Physical Quantities and Measurement

Mass = Density x Volume

Reason — The density of a substance is its mass per unit volume.

800 kg m^-3

Reason —

$$\begin{aligned}\text{Relative density of alcohol} = \frac{\text{Density of alcohol}}{\text{Density of water}} \\[1em] \text{0.8} = \frac{\text{Density of alcohol}}{\text{1000 kg m}^{-3}} \\[1em] \Rightarrow \text{Density of alcohol} = 0.8 \times 1000 = {\text{800 kg m}^{-3}}\end{aligned}$$

48 g

Reason —

$$\begin{aligned}\text{Density} = \frac{\text{Mass}}{\text{Volume}} \\[1em] \text{0.8} = \frac{\text{Mass}}{\text{60}} \\[1em] \Rightarrow \text{Mass} = 0.8 \times 60 = 48 \text{g}\end{aligned}$$

The mass of a certain volume of brass is more than the mass of equal volume of aluminium.

Reason —

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

So mass of certain volume of brass will be more than the mass of same volume of aluminium as density of brass is more than density of aluminium.

The density bottle will store 25 mL of any liquid in it.

Reason — A density bottle can store specific volume of any density of liquid. Since this bottle is marked 25 mL so it can store 25 mL of any density liquid in it.

The buoyant force on a body is equal to the weight of the liquid displaced by it.

Reason — According to Principle of Floatation, the buoyant force is equal to the weight of the liquid displaced by the immersed part of the body.

equal to the weight of the wood piece

Reason — According to Law of Floatation when a body floats on water the weight of the body is equal to the buoyant force.

sink

Reason — According to the Principle of Floatation, if the weight of the body is more than the buoyant force acting on the body the body will sink into the liquid.

Assertion is false but reason is true

Explanation

Assertion (A) is false because in order to come to the surface of water, fishes fill the bladder with air to decreases the average density of the whale and so it rises to the surface.

Reason (R) is true because a body of density greater than the density of liquid, sinks inside the liquid, while a body of density equal or less than the density of liquid, floats on the liquid.

Hence, assertion is false but reason is true.

(a) 1 kg is the mass of 1000 mL of water at 4°C.

(b) Mass = density x volume.

(c) The S.I. unit of density is kg m^-3.

(d) Density of water is 1000 kg m^-3.

(e) 1 g cm^-3 = 1000 kg m^-3.

(f) The density of a body which sinks in water is more than 1000 kg m^-3.

(g) A body sinks in a liquid A, but floats in a liquid B. The density of liquid A is less than the density of liquid B.

(h) A body X sinks in water, but a body Y floats on water. The density of the body X is more than the density of body Y.

(i) The buoyant force experienced by a body when floating in salt-water is equal to that when floating in pure water.

(j) The apparent weight of a body floating in a liquid is zero.

(a) False
Correct Statement — Equal volumes of two different substances have different masses.

(b) False
Correct Statement — The density of a piece of brass will not change by changing its size or shape.

(c) True

(d) True

(e) False
Correct Statement — Relative density has no unit.

(f) False
Correct Statement — When a body is immersed in a liquid, the buoyant force experienced by the body is equal to the weight of the liquid displaced by the immersed part of the body.

(g) True

(h) False
Correct Statement — The buoyant force experience by the body is different when floating or sinking in water.

(i) True

(j) True

The density of a substance is its mass per unit volume i.e.

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

The S.I. unit of density is kg m^-3 (kilogram per cubic metre).
1 g cm^-3 = 1000 kg m^-3.

The statement means one cubic centimeter volume of brass has mass of 8.4 g.

Cork < Water < Iron < Brass < Mercury.

The density of cork is the least and that of mercury is the most.

When temperature increases, volume of most of the liquid increases, so density decreases. Similarly when temperature decreases volume of liquids decreases so density increases. But exception is water which contracts on heating from 0°C to 4°C and expands on heating above 4°C. Water has maximum density at 4°C.

(a) Mass of the liquid remains unaffected on heating.

(b) Volume of the liquid increases with increase in temperature.

(c) Density of liquid decreases on heating because its volume increases and density is inversely proportional to volume.

The relative density of a substance is defined as the ratio of the density of a substance to the density of water. The symbol used for relative density is R.D. Thus,

$\text{R.D.} = \frac{\text{Density of the substance}}{\text{Density of water}}$

Relative density has no unit because it is the ratio of two densities.

Relative density of aluminium is 2.7 means a piece of aluminium of any volume has mass 2.7 times that of an equal volume of water.

The law of floatation states that — "When a body floats in a liquid, the weight of the liquid displaced by its immersed part is equal to the total weight of the body".

i.e., while floating

Weight of the floating body = Weight of the liquid displaced by its immersed part (i.e., buoyant force)

When a body floats in a liquid, the weight of the liquid displaced by its immersed part (i.e. the buoyant force) is equal to the total weight of the body.

The density of the material of a coin is determined by the following experiment:

1. We know density is mass per unit volume. So, first mass of the coin is measured by a common beam balance. Let it be M gram.
2. Volume of the coin is measured by a measuring cylinder as shown in the figure given below.
3. Take a measuring cylinder and fill it partly with water.
4. Note the level of water. Let it be V₁ mL.
5. Tie the coin with a thread and gently lower the coin inside the measuring cylinder. Take care that no water splashes out.
6. Note the level of water again. Let it be V₂ mL.
7. Find the volume of the coin (V) by subtracting V₁ from V₂.
   V = V₂ - V₁
   Density of coin is calculated by the formula:

$$\begin{aligned}\text{Density} = \frac{\text{Mass}}{\text{Volume}} \\[1em] \Rightarrow \text{D} = \frac{\text{M}}{\text{V}}{\text{ g cm}^{-3}}\end{aligned}$$

The density of a liquid is determined by first finding its mass and then its volume.

1. Take an empty beaker and measure its mass by using a common beam balance. Let the mass be M₁ g.
2. Take some amount of liquid and pour it into the empty beaker. Measure its mass again. Let it be M₂ g.
3. The difference between M₂ and M₁ gives the mass(M) of the liquid. M = M₂ - M₁.
4. Pour the liquid from beaker into a measuring cylinder. Note the volume of the liquid. Let it be V mL or V g cm^-3.
5. Calculate the density of the liquid using the following formula:

$$\begin{aligned}\text{Density} = \frac{\text{Mass}}{\text{Volume}} \\[1em] \Rightarrow \text{D} = \frac{\text{M}}{\text{V}}{\text{ g cm}^{-3}}\end{aligned}$$

A density bottle is a specially designed bottle which is used to determine the density of a liquid. The figure of density bottle is shown below:

Determination of Density of a Liquid using the Density Bottle

1. Measure the mass of empty density bottle using a beam balance. Let it be M₁ g.
2. Remove the stopper of the bottle and fill it with water. Replace the stopper, wipe the outside of bottle dry and measure its mass again. Let it be M₂ g.
3. Empty the bottle and dry it. Fill it with the given liquid, replace the stopper and wipe the outside of bottle dry. Measure its mass. Let it be M₃ g.
4. Calculate the mass of water (M₂ - M₁) and the mass of liquid (M₃ - M₁).
5. Since density of water is 1 g cm^-3, mass of water contained in bottle is the volume of bottle. Thus,
       Mass of liquid = M₃ - M₁
       Volume of liquid = M₂ - M₁

$$\begin{aligned}\therefore \text{Density of liquid} = \frac{\text{Mass of liquid}}{\text{Volume of liquid}} \\[1em] = \frac{\text{M}_{3}-\text{M}_{1}}{\text{M}_{2}-\text{M}_{1}}{\text{ g cm}^{-3}}\end{aligned}$$

If the density of a body is less than the density of liquid, the body will float on the surface of the liquid. But if the density of a body is more than the density of liquid the body will sink in the liquid.

An iron needle is solid and density of iron is more than that of water. Weight of iron needle is more than the buoyant force of water on it so it sinks in water. Although ship is made of iron but it has hollow space that is filled with air which make its average density less than that of water, so ship floats on water.

Sea water contains salts so its density is more than the density of river water. The weight of a person gets balanced by the less immersed part of his body in sea water as compared to that in river water. Thus, it is easier to swim in sea water than in river water.

Density of ice (0.9 g cm^-3) is less than the density of sea water (1.02 g cm^-3). So, an iceberg floats in sea water with its large portion submerged inside the water and only a little portion of it is above the surface of water. Thus, a ship can collide with the invisible part of the iceberg under the surface of water. Hence, it is dangerous.

When a stone is under water then its weight is less than its weight in air because it experiences an upward buoyant force. Due to apparent loss in weight, it is lighter in water which makes it easier to lift in water than in the air.

A submarine is a water tight boat which can travel under water like a ship and it is provided with water tanks.

To make the submarine dive, the tanks are filled with water so that the average density of the submarine becomes greater than the density of sea water, so it sinks into the water.

To make the submarine rise to the surface of water, these tanks are emptied. This makes the average density of the submarine less than the density of sea water, so the submarine rises up to the surface of water.

The density of hydrogen gas is less than the density of air. So the buoyant force experienced by the balloon due to air is greater than the weight of the balloon. Due to this net upward force balloon rises up in air.

The density of cork is less than the density of water so it floats on water surface but the density of iron nail is more than the density of water so it sinks in water.

1. Body A will float on water because its density (500 kg m^-3) is less than the density of water (1 g cm^-3 or 1000 kg m^-3).
2. Body B will sink in water because its density (2520 kg m^-3) is more than the density of water (1 g cm^-3 or 1000 kg m^-3).
3. Body C will sink in water because its density (1100 kg m^-3) is more than the density of water (1 g cm^-3 or 1000 kg m^-3).
4. Body D will float on water because its density (0.85 g cm^-3) is less than the density of water (1 g cm^-3).

(a) Iron will sink in water as density of iron (7.8 g cm^-3) is more than the density of water (1 g cm^-3).

(b) Iron will float in mercury as the density of iron (7.8 g cm^-3) is less than the density of mercury (13.6 g cm^-3).

(a) The buoyant force is same in each case as the weight of the body is same in each case and the buoyant force is equal to the weight of the liquid displaced by the immersed part of the body which balances the weight of the body.

(b) Liquid A has the least density as the maximum part of the body is immersed in liquid A.

(c) Liquid C has the highest density as the body immerses the least in liquid C.

Density of ice is 0.9 g cm^-3 and density of water is 1.0 g cm^-3. Since density of ice is less than density of water, so ice floats on water.

The egg sinks in fresh water because the density of the egg is greater than the density of fresh water. However, when salt is added to water, the density of the solution increases. In a strong salt solution, the density becomes greater than that of the egg, so the upward buoyant force becomes strong enough to support the egg, causing it to float.

Sea water is denser than river water because it contains dissolved salts. When the ship is in sea water, the buoyant force acting on it is greater due to the higher density. As the ship moves to river water, where the density is lower, the buoyant force decreases. To balance its weight, the ship must displace more water, causing it to submerge deeper in river water.

All three pieces—the disc, the square, and the L-section—have exactly the same density.
Density is an intrinsic property of a material. For a given substance (here, iron) it depends only on the material's nature, not on the object's shape or size. Since each piece is cut from the same sheet of iron, their density remains the constant density of iron, regardless of how that mass is distributed geometrically.

(a) The density of air is 1.28 g litre^-1.

1 litre = 1000 cm³

∴ Density of air in g cm^-3 = $\frac{\text{1.28}}{\text{1000}}$

Density of air = 0.00128 g cm^-3

(b) Density of air in kg m^-3 = Density of air in g cm ^-3 x 1000 = 0.00128 x 1000 = 1.28 kg m^-3

∴ Density of air in kg m^-3 = 1.28 kg m^-3.

Given,

• Density of air = 1.11 kg m^-3
• Volume of hall = 10 m x 7 m x 5 m = 350 m³
• Mass = ?

We know,

$$\begin{aligned}\text{Density}= \frac{\text{Mass}}{\text{Volume}} \\[1em] \Rightarrow \text{Mass} = \text{Density} \times \text{Volume} \\[1em] = 1.11 \times 350 \\[1em] = 388.5 \text{ kg}\end{aligned}$$

Hence, the mass of air in the hall = 388.5 kg.

Given,

• Density of Aluminium = 2.7 g cm^-3.

Density of aluminium in kg m^-3 = 2.7 x 1000 = 2700 kg m^-3

Hence, Density of aluminium = 2700 kg m^-3.

$$\begin{aligned}\text{Density of alcohol in g cm}^{-3} = \frac{\text{Density of alcohol in kg m}^{-3}}{\text{1000}} \\[1em] = \frac{600}{1000} \\[1em] = 0.6 \text{ g cm}^{-3}\end{aligned}$$

Hence, the density of alcohol in g cm^-3 = 0.6 g cm^-3.

Given,

• Mass of zinc = 438.6 g
• Volume of zinc = 86 cm³
• Density of zinc = ?

$$\begin{aligned}\text{Density of Zinc} = \frac{\text{Mass of Zinc}}{\text{Volume of Zinc}} \\[1em] = \frac{\text{438.6}}{\text{86}} \\[1em] = 5.1 \text{ g cm}^{-3}\end{aligned}$$

So, the density of Zinc = 5.1 g cm^-3.

Given,

• Mass of wood = 150 g
• Volume of wood = 200 cm³
• Density of wood = ?

(a) C.G.S. unit of density is g cm^-3.

$$\begin{aligned}\text{Density of Wood}= \frac{\text{Mass of Wood}}{\text{Volume of Wood}} \\[1em] = \frac{\text{150}}{\text{200}} \\[1em] = 0.75 \text{ g cm}^{-3}\end{aligned}$$

Density of wood = 0.75 g cm^-3.

(b) S.I. unit of density is kg m^-3.

Density of Wood in kg m^-3 = Density of Wood in g cm^-3 x 1000

= 0.75 x 1000

= 750 kg m^-3

So, the density of wood is 750 kg m^-3.

Given,

• Density of wood = 0.8 g cm^-3 = 800 kg m^-3
• Mass of wood = 6000 kg
• Volume of wood = ?

$$\begin{aligned}\text{Density of Wood}= \frac{\text{Mass of Wood}}{\text{Volume of Wood}} \\[1em] \Rightarrow \text{Volume of Wood}= \frac{\text{Mass of wood}}{\text{Density of wood}} \\[1em] = \frac{\text{6000}}{\text{800}} \\[1em] = 7.5 \text{ m}^3\end{aligned}$$

So, volume of wood = 7.5 m³.

Given,

• Mass of solid = 72 g
• Initial volume of water = 24 mL
• Final volume of water = 42 mL

Volume of solid = Final volume of water - Initial volume of water

= 42 - 24

= 18 mL

$$\begin{aligned}\text{Density of Solid}= \frac{\text{Mass of Solid}}{\text{Volume of Solid}} \\[1em] = \frac{\text{72}}{\text{18}} = 4\end{aligned}$$

So, density of solid = 4.0 g cm^-3.

Given,

• Mass of empty density bottle (M₁) = 21.8 g
• Mass of bottle with water (M₂) = 41.8 g
• Mass of bottle with liquid (M₃)= 40.6 g

(a) Mass of water in bottle = M₂ - M₁ = 41.8 - 21.8 = 20 g

Density of water is 1 g cm^-3, so volume of density bottle = 20 mL

(b) Mass of liquid in bottle = M₃ - M₁ = 40.6 - 21.8 = 18.8 g

$$\begin{aligned}\text{R.D. of liquid}= \frac{\text{Mass of Liquid}}{\text{Mass of eq. vol. of water}} \\[1em] = \frac{\text{18.8}}{\text{20}} \\[1em] = 0.94\end{aligned}$$

Relative density of liquid = 0.94.

Given,

• Mass of empty density bottle (M₁) = 22 g
• Mass of bottle + water (M₂) = 50 g
• Mass of bottle + brine solution (M₃) = 54 g

Mass of water in bottle = M₂ - M₁ = 50 - 22 = 28 g

Density of water is 1 g cm^-3, so volume of density bottle = 28 cm³.

Mass of brine solution in bottle = M₃ - M₁ = 54 - 22 = 32 g

$$\begin{aligned}\text{Density of brine soln.}= \frac{\text{Mass of brine soln.}}{\text{ Vol. of brine soln.}} \\[1em] = \frac{\text{32}}{\text{28}} \\[1em] = 1.14 \text{ g cm}^{-3}\end{aligned}$$

So, density of brine solution = 1.14 g cm^-3.

$$\begin{aligned}\text{R.D. of brine soln.}= \frac{\text{Mass of brine soln.}}{\text{Mass of eq. vol. of water}} \\[1em] = \frac{\text{32}}{\text{28}} \\[1em] = 1.14\end{aligned}$$

So, Relative density of brine solution is 1.14.

Given,

• Mass of empty density bottle (M₁) = 30 g
• Mass of bottle with water (M₂) = 75 g
• Mass of bottle with liquid (M₃)= 65 g

(a) Mass of water in bottle = M₂ - M₁ = 75 - 30 = 45 g

Density of water is 1 g cm^-3, so volume of density bottle = 45 mL.

(b) Mass of liquid in bottle = M₃ - M₁ = 65 - 30 = 35 g

$$\begin{aligned}\text{Density of liquid}= \frac{\text{Mass of liquid}}{\text{ Vol. of liquid}} \\[1em] = \frac{\text{35}}{\text{45}} \\[1em] = 0.77 \text{ g cm}^{-3}\end{aligned}$$

So, density of liquid = 0.77 g cm^-3.

(c) $$\begin{aligned}\text{R.D. of liquid}= \frac{\text{Mass of liquid}}{\text{Mass of eq. vol. of water}} \\[1em] = \frac{\text{35}}{\text{45}} \\[1em] = 0.77\end{aligned}$$

So, Relative density of liquid is 0.77.

(i) Floating or sinking does not depend on weight alone; it depends on density and buoyant force. A wooden log floats because the density of wood is less than the density of water and it can displace enough water to balance its weight. A stone sinks because its density is greater than the density of water.

(ii) A water-filled bottle has much greater mass with nearly the same volume, so its average density becomes close to that of water. Hence, it barely floats.
An empty, tightly capped bottle contains air inside it. This makes its average density much less than that of water, so it floats easily.

(iii) The stone wrapped in bubble wrap will float because the bubble wrap contains trapped air, which increases the overall volume without significantly increasing the mass and reduces the overall density of the system below the density of water, causing it to float.

(iv) Students should use the water displacement method in which the stone is submerged in water and the volume of water displaced is measured and then density is calculated using:

$\text{Density} = \frac{\text{Mass}}{\text{Volume}}$

(v) Yes, the water-filled bottle would float easily in mercury because mercury has a very high density (13,600 kg m^-3), which is much greater than the density of the water-filled bottle.

Since the bottle's density is much less than mercury's density, it will float comfortably on the surface of mercury