The weight of a body will be zero during free fall.

The weight of a body will be zero during free fall because in this condition the acceleration of falling body is equal to the acceleration due to gravity and it does not  feel any kind of reaction force.

l. A paratrooper does not get injured while jumping on the earth from a flying aeroplane.

When the paratrooper jump out of an aeroplane, s/he accelerate under the effect of gravity. As s/he open parachute, the air resistance slows down the speed. Once the force of air resistance balances the force of gravity, the paratrooper no longer accelerate, and come down slowly. So, the paratrooper does not get injured while jumping on the earth from a flying aeroplane.

m. When a coin and a feather are dropped from the roof of a building, the coin reaches the ground earlier.

When a coin and feather are dropped from the roof of a building, the coin reaches the ground earlier. It is  due to the larger surface area of the feather which experiences more air resistance and falling speed get decreased.

We all know that the sun revolves around the sun in the elliptical orbit which is oval in shape.

We also know that the gravitational force between different heavenly bodies is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers. In position 'A', the distance is longer; so the gravitational force is less. But in position 'B', the distance is shorter; so the gravitational force is higher than that in position 'A'.

Thus, the force of gravitation of the sun towards the earth increases as the earth moves from point A to point B.

a: The name of given experiment is feather and coin experiment.

b: The conclusion that can be drawn from the experiment is; acceleration due to gravity of freely falling bodies, whatever their masses, is the same in the absence of air resistance.

c: If the tube is filled with air, the coin will fall faster and feather will fall later, because the larger surface area of the feather experiences more air resistance and falling speed get decreased.

5. 3. Study the given diagram and find the gravitational force between two spherical objects.

Solution:

Thus, the gravitation force between two spherical object is 3.335 x 10⁻⁷ N. 

6. Numerical Problems.

a. A sphere of mass 40 kg is attracted by another sphere of mass 15 kg with a force of 9.8x10⁻⁷ N. Find the value of the universal gravitational constant if the centres of spheres are 20 cm apart.

Solution: 

In this case, the value of universal gravitational constant is 6.53 x 10⁻¹¹ Nm²Kg⁻².

b. A heavenly body has a mass equal to half of the mass of the earth and its radius half as that of the earth. If a stone weighs 100 N on the surface of the earth, find the weight of the stone on the heavenly body.

Solution:

Here, weight of stone on the  earth (W)= 100 N

The weight of the stone on the heavenly body is 200 N.

c.  Calculate the force of attraction between two bodies with their mass of 100 kg each and they are 1m apart on the surface of the earth. Will the force of attraction be different if the same bodies are taken on the moon, their separation remaining constant?

Solution:

Here,

Masses of two bodies (m₁)=(m₂)= 100 kg

Distance between their centers (d) = 1m

Universal gravitational constant (G) = 6.67 x 10⁻¹¹ Nm²kg⁻²

The gravitational force of attraction between them (F) =?

We know that, F = Gm₁m₂

                                      d²

                               =  6.67 x 10⁻¹¹ x 100 x 100

                                                 1²

                               = 6.67 x 10⁻¹¹ x 10⁴

                                                 1

                                = 6.67 x 10⁻¹¹ ⁺ ⁴

                                = 6.67 x 10⁻⁷ N

The gravitational force of attraction between them is 6.67 x 10⁻⁷ N.

The gravitational force of attraction between two bodies depends upon the masses of two bodies and the distance between their centers. Since mass is constant and distance of separation is kept the same on the moon as on the earth so the force of attraction won’t be different.

                                               

 

d. The acceleration due to gravity at the surface of the moon is 1.6 ms^-². If the radius of the moon is 1.7 x 10⁶ m, calculate the mass of the moon.

 

Solution:

Here,

acceleration due to gravity at the surface of the moon  (g) = 1.6 ms^-²

radius of moon (R) = 1.7 x 10⁶ m

Universal gravitational constant (G) = 6.67 x 10⁻¹¹ Nm²kg⁻²

mass of moon (M) = ?

We know that,

                   g = GM

                           R²

           Or,  1.6 =  6.67 x 10⁻¹¹ x M

                                (1.7 x 10⁶ )²

           Or, 1.6 x  (1.7 x 10⁶ )² = 6.67 x 10⁻¹¹ x M

           Or, 1.6 x 2.89 x 10¹² =  6.67 x 10⁻¹¹ x M

           Or,  4.624 x 10¹²  = M

                  6.67 x 10⁻¹¹

                   Or, M =   4.624 x 10¹²⁺¹¹  

                                         6.67

                   Or, M =  4.624 x 10²³  

                                         6.67

                  Or,  M =  46.24 x 10²²  

                                         6.67

                  Or, M = 6.93 x 10²² kg

Hence, the mass of moon is 6.93 x 10²² kg .

                                             

 

e. Do yourself ( similar to que d)

f. The masses of the sun and the earth are 2 x 10³⁰ kg and 6×10²⁴ kg respectively. If the distance between their centres is 1.5x 10¹¹ m, find the gravitational force between them.

Solution:

Here,

Mass of the sun (m₁) = 2 x 10³⁰ kg

Mass of the earth (m₂)= 6×10²⁴ kg

Distance between their centers (d) = 1.5x 10¹¹ m

Universal gravitational constant (G) = 6.67 x 10⁻¹¹ Nm²kg⁻²

The gravitational force of attraction between them (F) =?

 We know that,

             F = Gm₁m₂

                        d²

                 =  6.67 x 10⁻¹¹ x  2 x 10³⁰ x  6×10²⁴

                                     (1.5x 10¹¹ )²

 

                  =  80.04 × 10³⁰⁺²⁴⁻¹¹

                             2.25 x 10²²

 

                   =  80.04 × 10⁴³⁻²²

                                2.25

 

                    = 35.57  × 10²¹

                    = 3.557 × 10²² N

                    = 3.56 × 10²² N

The gravitational force of attraction between them is  3.56 × 10²² N.

 

 

g. A body weighs 63 N on the surface of the earth. What is the gravitational force on it due to the earth at a height of 3200 km from the earth's surface? (Radius of the earth is 6400 km)

 Solution:

Let, M be the mass of earth, m be the mass of body

Given,

radius of the earth = 6400 km

                                   = 6400 x 10³m

                                   = 6.4 x 10³ x 10³ 

                                   = 6.4 x 10⁶m

weight  of a body 

on the earth (W) = GMm  = 63 N

                                     R²

 

                    Or, GMm = 63 x R²

                    Or, GMm = 63 x (6.4 x 10⁶)²

                    Or, GMm = 63 x 40.96 x 10¹²

                    Or, GMm = 2580.48 x 10¹²

 

height (h) = 3200 km = 3.2 x 10⁶m

gravitational force at a height of 3200 km from the earth's surface is given by,

 

       F = GMm

             (R+h)²

 

          =          2580.48 x 10¹²         .

                  ( 6.4 x 10⁶+  3.2 x 10⁶)²

 

 

          =          2580.48 x 10¹²         .

                     {( 6.4 +  3.2 )x 10⁶}²

          

          =  2580.48 x 10¹²    .

               ( 9.6)² x 10¹²

 

         = 2580.48

               92.16

        =  28 N

 

The gravitational force on the body due to the earth at a height of 3200 km from the earth's surface is 28 N.

h. A man can lift a mass of 200 kg on the surface of the earth. What is the amount of mass he can lift on the surface of the moon?

Solution:

Here, on the earth,

Mass of the body on the earth (m) = 200 kg

Acceleration due to gravity of the earth (g) = 9.8ms⁻²

 Weight (W)=?

 

We have,

W= mxg = 200 x 9.8 = 1960 N

The person can lift 1960 N on the earth.

 

On the moon, we have

Again,

Weight (W)= 1960 N

Acceleration due to gravity of the moon (g_m) = 1.67 ms⁻²

Mass can be lifted on the moon (m_m) = ?

 

We know that,

      W= m_m x g_m

Or, m_m = W      = 1960  = 1173.6 Kg

                 g_m         1.67

Thus, the person can lift 1173.6 kg mass on the moon.

i. The mass of the Jupiter is 2 x 10²⁷ kg and its radius 6.5x10⁷ m. What is the acceleration due to gravity on the Jupiter? Also, calculate the weight of a person having the mass of 70 kg on the Jupiter.

 

Solution:

Here,

mass of Jupiter (M) = 2 x 10²⁷ kg

radius of Jupiter (R) = 6.5x10⁷  m

Universal gravitational constant (G) = 6.67 x 10⁻¹¹ Nm²kg⁻²

acceleration due to gravity on the Jupiter  (g) = ?

We know that,

            g = GM

                    R²

 

               = 6.67 x 10⁻¹¹ x 2 x 10²⁷

                         (6.5x10⁷)²

 

               = 13.34 x 10²⁷⁻¹¹

                     42.25 x 10¹⁴

 

               = 0.3157 x 10²⁷⁻¹¹⁻¹⁴

               = 0.3157 x 10²

               = 31.57 ms⁻²

 

Again,

mass of person on Jupiter (m) = 70 kg

acceleration due to gravity on the Jupiter  (g) = 31.57 ms⁻²

Weight of a person on Jupiter (W) = mg

                                                             = 70 x 31.57

                                                             = 2209.09 N

The acceleration due to gravity on the Jupiter is 31.57 ms⁻² and the weight of a person having the mass of 70 kg on the Jupiter is 2209.09 N.

j. The mass of the earth is 6x 10²⁴ kg and radius of the moon is 1.7x10⁶ m. Calculate the acceleration due to gravity of the new earth which is formed by the compression of the earth equal to the size of the moon.

 

Solution:

Here,

mass of the earth (M) = 6x 10²⁴ kg

radius of new earth (R) =1.7x10⁶ m

Universal gravitational constant (G) = 6.67 x 10⁻¹¹ Nm²kg⁻²

acceleration due to gravity on the new earth  (g) = ?

We know that,

            g = GM

                    R²

               = 6.67 x 10⁻¹¹ x 6x 10²⁴

                         (1.7x10⁶)²

 

              =  40.02 x 10²⁴⁻¹¹

                         2.89x10¹²

 

             = 13.8477 x 10¹³⁻¹²

             = 13.8477 x 10

             = 138.48 ms⁻²

 

Hence, the acceleration due to gravity of the new earth which is formed by the compression of the earth equal to the size of the moon is 138.48 ms⁻².

k. The mass of the Jupiter is 1.9×10²⁷ kg and that of the sun is 2×10³⁰ kg. If the magnitude of the gravitational force between these two masses is 4.166×10²³ N, find the distance between them.

 

Solution:

Here,

Mass of the Jupiter (m₁) =  1.9×10²⁷ kg

Mass of the sun (m₂)= 2×10³⁰ kg

Universal gravitational constant (G) = 6.67 x 10⁻¹¹ Nm²kg⁻²

The gravitational force of attraction between them (F) = 4.166×10²³ N

Distance between their centers (d) = ?

 

We know that, F = Gm₁m₂

                                      d²

Or, 4.166×10²³ =  6.67x10⁻¹¹ x1.9×10²⁷x2×10³⁰  

                                                      d²

 

              Or,  d²   =  25.346 × 10³⁰⁺²⁷⁻¹¹

                                        4.166×10²³

 

               Or,  d²  =  25.346 × 10⁴⁶

                                   4.166×10²³

 

               Or,  d² = 6.084 × 10⁴⁶⁻²³

               Or,  d² =  6.084 × 10²³

               Or,  d  =  √ 6.084 × 10²³

               Or,  d = 7.8 x 10¹¹ m

 

The distance between them is  7.8 x 10¹¹ m.

l. A man who weighs 75 kg is on the surface of the earth whose mass is 6x10 kg. If the radius of the earth is 6380 km, calculate the force of attraction between them. (Given, G=6.67 x 10⁻¹¹ Nm²kg⁻²).

Given,

Mass of the person (m) = 75 kg

Mass of the earth (M)= 6×10²⁴ kg

Radius of the earth (R) =6380 km

                                          = 6380×10³ m

                                          = 6.38 × 10⁶m

Universal gravitational constant (G) = 6.67 x 10⁻¹¹ Nm²kg⁻²

The gravitational force of attraction between them (F) =?

We know that, F = GMm

                                   R²

          =  6.67 x 10⁻¹¹ x 6×10²⁴ x  75

                           (6.38 × 10⁶)²

 

          =  3001.5 × 10²⁴⁻¹¹

               40.7044 x 10¹²

 

         =  73.738 x 10¹³⁻¹²

         =  73.738 x 10

         = 737.38 N

The force of attraction between them is 737.38 N.