Gravity and Buoyancy: A Comprehensive Guide

Posted on May 1, 2024 in Physics

Universal Law of Gravitation

Every object in the universe attracts other objects by a force of attraction, called gravitation, which is directly proportional to the product of masses of the objects and inversely proportional to the square of distance between them. This is called Law of Gravitation or Universal Law of Gravitation.

Formula to Find the Magnitude of the Gravitational Force

The formula to find the magnitude of the gravitational force between the earth and an object on the surface of the earth is:

Gravitational force, F = G × M × m / R²

where,

• G = gravitational constant
• M = mass of the Earth
• m = mass of the object
• R = radius of the earth

Free Fall

When an object falls from any height under the influence of gravitational force only, it is known as free fall. In the case of free fall no change of direction takes place but the magnitude of velocity changes because of acceleration. This acceleration acts because of the force of gravitation and is denoted by ‘g’.

Acceleration Due to Gravity

The acceleration gained by an object because of gravitational force is called its acceleration due to gravity. Its SI unit is m/s². Acceleration due to gravity is a vector, which means it has both a magnitude and a direction. It has a standard value defined as 9.80665 m/s².

Differences Between Mass and Weight

The following table highlights the differences between mass and weight:

Weight on the Moon

Weight is the force with which a body is pulled towards the centre of the earth or any other celestial body for that matter. The greater the force put by the celestial body, the greater the weight of an object on its surface. The gravitational attraction on the moon is about one sixth when compared to that on the earth. Hence, the weight of an object on the moon is 1/6th of the weight on the earth.

Difficulty in Holding a School Bag with a Thin Strap

We know that lesser is the surface area of an object the more is the pressure it exerts. Hence, it is difficult to hold a school bag having a strap made of thin and strong string, because the thin string has very less contacting area in case of a school bag having a strap made of thin and strong string which increases pressure which is uncomfortable to carry the school bag.

Buoyancy

When an object is immersed partially or fully in a liquid, it experiences an upward force. This upward force is known as buoyant force and this phenomenon is known as buoyancy. Buoyancy is also known as up thrust.

Floating or Sinking of an Object

An object will float on the surface of water when it has density less than that of water because it always displaces more weight of water than its own weight, and as the buoyant force is more than its own weight, therefore it floats. On the other hand, an object sinks in water when it has density more than that of water because it always displaces less weight of water than its own weight, and as the buoyant force is less than its own weight, it sinks in water.

Actual Weight

Weight is the gravitational force acting on the object in the downward direction. According to the rule of buoyancy, all fluids apply an up thrust force in the upward direction. Hence, the air exerts an upward force (buoyant force or up thrust) on our body and makes us slightly lighter than we actually are. So, the weight shown on the weighing scale would be less than 42 kg. Thus, we can say that our actual weight/mass will be slightly more.

Heavier Object Falling Faster

We know that, Measured weight of the body = actual weight of the body – buoyant force

Therefore, actual weight = measured weight + buoyant force

As measured weight of cotton and iron is same, actual weight will only depend on the buoyant force. Buoyant force is given by:

B = Volume of the body × g × Density of air.

Buoyant force is directly proportional to the volume of the body. As volume of iron is less than the volume of cotton, therefore buoyant force acting on the iron will be less than that of iron. Hence, actual weight of the cotton will be more than the iron.

Force of Gravitation Between Two Objects

Force between two objects of masses M and m separated by a distance r is given by:

F = GMm / r × r

Now according to question, New distance, r’ = r/2

So, new force between them F’ = 4GMm/r² = 4F

The force increases by factor 4.

Acceleration Due to Gravity

We know that force applied on an object is equal to the product of its mass and acceleration. Now, it is given to us that gravitational force acts on all objects in proportion to their masses, that is, force is proportional to mass. This is possible only if the acceleration (due to gravity) is constant for a heavy object as well as a lighter object. Since the acceleration is constant, therefore, all the objects (heavy or light) fall at the same speed.

Gravitational Force Between Earth and a 1 kg Object

Firstly, we will calculate the magnitude of gravitational force by using the formula:

F = G × m1 × m2 / r²

We know that,

• Gravitational constant, G = 6.7 × 10^-11 Nm² /kg²
• Given, Mass of earth, m1 = 6.4 × 10²⁴ kg
• Mass of object, m2 = 1 kg
• Distance between centre of earth and object, Radius of earth, r = 6.4× 10⁶ m

Putting all the values in (i), we get:

F = 6.7 × 10-11 × 6 × 1024 × 1 / (6.4 × 106 ) 2 = 9.8 N Therefore, the magnitude of gravitational force is 9.8 N 

Q.4. The earth and the moon are attracted to each other by gravitational force. Does the earth attract the moon with a force that is greater or smaller or the same as the force with which the moon attracts the earth? Ans. Yes, the moon do attracts the earth but the earth also attracts the moon. The gravitational pull of earth is too much as mass is directly proportional to the gravity and mass of earth is extremely greater than the mass of moon. So, the attraction exerted by the moon on the earth becomes negligible. Q.5. If the moon attracts the earth, why does the earth not move towards the moon? Ans. The Earth and the moon experience equal gravitational forces from each other. However, the mass of the Earth is much larger than the mass of the moon. Hence, it accelerates at a rate lesser than the acceleration rate of the moon towards the Earth. For this reason, the Earth does not move towards the moon. Q.6. What happens to the force between two objects, if (i) the mass of one object is doubled? (ii) the distance between the objects is doubled and tripled? (iii) the masses of both objects are doubled? Ans. We know that Gravitational force, F = GMm / r 2 (i) When the mass of 1 object is doubled F’ = G (2M) m / r 2 F’ = 2GMm / r 2 F’ = 2F, the force will become 2 times. (ii) Distance between two objects is doubled F’ = GMm / (2r)2 F’ = GMm / 4r2 F’ = 1/4 F, the force will become 1/4 times Distance between two objects is tripled F’ = GMm / (3r)2 F’ = GMm / 9r2 F’ = 1/9F, the force will become 1/9 times (iii) When masses of both objects is doubled F’ = G (2M) (2m) / r 2 F’ = 4GMm / r 2 F’ = 4F, the force will become 4 times. Q.7. What is the importance of the universal law of gravitation? Ans. The importance of universal law of gravitation is that it explains:  The gravitational force of attraction of the Earth binds all terrestrial objects to the Earth.  The gravitational force of Earth keeps the atmosphere close to Earth.  The moon revolves around the Earth.  The gravitational pull of the Sun on the planet keeps them revolving around the Sun.  The tides formed by the rising & falling of water level in the oceans are due to the gravitational force of Sun & moon on the water. Q.8. What is the acceleration of free fall? Ans. As, the only force which is acting on body on free fall is gravity. Hence, its acceleration will also be relative to gravity. If gravity is the only force acting on an object, then we find the object will accelerate at a rate of 9.8 m/s2 down toward the centre of the Earth (this is often rounded to 10 m/s). In fact, all objects will fall at this rate if the air does not act to slow them down. Q.9. What do we call the gravitational force between the earth and an object? Ans. The gravitational force between the Earth and an object is called ‘earth’s gravity’. Gravity is the force that attracts a body towards the centre of the Earth, or towards any other physical body having mass. Q.10. Amit buys few grams of gold at the poles as per the instructions of one of his friends. He hands over the same when he meets him at the equator. Will his friend agree with the weight of gold bought? If not, why? [Hint. The value of g is greater at the poles than at the equator.] Ans. Weight of a body on the Earth is given by: W = mg Where, m = mass of the body g = acceleration due to gravity The value of g is greater at poles than at the equator. Therefore, gold at the equator weighs less than at the poles. Hence, Amit’s friend will not agree with the weight of the gold bought.

Q.11. Why will a sheet of paper fall slower than one that is crumpled into a ball? Ans. Sheet of paper will fall slower because due to larger surface area the amount of upward gravitational force causes resistance to its movement. The air resistance would make the actual force lesser than that imparted by the gravity of Earth and hence slowing down the speed of sheet of paper. Q.12. Gravitational force on the surface of moon is only 1/6 as strong as gravitational force on the earth. What is the weight in newtons of a 10 kg object on the moon and on the earth? Ans. We know that, acceleration due to gravity on the surface of Earth = 9.8 m/s2 So, the acceleration due to gravity on the surface of moon will be 1/6 of this value. = 9.8 × 1/6 m/s2 Firstly, we will calculate the weight on the moon: Mass of objects, m = 10 kg Acceleration due to gravity on moon, g = 9.8 × 1/6 m/s2 Weight of object (on moon), W = m × g = 10 × 9.8 × 1/6 = 16.3 N Now, we will calculate the weight of the object on the earth surface: Mass of object, m = 10 kg Acceleration due to gravity on earth, g = 9.8 m/s2 Weight of object (on earth), W = m × g = 10 × 9.8 = 98 N Q.13. A ball is thrown vertically upwards with a velocity of 49 m/s. Calculate (i) the maximum height to which it rises, (ii) the total time it takes to return to the surface of the Earth. Ans. (i) The value of acceleration due to gravity g will be taken as negative as the ball is going upwards. Given, initial velocity of ball, u = 49 m/s Final velocity of ball, v = 0 (as the ball stops at top) Acceleration due to gravity, g = – 9.8 m/s2 Height, h =? We know that, v 2 = u2 + 2gh (0)2 = (49)2 + 2 × (-9.8) × h 0 = 2401 – 19.6 h 19.6 h = 2401 h = 2401 / 19.6 = 122.5 m (ii) Now, we will calculate the time taken by the ball to reach at the highest point: Given, final velocity, v = 0 (as the ball stops at top) Initial velocity, u = 49 m/s Acceleration due to gravity, g = – 9.8 m/s2 Time taken, t =? 0 = 49 + (-9.8) × t 0 = 49 – 9.8 t 9.8 t = 49 t = 49 / 9.8 t = 5 s The ball will take equal time to return to the surface of earth. Therefore, total time = 5 + 5 = 10 seconds   Q.14. A stone is released from the top of a tower of height 19.6 m. Calculate its final velocity just before touching the ground. Ans. Given, initial velocity, u = 0 Final velocity, v = ? Acceleration due to gravity, g = 9.8 m/s2 Height, h = 19.6 m We know that, v 2 = u2 + 2gh = (0)2 + 2 × 9.8 × 19.6 = 19.6 × 19.6 = (19.6)2 = 19.6 m/s Therefore, its final velocity just before touching the ground is 19.6 m/s. Q.15. A stone is thrown vertically upward with an initial velocity of 40 m/s. Taking g = 10 m/s2 , find the maximum height reached by the stone. What is the net displacement and the total distance covered by the stone when it falls back to the ground? Ans. Given, initial velocity, u = 40 m/s Final velocity, v = 0 (as the stone stops) Acceleration due to gravity, g = – 10 m/s2 Height, h = ? v 2 = u2 + 2gh (0)2 = (40)2 + 2 × (-10) × h 0 = 1600 – 20 h 20 h = 1600 h = 1600 / 20 h = 80 m The maximum height to which stone is thrown up is 80 m. As the stone is thrown up from the ground and after reaching to a maximum height of 80 m it falls back to the ground. Therefore, the net displacement of the stone is zero. The distance covered by the stone in reaching the maximum height is 80 m. The stone will cover the same distance of 80 m in coming back to ground. So, the total distance covered by the stone = 80 + 80 = 160 m 

Q.16. Calculate the force of gravitation between the Earth and the Sun, given that the mass of the Earth = 6 × 1024 kg and of the Sun = 2 × 1030 kg. The average distance between the two is 1.5 × 1011 m. Ans. We know that, F = G × m1 × m2 / r2 (i) Gravitational constant, G = 6.7 × 10-11 Nm2 /kg2 Mass of earth, m1 = 6 × 1024 kg Mass of sun, m2 = 2 × 1030 kg Distance between the earth and the sun, r = 1.5 × 1011 m Now, putting all these values in (i), we get: F = 6.7 × 10-11 × 6 × 1024 × 2 × 1030 / (1.5 × 1011) 2 F = 3.57 × 1022 N Q.17. A stone is allowed to fall from the top of a tower 100 m high and at the same time another stone is projected vertically upwards from the ground with a velocity of 25 m/s. Calculate when and where the two stones will meet. Ans. Firstly, for the stone falling from top of tower: Height, h = (100 – x) Initial velocity, u = 0 Time, t = ? g = 9.8 m/s2 We know that, h = ut + ½ gt2 100 – x = 0 × t + ½ × 9.8 × t2 100 – x = 4.9 t2 (i) Now, for stone projected vertically upwards: Height, h = x Initial velocity, u = 25 m/s Time, t = ? g = – 9.8 m/s2 (as the stone goes up) We know that, s = ut + ½ gt2 x = 25 × t + ½ × (-9.8) × t2 x = 25t – 4.9 t2 (ii) Now, by adding (i) and (ii), we get: 100 – x + x = 4.9 t2 + 25 t – 4.9 t2 100 = 25 t t = 100 / 25 t = 4 s Therefore, the 2 stones will meet after a time of 4 seconds. Now, putting the value of t in (i), we get: 100 – x = 4.9 × (4)2 100 – x = 4.9 × 16 100 – x = 78.4 x = 21.6 m Therefore, the two stones will meet at a height of 21.6 m above the ground.Q.18. A ball thrown up vertically returns to the thrower after 6 s. Find (a) the velocity with which it was thrown up, (b) the maximum height it reaches, and (c) its position after 4 s. Ans. Since the ball thrown up vertically returns to the thrower in 6 seconds, this means that the ball will take half of this time. 6/2 = 3 s (a) Given, final velocity, v = 0 Initial velocity, u = ? Acceleration due to gravity, g = – 9.8 m/s2 (as the ball goes up) Time taken, t = 3 s We know that, v = u + gt 0 = u + (-9.8) × 3 0 = u – 29.4 u = 29.4 m/s (b) Now, calculation of maximum height: We know that, v 2 = u2 + 2gh (0)2 = (29.4)2 + 2 × (-9.8) × h 0 = 864.36 – 19.6 h 19.6 h = 864.36 h = 44.1 m (c) Finally, we have to calculate the position of ball after 4 seconds: We know that, h = ut + ½ gt2 h = 0 × 1 + ½ × 9.8 × (1)2 (because t = 1 s) = 0 + 4.9 ×1 = 4.9 m Q.19. In what direction does the buoyant force on an object immersed in a liquid act? Ans. The buoyant force on an object immersed in a liquid acts in upward direction (i.e., opposite to weight of the object). Or in other words we can say that the buoyant force on an object submerged in a fluid is caused by the pressure difference between the top and bottom of the object. To overcome the gravitational force, the buoyant force acts in the upward direction. The larger pressure at greater depth pushes upward on the object. Q.20. Why does a block of plastic released under water come up to the surface of water? Ans. The density of plastic is less than that of water, so the force of buoyancy on plastic block will be greater than the weight of plastic block displaced. Hence, the acceleration of plastic block will be in upward direction, and comes up to the surface of water. 

Q.21. The volume of 50 g of a substance is 20 cm3 . If the density of water is 1 g cm-3 , will the substance float or sink? Ans. Given, mass of substance = 50 g Volume of substance = 20 cm3 We know that, density of substance = mass of substance / volume of substance = 50 g / 20 cm3 = 1.42 g cm-3 Clearly, the density of packet is more than the density of water. Therefore, it will sink. The packet which sinks in water will displace water equal to its own volume. Given, volume of packet = 350 cm3 So, it will displace 350 cm3 of water. Now we have to calculate the mass of 350 cm3 of water. Density of water = mass of water / volume of water 1 g cm-3 = mass of water / 350 cm3 Mass of water = 1 g cm-3 × 350 cm3 = 350 g Q.22. The volume of a 500 g sealed packet is 350 cm3 . Will the packet float or sink in water if the density of water is 1 g cm–3 ? What will be the mass of the water displaced by this packet? Ans. Given, mass of sealed packet = 500 g Volume of sealed packet = 350 cm3 We know that, density of sealed packet = mass of sealed packet / volume of sealed packet = 500/350 = 1.42 g/cm3 Density of sealed packet is greater than density of water. Therefore, the packet will sink. Considering Archimedes Principle, displaced water volume = Force exerted on the sealed packet. Volume of water displaced = 350 cm3 Therefore, displaced water mass = ρ x V = 1 × 350 Mass of displaced water = 350 g.