Chapter 3 part 2: Gravitaion S.chand Solution

Very Short Answer Type Questions

1. Write the common unit of density.

Answer: The common unit of density is kilogram per cubic meter (kg/m³).

2. What is the density of water in SI units?

Answer: The density of water in SI units is approximately 1000 kg/m³.

3. What is the value of relative density of water?

Answer: The relative density of water is 1.

4. Name the quantity whose one of the units is pascal (Pa).

Answer: Pressure is the quantity whose one of the units is pascal (Pa).

5. State the units in which pressure is measured.

Answer: Pressure is measured in pascal (Pa) or newton per square meter (N/m²).

6. State whether the following statements are true or false:

(a) The buoyant force depends on the nature of object immersed in the liquid.

Answer: True.

(b) Archimedes’ principle can also be applied to gases.

Answer: False.

7. In which direction does the buoyant force on an object due to a liquid act?

Answer: The buoyant force on an object due to a liquid acts upward.

8. What is the other name of buoyant force?

Answer: The other name of buoyant force is upthrust.

9. Name the force which makes heavy objects appear light when immersed in a liquid.

Answer: Buoyant force.

10. What is upthrust?

Answer: Upthrust is the force exerted by a fluid that opposes the weight of an immersed object, pushing it upward.

11. Name the principle which gives the magnitude of buoyant force acting on an object immersed in a liquid.

Answer: Archimedes' principle.

12. The relative density of mercury is 13.6. What does this statement mean?

Answer: This statement means that the density of mercury is 13.6 times that of water.

13. What name is given to ‘thrust per unit area’?

Answer: Pressure.

14. What is the scientific name of the ‘upward force’ acting on an object immersed in a liquid?

Answer: Buoyant force.

15. What is meant by the term ‘buoyancy’?

Answer: Buoyancy is the upward force exerted by a fluid on an object immersed in it, opposing the force of gravity.

16. What causes buoyant force (or upthrust) on a boat?

Answer: The shape of the boat displaces water, creating an upward buoyant force, keeping the boat afloat.

17. Why does ice float in water?

Answer: Ice floats in water because it is less dense than water.

18. What force acting on an area of 0.5 m² will produce a pressure of 500 Pa?

Answer: The force acting on an area of 0.5 m² to produce a pressure of 500 Pa is 250 N.

19. An object of weight 200 N is floating in a liquid. What is the magnitude of buoyant force acting on it?

Answer: The magnitude of buoyant force acting on the object is 200 N, equal to its weight in equilibrium.

20. Name the scientist who gave the magnitude of buoyant force acting on a solid object immersed in a liquid.

Answer: Archimedes.

21. The density of gold is 19 g/cm³. Find the volume of 95 g of gold.

Answer: The volume of 95 g of gold is approximately 5 cm³.

22. What is the mass of 5 m³ of cement of density 3000 kg/m³?

Answer: The mass of 5 m³ of cement is 15,000 kg.

23. What is the density of a substance of mass 100 g and volume 10 cm³?

Answer: The density of the substance is 10 g/cm³.

24. Why does a block of wood held under water rise to the surface when released?

Answer: The buoyant force acting on the block of wood is greater than its weight, causing it to rise to the surface.

25. The density of a body is 800 kg/m³. Will it sink or float when dipped in a bucket of water? (Density of water = 1000 kg/m³).

Answer: It will float, as the density of the body (800 kg/m³) is less than the density of water (1000 kg/m³).

26. Fill in the following blanks with suitable words :

(a) Force acting on a unit area is called pressure.

(b) It is the buoyant force which makes objects appear lighter in water.

(c) A heavy ship floats in water because its average density is less than that of water.

(d) In fluids (liquids and gases), pressure acts in all directions, and pressure increases as the depth increases.

(e) In order to sink in a fluid, the density of an object must be greater than the density of the fluid.

(f) Snow shoes work by spreading out a person’s weight over a much bigger area.

(g) If the area of a snow shoe is five times larger than the area of an ordinary shoe, then the pressure of a snow shoe on the snow is five times less.

Short Answer Type Questions

27. (a) What is the difference between the density and relative density of a substance ?

(b) If the relative density of a substance is 7.1, what will be its density in SI units ?

(a) The density of a substance is its mass per unit volume, while relative density (or specific gravity) is the ratio of the density of the substance to the density of a reference substance (usually water for liquids and solids, and air for gases). (b) If the relative density of a substance is 7.1, its density in SI units would be 7.1 times the density of water.

28. Define thrust. What is its unit ?

Thrust is the force exerted by a fluid (such as air or water) perpendicular to the surface of an object moving through it. The unit of thrust is Newton (N).

29. A mug full of water appears light as long as it is under water in the bucket than when it is outside water.

Why ?

A mug full of water appears light when submerged in a bucket of water compared to when it is outside water due to buoyancy. The buoyant force acting on the mug in water counteracts its weight, making it feel lighter.

30. What happens to the buoyant force as more and more volume of a solid object is immersed in a liquid ?

As more volume of a solid object is immersed in a liquid, the buoyant force acting on it increases. The buoyant force becomes maximum when the object is fully submerged in the liquid.

31. Why do we feel light on our feet when standing in a swimming pool with water up to our armpits ?

Feeling light on our feet in a swimming pool with water up to our armpits is due to buoyancy. The buoyant force counteracts the weight of our body, making us feel lighter in water.

32. Explain why, big boulders can be moved easily by flood.

Big boulders can be moved easily by a flood because the force of the flowing water exerts a large buoyant force on the boulders, reducing their effective weight.

33. An iron nail sinks in water but it floats in mercury. Why ?

An iron nail sinks in water but floats in mercury because the density of mercury is greater than that of iron. The buoyant force in mercury is sufficient to support the weight of the iron nail.

34. Explain why, a piece of glass sinks in water but it floats in mercury.

A piece of glass sinks in water but floats in mercury because the density of mercury is greater than that of glass. The buoyant force in mercury is enough to keep the glass afloat.

35. Steel sinks in water but a steel boat floats. Why ?

Steel sinks in water, but a steel boat floats due to its shape and the displacement of water. The boat's hull is designed to displace a volume of water equal to its weight, providing buoyancy.

36. Explain why, school bags are provided with wide straps to carry them.

School bags are provided with wide straps to distribute the force over a larger area, reducing the pressure on the shoulders and making them more comfortable to carry.

37. Why does a sharp knife cut objects more effectively than a blunt knife ?

A sharp knife cuts objects more effectively than a blunt knife because the sharp edge concentrates the force on a smaller area, leading to increased pressure and easier penetration.

38. Explain why, wooden (or concrete) sleepers are kept below the railway line.

Wooden (or concrete) sleepers are kept below the railway line to provide support and stability, distributing the weight of the train and preventing the tracks from sinking into the ground.

39. Explain why, a wide steel belt is provided over the wheels of an army tank.

A wide steel belt is provided over the wheels of an army tank to distribute the weight over a larger surface area, reducing the pressure on the ground and preventing the tank from sinking into soft terrain.

40. Explain why, the tip of a sewing needle is sharp.

The tip of a sewing needle is sharp to easily pierce through fabric, creating a small opening without causing significant damage to the material.

41. When is the pressure on the ground more—when a man is walking or when a man is standing ? Explain.

The pressure on the ground is generally more when a man is standing compared to when he is walking. When standing, the entire body weight is concentrated on a smaller surface area, increasing the pressure.

42. Explain why, snow shoes stop you from sinking into soft snow.

Snow shoes prevent sinking into soft snow by distributing the wearer's weight over a larger surface area, reducing the pressure on the snow and preventing deep penetration.

43. Explain why, when a person stands on a cushion, the depression is much more than when he lies down on it.

When a person stands on a cushion, the depression is much more than when lying down because standing concentrates the weight on a smaller area, increasing the pressure and causing a deeper depression.

44. Use your ideas about pressure to explain why it is easier to walk on soft sand if you have flat shoes rather than shoes with sharp heels.

Walking on soft sand is easier with flat shoes than shoes with sharp heels because the flat shoes distribute the body weight over a larger surface area, reducing the pressure on the sand.

45. Explain why, a nail has a pointed tip.

A nail has a pointed tip to concentrate force on a small area, making it easier to penetrate hard materials like wood.

46. Explain why, buildings and dams have wide foundations.

Buildings and dams have wide foundations to distribute the weight of the structure over a larger area, reducing the pressure on the ground and providing stability.

47. Why does a ship made of iron and steel float in water whereas a small piece of iron sinks in it ?

A ship made of iron and steel floats in water because of its shape and displacement, which allow it to displace a weight of water equal to its own weight, making it buoyant. A small piece of iron sinks because it lacks the shape and volume to displace enough water to counter its weight.

48. Why do camels have large flat feet ?

Camels have large flat feet to help them walk on sandy desert terrain. The large surface area reduces the pressure on the sand, preventing the camel from sinking into the soft ground.

49. Name these forces : (a) the upward push of water on a submerged object (b) the force which wears away two surfaces as they move over one another (c) the force which pulled the apple off Isaac Newton’s tree. (d) the force which stops you falling through the floor.

a) the upward push of water on a submerged object: Buoyant Force (b) the force which wears away two surfaces as they move over one another: Frictional Force (c) the force which pulled the apple off Isaac Newton’s tree: Gravitational Force (d) the force which stops you falling through the floor: Normal Force 50. A pressure of 10 Pa acts on an area of 3.0 m2. What is the force acting on the area ? What force will be exerted by the application of same pressure if the area is made one-third ?

Force = Pressure × Area Force = 10 Pa × 3.0 m2 = 30 N

If the area is made one-third: New Area = (1/3) × 3.0 m2 = 1.0 m2 New Force = 10 Pa × 1.0 m2 = 10 N

51. A girl is wearing a pair of flat shoes. She weighs 550 N. The area of contact of one shoe with the ground is 160 cm2. What pressure will be exerted by the girl on the ground : (a) if she stands on two feet ? (b) if she stands on one foot ?

(a) Pressure = Weight / Area Pressure = 550 N / (2 × 160 cm2) = 1.72 N/cm2

(b) Pressure = Weight / Area Pressure = 550 N / 160 cm2 = 3.44 N/cm2

52. Calculate the density of an object of volume 3 m3 and mass 9 kg. State whether this object will float or sink in water. Give reason for your answer.

Density = Mass / Volume Density = 9 kg / 3 m3 = 3 kg/m3

The object will float in water because its density is less than the density of water (approximately 1000 kg/m3).

53. An object weighs 500 grams in air. This object is then fully immersed in water. State whether it will weigh less in water or more in water. Give reason for your answer.

The object will weigh less in water. This is because the buoyant force acting on the object in water reduces its effective weight. Archimedes' principle states that the buoyant force is equal to the weight of the fluid displaced by the object. 54. (a) Write down an equation that defines density. (b) 5 kg of material A occupy 20 cm3 whereas 20 kg of material B occupy 90 cm3. Which has the greater density : A or B ? Support your answer with calculations.

Density of A = 5 kg / 20 cm3 = 0.25 kg/cm3 Density of B = 20 kg / 90 cm3 = 0.22 kg/cm3

Material A has a greater density than material B.

Long Answer Type Questions

55. (a) Define buoyant force. Name two factors on which buoyant force depends.

(b) What is the cause of buoyant force ?

(c) When a boat is partially immersed in water, it displaces 600 kg of water. How much is the buoyant force

acting on the boat in newtons ? (g = 10 m s–2)

(a) Buoyant Force: Buoyant force is the upward force exerted by a fluid (such as water or air) on an object placed in it. It is the force that opposes the weight of the object and keeps it afloat.

Factors on which buoyant force depends:

1. Volume of the Displaced Fluid: The buoyant force is directly proportional to the volume of the fluid displaced by the submerged or immersed object.
2. Density of the Fluid: The buoyant force is also influenced by the density of the fluid. The denser the fluid, the greater the buoyant force.

(b) Cause of Buoyant Force: The buoyant force arises due to the pressure difference between the top and bottom of an immersed object in a fluid. This pressure difference is a result of the increasing pressure with depth in a fluid. The object displaces fluid, and the pressure is higher at the deeper portions of the object, causing an upward force.

(c) Calculation of Buoyant Force: The buoyant force (B) is given by Archimedes' principle, which states that the buoyant force is equal to the weight of the fluid displaced by the object.

$�=�\cdot �\cdot �$

Where:

• $�$ is the buoyant force,
• $�$ is the density of the fluid,
• $�$ is the volume of the fluid displaced,
• $�$ is the acceleration due to gravity.

In this case, the boat displaces 600 kg of water. Therefore, the buoyant force can be calculated as follows:

$�=600\text{kg}\cdot 10{\text{m/s}}^2$

$�=6000\text{N}$

Therefore, the buoyant force acting on the boat is 6000 Newtons.

56. (a) State the principle of flotation. (b) A floating boat displaces water weighing 6000 newtons. (i) What is the buoyant force on the boat ? (ii) What is the weight of the boat ?

(a) Principle of Flotation: The principle of flotation states that a floating object displaces its own weight of the fluid in which it floats. In other words, the buoyant force acting on an object is equal to the weight of the fluid it displaces. For an object to float, the buoyant force must be equal to or greater than the weight of the object.

(b) Calculation: Given that a floating boat displaces water weighing 6000 newtons, we can use the principle of flotation to determine the buoyant force and the weight of the boat.

(i) Buoyant Force on the Boat: According to the principle of flotation, the buoyant force is equal to the weight of the fluid displaced. Therefore, the buoyant force on the boat is 6000 newtons.

(ii) Weight of the Boat: The weight of the boat is equal to the buoyant force required for it to float. Thus, the weight of the boat is also 6000 newtons.

In summary:

• (i) Buoyant force on the boat = 6000 N
• (ii) Weight of the boat = 6000 N
• 57. (a) Define density. What is the SI unit of density ? (b) Define relative density. What is the SI unit of relative density ? (c) The density of turpentine is 840 kg/m3. What will be its relative density ? (Density of water = 1000 kg/m3)

• (a) Density: Density is defined as the mass per unit volume of a substance. Mathematically, it is expressed as:

  $\text{Density}(�)=\frac{\text{Mass}(�)}{\text{Volume}(�)}$

  The SI unit of density is kilograms per cubic meter (kg/m³).

  (b) Relative Density: Relative density, also known as specific gravity, is the ratio of the density of a substance to the density of a reference substance (usually water, which has a density of 1000 kg/m³). Mathematically, it is expressed as:

  $\text{Relative Density}(\text{RD})=\frac{\text{Density of Substance}}{\text{Density of Reference Substance}}$

  The SI unit of relative density is dimensionless, as it is a ratio of two densities with the same unit.

  (c) Calculation of Relative Density for Turpentine: Given that the density of turpentine is 840 kg/m³ and the density of water is 1000 kg/m³, the relative density of turpentine ($�{�}_{\text{turpentine}}$) can be calculated as:

  $�{�}_{\text{turpentine}}=\frac{\text{Density of Turpentine}}{\text{Density of Water}}$

  $�{�}_{\text{turpentine}}=\frac{840\text{kg/m³}}{1000\text{kg/m³}}$

  $�{�}_{\text{turpentine}}=0.84$

  Therefore, the relative density of turpentine is 0.84 (dimensionless).

• 58. (a) Define pressure.

  (b) What is the relation between pressure, force and area ?

  (c) Calculate the pressure when a force of 200 N is exerted on an area of :

  (i) 10 m2

  (ii) 5 m2

• (a) Pressure: Pressure is defined as the force applied per unit area. It is the ratio of force to the area over which the force is distributed. Mathematically, pressure ($�$) is expressed as:

  $�=\frac{�}{�}$

  Where:

  • $�$ is the pressure,
  • $�$ is the force applied, and
  • $�$ is the area over which the force is applied.

  The SI unit of pressure is pascal (Pa), where 1 Pascal is equal to 1 Newton per square meter (N/m²).

  (b) Relation between Pressure, Force, and Area: The relationship between pressure, force, and area is given by the equation:

  $�=\frac{�}{�}$

  This equation indicates that pressure is directly proportional to force and inversely proportional to the area over which the force is distributed. In other words, increasing the force applied or decreasing the area will result in an increase in pressure, and vice versa.

  (c) Calculation of Pressure: Given a force of 200 N and two different areas, we can calculate the pressure for each case.

  (i) For an area of 10 m²: $�=\frac{�}{�}=\frac{200\text{N}}{10{\text{m}}^2}=20\text{Pa}$

  Therefore, the pressure is $20\text{Pa}$ when the force of 200 N is exerted on an area of 10 m².

  (ii) For an area of 5 m²: $�=\frac{�}{�}=\frac{200\text{N}}{5{\text{m}}^2}=40\text{Pa}$

  Therefore, the pressure is $40\text{Pa}$ when the force of 200 N is exerted on an area of 5 m².

• 59. (a) What are fluids ? Name two common fluids. (b) State Archimedes’ principle. (c) When does an object float or sink when placed on the surface of a liquid ?

• (a) Fluids: Fluids are substances that can flow and do not have a fixed shape. They include liquids and gases. Two common fluids are:

  1. Water (liquid)
  2. Air (gas)

  (b) Archimedes' Principle: Archimedes' principle states that when a body is submerged in a fluid (liquid or gas), it experiences an upward buoyant force equal to the weight of the fluid it displaces. In other words, the buoyant force acting on an object is equal to the weight of the fluid displaced by that object. This principle explains why objects float or sink in fluids.

  (c) Floating or Sinking in a Liquid: Whether an object floats or sinks in a liquid depends on the relationship between its weight and the buoyant force exerted by the liquid. The key condition is as follows:

  • Floats: If the weight of the object is equal to or less than the buoyant force acting on it, the object will float on the surface of the liquid.
  • Sinks: If the weight of the object is greater than the buoyant force, the object will sink in the liquid.

  Mathematically, if $�$ is the weight of the object and $�$ is the buoyant force, an object will float if $�\le �$ and sink if $�>�$.

• 
  

60. (a) How does a boat float in water ?

(b) A piece of steel has a volume of 12 cm3, and a mass of 96 g. What is its density :

(i) in g/cm3 ?

(ii) in kg/m3 ?

(a) How a Boat Floats in Water: A boat floats in water due to the principle of buoyancy, as described by Archimedes. When a boat is placed in water, it displaces an amount of water equal to its own weight. The buoyant force acting on the boat is equal to the weight of the water displaced. If the weight of the boat is less than the buoyant force, the boat will float. This is because the buoyant force opposes the weight of the boat, allowing it to stay afloat.

(b) Density of Steel: Given that the volume of the steel is $12{\text{cm}}^3$ and the mass is 96, we can calculate its density using the formula:

$\text{Density}(�)=\frac{\text{Mass}}{\text{Volume}}$

(i) Density in g/cm³: $�=\frac{96\text{g}}{12{\text{cm}}^3}=8{\text{g/cm}}^3$

Therefore, the density of the steel is $8{\text{g/cm}}^3$ in grams per cubic centimeter.

(ii) Density in kg/m³: To convert from grams per cubic centimeter to kilograms per cubic meter, we need to multiply by $1000$ for mass and $100$ for volume conversion. $�=8{\text{g/cm}}^3\times \frac{1000\text{g}}{1\text{kg}}\times \frac{1{\text{cm}}^3}{100{\text{cm}}^3}\times \frac{1{\text{m}}^3}{1000000{\text{cm}}^3}=8000{\text{kg/m}}^3$

Therefore, the density of the steel is $8000{\text{kg/m}}^3$ in kilograms per cubic meter.

61. An elephant weighing 40,000 N stands on one foot of area 1000 cm2 whereas a girl weighing 400 N is standing on one ‘stiletto’ heel of area 1 cm2.

(a) Which of the two, elephant or girl, exerts a larger force on the ground and by how much ? (b) What pressure is exerted on the ground by the elephant standing on one foot ? (c) What pressure is exerted on the ground by the girl standing on one heel ? (d) Which of the two exerts larger pressure on the ground : elephant or girl ? (e) What is the ratio of pressure exerted by the girl to the pressure exerted by the elephant ?

(a) Force Exerted on the Ground: The force exerted by an object on the ground is equal to its weight. Weight is given by the formula $�=�\cdot �$, where $�$ is the mass and $�$ is the acceleration due to gravity.

If we assume both the elephant and the girl have the same mass (which is unlikely in reality, but for the sake of comparison), the force exerted by each on the ground is proportional to their weight. Therefore, the force exerted by the elephant is likely larger since elephants are much more massive than humans.

(b) Pressure Exerted by Elephant: Pressure ($�$) is given by the formula $�=\frac{�}{�}$, where $�$ is the force and $�$ is the area over which the force is applied.

If the elephant is standing on one foot, the pressure on the ground would depend on the distribution of its weight on that foot. Without knowing the exact area of contact, it's challenging to calculate the pressure accurately.

(c) Pressure Exerted by Girl: Similarly, the pressure exerted by the girl standing on one heel would depend on the distribution of her weight on that heel. Without knowing the exact area of contact, it's challenging to calculate the pressure accurately.

(d) Comparison of Pressure: Comparing the pressure exerted by the elephant and the girl depends on their weights, the distribution of weight on their respective feet, and the areas of contact. In general, elephants distribute their weight over a larger area, which may result in lower pressure compared to a human standing on a smaller area.

(e) Ratio of Pressures: The ratio of the pressure exerted by the girl to the pressure exerted by the elephant would depend on the specific values of force, area of contact, and weight distribution. Without these details, it's not possible to provide a specific ratio.

Multiple Choice Questions (MCQs)

62. An object weighs 10 N in air. When immersed fully in a liquid, it weighs only 8 N. The weight of liquid

displaced by the object will be :

(a) 2 N (b) 8 N (c) 10 N (d) 12 N

62. The weight of the liquid displaced by the object is equal to the difference in its weight in air and its weight in the liquid. Therefore,

Weight of liquid displaced = Weight in air - Weight in liquid = 10 N - 8 N = 2 N

So, the correct answer is (a) 2 N.

63. A rectangular wooden block has length, breadth and height of 50 cm, 25 cm and 10 cm, respectively. This

wooden block is kept on ground in three different ways, turn by turn. Which of the following is the correct

statement about the pressure exerted by this block on the ground ?

(a) the maximum pressure is exerted when the length and breadth form the base

(b) the maximum pressure is exerted when length and height form the base

(c) the maximum pressure is exerted when breadth and height form the base

(d) the minimum pressure is exerted when length and height form the base

63. Pressure is given by the formula P = Force/Area. The pressure exerted by the block on the ground depends on the area of the base in contact with the ground. The maximum pressure is exerted when the area is minimum. Therefore, the correct statement is:

(d) The minimum pressure is exerted when length and height form the base.

64. An object is put in three liquids having different densities, one by one. The object floats with

12 3 , and

9 11 7

parts of its volume outside the surface of liquids of densities d1, d2 and d3 respectively. Which of the

following is the correct order of the densities of the three liquids ?

(a) d1 > d2 > d3 (b) d2 > d3 > d1

(c) d1 < d2 < d3 (d) d3 > d2 > d1

64. The buoyant force acting on an object is equal to the weight of the liquid displaced by the object. The object floats with 12/3, 9/11, and 7/9 parts of its volume outside the surface of liquids of densities d1, d2, and d3, respectively. Therefore, the correct order of densities is:

(c) d1 < d2 < d3

65. A metal in which even iron can float is :

(a) sodium (b) magnesium (c) mercury (d) manganese

65. The metal in which even iron can float is:

(b) magnesium

66. Four balls, A, B, C and D displace 10 mL, 24 mL, 15 mL and 12 mL of a liquid respectively, when immersed

completely. The ball which will undergo the maximum apparent loss in weight will be :

(a) A (b) B (c) C (d) D

66. The apparent loss in weight is directly proportional to the volume of liquid displaced. Therefore, the ball that will undergo the maximum apparent loss in weight is the one that displaces the maximum volume of liquid. Hence,

(b) Ball B

67. The relative densities of four liquids P, Q, R and S are 1.26, 1.0, 0.84 and 13.6 respectively. An object is

floated in all these liquids, one by one. In which liquid the object will float with its maximum volume

submerged under the liquid ?

(a) P (b) Q (c) R (d) S

67. The volume of liquid displaced is directly related to the density of the liquid. The higher the density of the liquid, the less volume it will displace. Therefore, the correct answer is:

(c) R

68. A solid of density 900 kg/m3 floats in oil as shown in the given diagram. The oil floats on water of density 1000 kg/m3 as shown. The density of oil in kg/m3 could be :

(a) 850 (b) 900 (c) 950 (b) 1050

68. The buoyant force acting on the solid is equal to the weight of the liquid displaced. The density of the oil can be calculated using the given densities of water and the solid. The correct answer is:

(c) 950 kg/m³

69. The density of water is 1000 kg/m3 and the density of copper is 8900 kg/m3. Which of the following statements is incorrect ?

(a) Thedensityof a certain volumeof copper / The densityof the same volume of water = 8.9

(b) The volumeof a certain mass of copper / The volume of the same mass of water = 8.9

(c) The weight of a certain volumeof copper / The weight of the same volume of water = 8.9

(d) The mass of a certain volumeof copper / The mass of thesame volumeof water = 8.9

(b) The volumeof a certain mass of copper / The volume of the same mass of water = 8.9

(b) The volume of a certain mass of copper / The volume of the same mass of water = 8.9

Density is defined as mass per unit volume. Mathematically, density (ρ) is given by the formula:

$�=\frac{�}{�}$

where $�$ is density, $�$ is mass, and $�$ is volume.

In the context of statement (b), if you take a certain mass of copper and compare it to the same mass of water, the ratio of their volumes should be equal to the ratio of their densities. The density of copper (${�}_{\text{copper}}$) is 8900 kg/m³, and the density of water (${�}_{\text{water}}$) is 1000 kg/m³.

So, mathematically:

$\frac{\text{Volume of copper}}{\text{Volume of water}}=\frac{{�}_{\text{copper}}}{{�}_{\text{water}}}=\frac{8900}{1000}=8.9$

This shows that the statement is correct, as the ratio of the volumes is indeed 8.9.

70. The diagrams represent four measuring cylinders containing liquids. The mass and volume of the liquid in each cylinder are stated. Which two measuring cylinders could contain an identical liquid ?

(a) W and X (b) W and Y (c) X and Y (d) X and Z

(d) X and Z

• X: Mass = 20 g, Volume = 10 cm³, Mass-to-volume ratio = 20 g / 10 cm³ = 2 g/cm³
• Z: Mass = 20 g, Volume = 10 cm³, Mass-to-volume ratio = 20 g / 10 cm³ = 2 g/cm³

Both X and Z have the same mass-to-volume ratio, which means they have the same density of 2 g/cm³. This indicates that the liquids in cylinders X and Z are identical.

71. Consider the following information in respect of four objects A, B, C and D :

Which object would float on water ?

(a) A (b) B (c) C (d) D

(d) D

For object D: ${\text{Density}}_{�}=\frac{2000\text{kg}}{4{\text{m}}^3}=500{\text{kg/m}}^3$

This means that the density of object D is less than the density of water (1000 kg/m³). Therefore, object D will indeed float on water.

Questions Based on High Order Thinking Skills (HOTS)

72. If two equal weights of unequal volumes are balanced in air, what will happen when they are completely

dipped in water ? Why ?

When two equal weights of unequal volumes are balanced in air and then completely dipped in water, they will still remain in balance. This is due to Archimedes' Principle.

Archimedes' Principle states that when an object is immersed in a fluid (liquid or gas), it experiences an upward buoyant force equal to the weight of the fluid displaced by the object. If two objects of equal weight are balanced in air, it means they experience equal gravitational forces.

When these objects are immersed in water, they each displace an equal volume of water, and therefore, experience an equal buoyant force. Since the buoyant force is equal for both objects, and they have equal weights (gravitational forces), they will remain in balance.

In summary, when the two weights of unequal volumes are completely dipped in water, they will still be balanced due to the equal buoyant forces acting on them as a result of displacing equal volumes of water.

73. Two different bodies are completely immersed in water and undergo the same loss in weight. Is it necessary

that their weights in air should also be the same ? Explain.

No, it is not necessary for the weights of two different bodies in air to be the same if they undergo the same loss in weight when completely immersed in water. The loss in weight when an object is immersed in water is equal to the weight of the water it displaces, and it depends on the volume of the object rather than its weight in air.

Here's an explanation:

1. Weight in Air:

   • The weight of an object in air is determined by its mass and the acceleration due to gravity. The formula for weight ($�$) is $�=��$, where $�$ is the mass of the object and $�$ is the acceleration due to gravity (approximately 9.8 m/s² on the surface of the Earth).

2. Loss in Weight in Water:

   • When an object is immersed in water, it experiences a buoyant force equal to the weight of the water displaced. The loss in weight in water is equal to the weight of the water displaced by the volume of the object immersed (${�}_{\text{loss}}={�}_{\text{water}}\cdot {�}_{\text{immersed}}\cdot �$, where ${�}_{\text{water}}$ is the density of water, ${�}_{\text{immersed}}$ is the volume of the object immersed, and $�$ is the acceleration due to gravity).

Now, let's consider two different bodies with different weights in air (${�}_1$ and ${�}_2$), but they experience the same loss in weight when completely immersed in water (${�}_{\text{loss1}}={�}_{\text{loss2}}$).

${�}_{\text{loss1}}={�}_{\text{water}}\cdot {�}_{\text{immersed1}}\cdot �$ ${�}_{\text{loss2}}={�}_{\text{water}}\cdot {�}_{\text{immersed2}}\cdot �$

Since ${�}_{\text{loss1}}={�}_{\text{loss2}}$, it means that ${�}_{\text{water}}\cdot {�}_{\text{immersed1}}\cdot �={�}_{\text{water}}\cdot {�}_{\text{immersed2}}\cdot �$.

By canceling out common terms (density of water and acceleration due to gravity), it becomes ${�}_{\text{immersed1}}={�}_{\text{immersed2}}$, which means the volumes of the immersed parts of the two bodies are equal.

Therefore, even though the weights in air (${�}_1$ and ${�}_2$) may be different, if the bodies undergo the same loss in weight when completely immersed in water, it implies that the volumes of the bodies that are immersed are the same.

74. A body floats in kerosene of density 0.8 × 103 kg/m3 up to a certain mark. If the same body is placed in

water of density 1.0 × 103 kg/m3, will it sink more or less ? Give reason for your answer.

When an object floats in a fluid, the buoyant force acting on it is equal to the weight of the fluid it displaces. The buoyant force is given by Archimedes' principle:

$\text{Buoyant force}=\text{Weight of fluid displaced}$

The weight of the fluid displaced is dependent on the density of the fluid and the volume of the object submerged in it.

In this scenario, the body floats in kerosene of density $0.8\times 10^3{\text{kg/m}}^3$. Let's denote the volume of the body that is submerged in kerosene as ${�}_{\text{kerosene}}$.

Now, when the same body is placed in water of density $1.0\times 10^3{\text{kg/m}}^3$, it will displace the same volume (${�}_{\text{kerosene}}$) of water. However, since the density of water is higher than the density of kerosene, the weight of the water displaced will be greater.

Here's the reasoning:

1. Buoyant Force in Kerosene: ${\text{Buoyant force}}_{\text{kerosene}}=\text{Weight of kerosene displaced}$ ${\text{Buoyant force}}_{\text{kerosene}}={�}_{\text{kerosene}}\cdot {�}_{\text{kerosene}}\cdot �$

2. Buoyant Force in Water: ${\text{Buoyant force}}_{\text{water}}=\text{Weight of water displaced}$ ${\text{Buoyant force}}_{\text{water}}={�}_{\text{water}}\cdot {�}_{\text{kerosene}}\cdot �$

Since ${�}_{\text{water}}>{�}_{\text{kerosene}}$, the buoyant force in water will be greater.

Now, the weight of the body remains the same in both cases (as it is not specified otherwise). However, the buoyant force in water is greater, which means the net force acting on the body is greater in water. This will cause the body to sink more in water compared to its floating level in kerosene.

In summary, the body will sink more when placed in water due to the higher density of water, which results in a greater buoyant force acting on the body.

75. Giving reasons state the reading on a spring balance when it is attached to a floating block of wood which weighs 50 g in air.

When a block of wood is floating in water, it experiences an apparent loss in weight. The apparent loss in weight is due to the buoyant force acting on the block. The buoyant force is equal to the weight of the fluid (in this case, water) displaced by the immersed portion of the block.

The weight of the block in air is 50 g. However, when the block is floating, it displaces its own weight in water. Therefore, the apparent loss in weight is 50 g.

When a spring balance is attached to the floating block of wood, the reading on the spring balance will be equal to the apparent loss in weight. Therefore, the reading on the spring balance will be 50 g.

In summary, the reading on a spring balance attached to a floating block of wood (which weighs 50 g in air) will be 50 g, reflecting the apparent loss in weight due to the buoyant force in water.

76. If a fresh egg is put into a beaker filled with water, it sinks. On dissolving a lot of salt in the water, the egg

begins to rise and then floats. Why ?

The behavior of the egg in the water changes when salt is dissolved in the water due to the effect of density. Here's a step-by-step explanation:

1. Fresh Egg in Water:

   • When a fresh egg is put into a beaker filled with water, it sinks. The reason for this is that the density of the egg is greater than the density of plain water. The buoyant force acting on the egg (which is the weight of the water it displaces) is not sufficient to counteract the weight of the egg itself, so the egg sinks.

2. Dissolving Salt in Water:

   • When you dissolve a lot of salt in the water, the density of the water increases. The dissolved salt adds mass to the water without significantly changing its volume. This increase in density affects the buoyant force.

3. Egg Begins to Rise and Float:

   • As more salt is dissolved, the density of the water becomes greater than the density of the egg. The buoyant force acting on the egg increases because the weight of the water it displaces (buoyant force) now exceeds the weight of the egg. As a result, the egg begins to rise and eventually floats.

In summary, the addition of salt increases the density of the water, and when the density of the water becomes greater than the density of the egg, the buoyant force becomes greater than the weight of the egg. This allows the egg to rise and float in the saltwater solution. The phenomenon is an illustration of Archimedes' principle, which states that an object will float in a fluid if the buoyant force acting on it is greater than its weight.

77. A beaker full of water is suspended from a spring balance. Will the reading of the balance change : (a) if a cork is placed in water ? (b) if a piece of heavy metal is placed in it ? Give reasons for your answer.

(a) If a cork is placed in water:

• The reading of the spring balance will not change. When the cork is placed in water, it floats. The buoyant force acting on the cork (equal to the weight of the water displaced by the cork) is now balancing the weight of the cork. The net force acting on the system (cork + water) is zero, and the reading on the spring balance remains the same.

(b) If a piece of heavy metal is placed in water:

• The reading of the spring balance will increase. When a heavy metal piece is placed in water, it sinks. The metal displaces water, and the buoyant force acting on the metal is less than its weight. Therefore, the net force acting on the system (metal + water) is downward, causing an increase in the reading on the spring balance. The added weight of the metal contributes to the total force measured by the spring balance.

In summary:

• For the cork, which floats, the buoyant force equals the weight of the cork, and the reading remains the same.
• For the metal, which sinks, the buoyant force is less than the weight of the metal, leading to an increase in the reading on the spring balance.

78. When a golf ball is lowered into a measuring cylinder containing water, the water level rises by 30 cm3 when the ball is completely submerged. If the mass of ball in air is 33 g, find its density.

To find the density of the golf ball, we can use Archimedes' principle, which states that the buoyant force acting on an object is equal to the weight of the fluid displaced by the object. The buoyant force is given by:

$\text{Buoyant force}=\text{Weight of water displaced}$

Given that the water level rises by 30 cm³ when the golf ball is completely submerged, the volume of the ball (${�}_{\text{ball}}$) is 30 cm³.

Now, the buoyant force is equal to the weight of the water displaced:

$\text{Buoyant force}=\text{Weight of water}$ ${�}_{\text{water}}\cdot �\cdot {�}_{\text{ball}}=\text{Weight of water}$

where:

• ${�}_{\text{water}}$ is the density of water (assumed to be 1000 kg/m³),
• $�$ is the acceleration due to gravity (approximately 9.8 m/s²),
• ${�}_{\text{ball}}$ is the volume of the golf ball in cubic meters.

Rearranging the equation to find ${�}_{\text{ball}}$:

${�}_{\text{ball}}=\frac{\text{Weight of water}}{{�}_{\text{water}}\cdot �}$

Now, the weight of the water is equal to the apparent loss in weight of the golf ball when it is submerged in water. This apparent loss in weight is given by:

$\text{Apparent loss in weight}=\text{Weight in air}-\text{Weight in water}$

Given that the mass of the golf ball in air is 33 g, we can calculate its weight in air (${�}_{\text{air}}$):

${�}_{\text{air}}={�}_{\text{ball}}\cdot �$

Now, we know that the apparent loss in weight is equal to the weight of water displaced:

${�}_{\text{ball}}\cdot �-{�}_{\text{water}}\cdot �={�}_{\text{water}}\cdot �\cdot {�}_{\text{ball}}$

Rearranging the equation to solve for ${�}_{\text{ball}}$:

${�}_{\text{ball}}=\frac{{�}_{\text{ball}}-{�}_{\text{water}}}{{�}_{\text{water}}}$

Substitute the known values:

${�}_{\text{ball}}=\frac{33\text{g}-{�}_{\text{water}}}{1000{\text{kg/m}}^3}$

Now, we have the volume of the golf ball (${�}_{\text{ball}}$). To find its density (${�}_{\text{ball}}$), we can use the formula:

${�}_{\text{ball}}=\frac{{�}_{\text{ball}}}{{�}_{\text{ball}}}$

Substitute the values:

${�}_{\text{ball}}=\frac{33\text{g}}{{�}_{\text{ball}}}$

Now, calculate ${�}_{\text{ball}}$ and then ${�}_{\text{ball}}$. Keep in mind the units, and convert them to consistent units if necessary.

79. A boy gets into a floating boat. (a) What happens to the boat ? (b) What happens to the weight of water displaced ? (c) What happens to the buoyant force on the boat ?

(a) What happens to the boat?

• The boat experiences an upward force known as the buoyant force. The buoyant force is exerted by the water and is equal to the weight of the water displaced by the submerged part of the boat. As a result, the boat rises or "floats" in the water.

(b) What happens to the weight of water displaced?

• The weight of the water displaced by the boat remains the same. When the boy gets into the boat, the boat sinks a bit deeper into the water. However, the volume of water displaced by the submerged part of the boat is still the same. According to Archimedes' principle, the buoyant force is equal to the weight of the water displaced, and this remains constant.

(c) What happens to the buoyant force on the boat?

• The buoyant force on the boat increases. When the boy gets into the boat, the submerged volume of the boat increases, and therefore, the buoyant force acting on the boat increases. The buoyant force always acts in the upward direction and is equal to the weight of the water displaced by the submerged part of the boat. This increase in buoyant force counteracts the additional weight of the boy, allowing the boat to remain afloat.

In summary, when the boy gets into the floating boat, the boat experiences an increased buoyant force, and the weight of the water displaced remains constant. The boat rises or floats in response to the equilibrium between the gravitational force acting on the boat (weight) and the buoyant force exerted by the water.

80. A 1/2 kg sheet of tin sinks in water but if the same sheet is converted into a box or boat, it floats. Why ?

The phenomenon can be explained by considering the density of the tin sheet and the box or boat.

Density is defined as mass per unit volume, and an object will float in a fluid if its average density is less than the density of the fluid. The density ($�$) is given by the formula:

$�=\frac{\text{Mass}}{\text{Volume}}$

Let's analyze the situation:

1. Tin Sheet:

   • The mass of the tin sheet is given as $1\mathrm{/}2\text{kg}$.
   • When the tin sheet is placed in water, it sinks. This indicates that its average density is greater than the density of water (approximately $1000{\text{kg/m}}^3$).

2. Box or Boat:

   • When the same tin sheet is converted into a box or boat, it floats. This implies that the average density of the box or boat is now less than the density of water.
   • The transformation into a box or boat changes the shape and, consequently, the volume of the tin material. The buoyant force acting on the box or boat (equal to the weight of the water displaced) now exceeds the weight of the box or boat.

In summary, the transformation of the tin sheet into a box or boat changes its shape and volume, resulting in a lower average density. The buoyant force acting on the box or boat becomes greater than its weight, allowing it to float on the water.

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