Physical Quantities and Measurement | ICSE Class 7 Physics Notes

This note covers physical quantities and units, volume and unit cubes, measuring liquids and solids, estimating irregular areas with graph paper, density, and calculations involving speed, distance and time.

What does it mean to measure a physical quantity?

A physical quantity is a property that can be measured. Measurement expresses a quantity using a number and a unit, an agreed quantity used for comparison. The number tells us how much; the unit tells us what that number represents.

Length is the distance between two points. Mass is the quantity of matter in an object. Matter is anything that has mass and occupies space. A time interval is the duration between events. These measurements help us find further quantities, including the space occupied by an object and how fast it moves.

Which units and symbols will we use?

A unit symbol is the short form written after a measurement. A centimetre is written as cm, a metre as m, and a kilometre as km. The symbols g and kg mean gram and kilogram. The symbols s, min and h mean second, minute and hour.

The International System of Units, abbreviated as SI, provides internationally agreed units. The SI unit of length is the metre. The SI unit of mass is the kilogram. The SI unit of time is the second.

In calculations, × means multiplication, ÷ means division, − means subtraction and = means equality.

Keep the unit beside the numerical value throughout a calculation. A measurement expressed in one unit may need conversion before it can be combined with another. For length, 1 km = 1000 m. For time, 1 min = 60 s and 1 h = 60 min.

Changing the unit changes the number used to express the measurement, not the quantity being measured. Before comparing measurements, check that they use compatible units. In particular, a distance given in kilometres and a time given in minutes cannot directly produce a speed in metres per second.

Note: Write time symbols as s, min and h. Leave a space between the number and its unit symbol. The symbols do not become plural when the number increases.

What is volume, and why is it measured in cubic units?

Definition: Volume is the amount of space occupied by an object or a substance. It describes a three-dimensional quantity, involving length, width and height.

Three-dimensional means extending in three directions. A solid occupies space rather than merely covering a flat region. To measure that space, imagine dividing the solid into small cubes and counting how many cubes it contains.

What does one cubic centimetre mean?

A unit cube is a cube whose edge is one unit long. A cube has equal edges and square faces. A cube with each edge 1 cm long has a volume of one cubic centimetre, written as 1 cm³.

The raised 3 in cm³ indicates a cubic unit: 1 cm × 1 cm × 1 cm. It does not mean 3 cm. Similarly, a cubic metre, written as m³, is the volume of a cube with each edge 1 m long. The SI unit of volume is the cubic metre.

Capacity is the quantity a container can hold. A container's capacity can be expressed using the same volume units used for its contents. For liquids, the litre, written as L, and millilitre, written as mL, are convenient units.

These relationships connect liquid measurements with solid volumes. When a measuring cylinder gives a displaced-water volume in millilitres, the same numerical value can be written in cubic centimetres for the solid. The unit changes, but the measured space remains the same.

How do we measure the volume of a liquid?

A graduated vessel has a scale marked on it. A graduated cylinder, also called a measuring cylinder, is a narrow transparent container with volume markings. A graduated beaker is another marked container used to measure liquid volume.

Before taking a reading, identify the unit on the vessel, its maximum marked volume, and the value represented by each small division. Do not assume that every cylinder or beaker has the same divisions. Read the scale supplied on the particular vessel.

How much does one division represent?

For a cylinder with ten equal intervals between the 10 mL and 20 mL marks, the volume difference is 10 mL. Each interval therefore represents 10 ÷ 10 = 1 mL. This is the smallest marked volume interval on that cylinder.

The drawing shows a tall, narrow cylinder on a broad base. Its scale is labelled in mL and extends to 100 mL, with smaller divisions between numbered marks.

See Fig. 9.16 in your NCERT textbook

How should the water surface be read?

The meniscus is the curved surface of a liquid in a container. For water, read the bottom of this curve with your eyes at the same level. Looking from above or below does not give the correct alignment with the scale.

1. Place a clean, dry measuring cylinder on a flat surface.
2. Pour water slowly towards the required mark.
3. Use a dropper, a device for adding or removing small drops, to adjust the amount if necessary.
4. Bring your eyes level with the bottom of the water meniscus.
5. Read the corresponding scale mark and record the volume with its unit.

A curved water surface is labelled “Meniscus”. Three eye positions are shown beside the cylinder. The middle eye is level with the bottom of the curve; the other eyes are above and below it.

See Fig. 9.18 in your NCERT textbook

Choose a suitable capacity. Measuring 70 mL in a 50 mL cylinder requires separate measurements of 50 mL and 20 mL. A 100 mL cylinder permits one measurement; the larger divisions of a 250 mL or 500 mL cylinder reduce accuracy for this task.

How can we calculate the volume of a regular solid?

A regular solid has a definite geometrical shape whose volume can be calculated from suitable measurements. A cuboid is a box-shaped solid with rectangular faces. Its length, width and height are measured along three mutually perpendicular directions, meaning directions at right angles.

Measure all three dimensions in the same length unit. Multiplying measurements in centimetres gives a volume in cubic centimetres. The unit records that three lengths have been multiplied, rather than added.

What formula applies to a cuboid?

Volume = length × width × height

The formula can be understood by imagining layers of unit cubes. The length and width determine the cubes in a layer; the height determines how many such layers make up the cuboid. All of the layers contribute to the occupied space.

Worked example 1. A notebook has length 25 cm, width 18 cm and height 2 cm. Calculate its volume.

Formula: Volume = length × width × height.

Substitute: Volume = 25 cm × 18 cm × 2 cm.

Answer: The notebook's volume is 900 cm³.

How does the formula change for a cube?

A cube is a special cuboid with equal length, width and height. Its volume is found by multiplying the length of an edge by itself and then by itself again. The word “side” in the following formula means the length of an edge.

Volume = side × side × side

Worked example 2. Find the volume of a cube with a side of 4 cm.

Formula: Volume = side × side × side.

Substitute: Volume = 4 cm × 4 cm × 4 cm.

Answer: The volume is 64 cm³.

Keep area and volume distinct. Multiplying two perpendicular lengths describes a rectangular area. A cuboid's volume requires the third dimension as well. Measuring a notebook's cover alone therefore does not establish the volume of the notebook.

How is the volume of an irregular solid measured?

An irregular solid, such as a stone, does not have a simple regular shape for which measured length, width and height give its volume. Its volume can instead be found by water displacement, the movement of water when an object occupies space in it.

What readings are needed?

The initial reading is the water volume before the solid is added. The final reading is the cylinder reading after the solid is under water. The increase gives the volume displaced by the solid.

Solid volume = final reading − initial reading

1. Pour water into a graduated cylinder and record its initial volume.
2. Tie the stone with a thread and lower it slowly into the water.
3. Ensure that the stone is completely under water and record the final reading.
4. Subtract the initial reading from the final reading.
5. Express the resulting volume in mL, or use the equivalent value in cm³ for the solid.

Two measuring cylinders are drawn side by side. The first contains water without the object. The second contains a stone suspended from a thread, and its water level is higher.

See Fig. 9.19 in your NCERT textbook

Worked example 3. A stone is completely immersed in water. The initial cylinder reading is 50 mL and the final reading is 55 mL. Find its volume, using 1 mL = 1 cm³.

Formula: Solid volume = final reading − initial reading.

Substitute: Solid volume = 55 mL − 50 mL.

Answer: The displaced volume is 5 mL, so the stone's volume is 5 cm³.

The final reading is not the stone's volume: it includes the effect of the water already present. The subtraction isolates the increase. Use a solid that does not dissolve in or absorb the water, keep it fully submerged, and avoid spilling water or trapping air bubbles.

Note: “Submerged” means completely below the liquid surface. If part of the solid remains above water, the displaced volume does not represent the whole solid's volume.

How do we estimate the area of an irregular shape?

Area is the amount of region enclosed by a closed figure. It describes the size of a flat region. A closed figure has a boundary that encloses a region without a gap. Its area is different from the length of its boundary.

Graph paper has a grid of equal squares. It can be used to estimate areas even when the boundary is irregular. An estimate is an approximate value. Here the approximation arises when parts of squares near the boundary are counted using agreed rules.

What does a square unit represent?

A square measuring 1 unit along each side has an area of one square unit. A square with sides of 1 cm has an area of one square centimetre, written as cm². The raised 2 indicates an area unit.

Trace the shape onto transparent paper and place the tracing over squared paper. First identify the area represented by each small square. Counting squares is meaningful only when the unit area of a square is known.

How should full and partly covered squares be counted?

1. Count each complete small square inside the boundary as one square unit.
2. Ignore a covered portion smaller than half a square.
3. Count a covered portion larger than half a square as one square unit.
4. Count an exactly half-covered square as half a square unit.
5. Add these contributions and express the result as an estimated area.

Keep “estimated” in the result. The boundary may cross many squares, and the counting convention rounds their contributions. Do not describe the answer as exact merely because the final addition is exact. The addition combines values that already include approximations.

Area uses square units because it covers a region. Volume uses cubic units because it occupies space. Estimating the outline of an irregular object on graph paper gives the area of that outline, not the volume of the solid object.

What is density, and how does it differ from mass?

Definition: Density is the mass present in a unit volume of a substance. It relates the amount of matter to the space that matter occupies.

Density = mass ÷ volume

Mass alone does not tell us density. To calculate density, we also need volume. For objects with the same volume, the one with greater mass has greater density. Comparing objects of different sizes requires calculating their mass per unit volume.

What do density units mean?

When mass is measured in grams and volume in cubic centimetres, density is measured in grams per cubic centimetre, written as g/cm³. The slash means “per”, or division by. For liquids measured in millilitres, grams per millilitre is written as g/mL.

Worked example 4. An aluminium block has a mass of 27 g and a volume of 10 cm³. Calculate its density.

Formula: Density = mass ÷ volume.

Substitute: Density = 27 g ÷ 10 cm³.

Answer: Its density is 2.7 g/cm³. This expresses the mass associated with each cubic centimetre of the aluminium.

How can objects of different masses be compared?

An object labelled A has mass 200 g and volume 40 cm³. Another object, labelled B, has mass 240 g and volume 60 cm³. The labels A and B identify the objects; they do not represent units.

Object B has greater mass, but object A has greater density. The comparison shows why “heavier” does not by itself mean “denser”. Dividing each mass by its own volume makes the comparison on a common basis.

The mass of 1 mL of water is close to 1 g at room temperature. Keep this qualification when using water as a comparison. “Close to” is an approximation, not a statement that its mass is exactly that value under every condition.

How do we determine the density of regular and irregular solids?

Finding density requires two measurements: mass and volume. A balance supplies the mass. The method used to find volume depends on the shape of the solid. Once both quantities are known, the same density formula applies.

How should the mass be measured?

A digital balance displays a measured mass electronically. Its tare control resets the reading to zero after a container has been placed on the pan. This allows the mass of the added object to be read without including the container.

1. Switch on the balance and check that the initial display reads zero.
2. Place clean, dry butter paper on the pan.
3. Press the tare or reset control to return the display to zero.
4. Place the solid on the paper and record its mass with the displayed unit.

For a cuboid, measure length, width and height and multiply them to obtain volume. For a suitable irregular solid, use the increase in water level. In both cases, divide the measured mass by the volume of that same object.

Worked example 5. A stone has mass 16.400 g. Complete immersion changes a cylinder reading from 50 mL to 55 mL, without water loss. Calculate its density, using 1 mL = 1 cm³.

Formula: Volume = final reading − initial reading; Density = mass ÷ volume.

Substitute: Volume = 55 − 50 = 5 mL = 5 cm³. Density = 16.400 g ÷ 5 cm³.

Answer: The density of the stone is 3.28 g/cm³.

Can the density formula be rearranged?

If density and volume are known, multiplying them gives mass. If mass and density are known, dividing mass by density gives volume. These are rearrangements of the same relationship, not separate properties of the substance.

Derivation: How can mass and volume be found from density?

1. Start with density = mass ÷ volume, the definition of density.
2. Multiply both sides by volume. This gives density × volume = mass.
3. To isolate volume instead, divide this last equality by density. This gives volume = mass ÷ density.

Mass = density × volume

Volume = mass ÷ density

Check that the units match the relationship. A density in g/cm³ combined with a volume in cm³ gives a mass in g. A mass in g divided by density in g/cm³ gives volume in cm³.

Note: In a displacement calculation, first find the solid's volume. Dividing its mass by the final cylinder reading would include the original water and would not calculate the solid's density.

What does speed tell us about motion?

Motion is a change in position with time relative to a reference point, an object or place used for comparison. Distance travelled is the length of the path covered. Speed tells us the distance covered in a unit time. It helps compare how fast or slowly objects move.

When two objects travel for the same time, the one covering a greater distance has greater speed over that interval. When two objects cover the same distance, the one taking less time has greater speed. Each comparison holds the other quantity the same.

How is speed calculated?

Speed = distance ÷ time

Use the distance covered and the time taken to cover that distance. The unit follows from those measurements. Metres divided by seconds give metres per second, written as m/s. Kilometres divided by hours give kilometres per hour, written as km/h. The SI unit of speed is the metre per second.

Does one speed describe every instant of a journey?

Average speed is the total distance covered divided by the total time taken. An object might have travelled sometimes more slowly and sometimes more quickly. The speed calculated for the whole journey therefore need not describe its speed at every instant.

When using a whole journey's distance and time in the speed formula, interpret the answer as an average. Do not conclude that the object maintained that speed throughout unless constant speed, meaning unchanging speed, is given.

This also matters when predicting a later distance or time. The calculation must use the speed specified for that part of the journey. A measured average for an earlier journey does not by itself establish an unchanging speed for a later one.

How do we solve speed, distance and time problems?

Begin by identifying what is given and what must be found. Write the formula before substituting values. Check that distance and time are expressed in units compatible with the requested answer, then calculate and retain the unit in the result.

How can speed be found after converting units?

Worked example 6. Swati cycles 3.6 km from home to school in 15 min. Find her average speed in m/s. Use 1 km = 1000 m and 1 min = 60 s.

Formula: Speed = distance ÷ time.

Substitute: Distance = 3.6 × 1000 = 3600 m. Time = 15 × 60 = 900 s. Speed = 3600 ÷ 900.

Answer: The bicycle's average speed is 4 m/s.

Dividing the original numbers without conversion would give a speed in kilometres per minute. Writing m/s after that number would not change its unit correctly. Convert the quantities used in the division before labelling the result.

How are distance and time calculated?

Multiplying speed by the travel time gives distance. Dividing distance by speed gives the time needed. Use the following forms when the stated speed applies to the journey being considered.

Derivation: How do we rearrange the speed formula?

1. Start with speed = distance ÷ time for the journey being considered.
2. Multiply both sides by time. This gives speed × time = distance.
3. For a non-zero speed, divide the last equality by speed to isolate time: time = distance ÷ speed.

Distance = speed × time

Time = distance ÷ speed

Worked example 7. A bus travels at 50 km/h for 2 h. Find the distance covered.

Formula: Distance = speed × time.

Substitute: Distance = 50 × 2.

Answer: The bus covers 100 km.

Worked example 8. A train travels at 90 km/h. How long does it take to cover 360 km?

Formula: Time = distance ÷ speed.

Substitute: Time = 360 ÷ 90.

Answer: The train takes 4 h.

How can one result be used in the next step?

Worked example 9. A train covers 180 km in 3 h. Find its speed in km/h and the distance it would cover in 4 h if it maintains that speed throughout.

Formula: Speed = distance ÷ time; Distance = speed × time.

Substitute: Speed = 180 ÷ 3 = 60 km/h. Distance = 60 × 4.

Answer: Its speed is 60 km/h, and the distance in 4 h is 240 km under the stated condition.

The condition in the final example matters. The predicted distance uses a maintained speed. If the train speeds up, slows down or stops during that later interval, the prediction does not automatically follow from the earlier average.

Glossary

• Physical quantity — A measurable property expressed using a numerical value and an appropriate unit.
• Unit — An agreed quantity used as the standard for measuring another quantity of the same kind.
• Volume — The amount of three-dimensional space occupied by an object or a substance.
• Unit cube — A cube with an edge of one unit, used to represent a unit of volume.
• Capacity — The quantity that a container can hold, expressible in units of volume.
• Graduated cylinder — A transparent container with scale markings used to measure the volume of a liquid.
• Meniscus — The curved liquid surface whose bottom is read when measuring water in a cylinder.
• Water displacement — The movement of water when an immersed object takes up space within it.
• Area — The amount of region enclosed by a closed figure, measured in square units.
• Density — The mass present in a unit volume of a substance, calculated as mass divided by volume.
• Tare — The balance control that resets its display to zero after a container has been added.
• Average speed — The total distance covered by an object divided by the total time taken.

Common errors and misconceptions

• Misconception: Volume is measured in cm. Correct: Centimetres measure length. A solid's volume may be measured in cm³, while its area may be measured in cm².
• Misconception: The final cylinder reading is the stone's volume. Correct: Subtract the initial water reading from the final reading after complete immersion.
• Misconception: Water volume can be read from any eye position. Correct: Keep the eye level with the bottom of the water meniscus when reading the scale.
• Misconception: Every partly covered graph square counts as a full square. Correct: Apply the separate conventions for less than half, more than half and exactly half coverage.
• Misconception: The object with greater mass must be denser. Correct: Density compares mass per unit volume, so calculate using both the mass and volume of each object.
• Misconception: Distance divided by time proves that speed stayed constant. Correct: Total distance divided by total time gives average speed; speed during the journey might have varied.
• Misconception: Kilometres divided by minutes directly gives m/s. Correct: Convert kilometres to metres and minutes to seconds before calculating a speed in m/s.

Exam-style questions with model answers

Q1. Define volume and explain what one cubic centimetre represents. [2 marks]

1. Volume is the amount of space occupied by an object or a substance.
2. One cubic centimetre, written as 1 cm³, is the volume of a cube whose edges are each 1 cm long.

Q2. A measuring cylinder has ten equal intervals between 10 mL and 20 mL. Find the value of each interval and explain how to read the water meniscus correctly. [3 marks]

1. The marked difference is 20 − 10 = 10 mL. Dividing by ten equal intervals gives 1 mL for each interval.
2. Place the cylinder on a flat surface and identify the bottom of the meniscus, which is the curved surface of the water.
3. Keep the eyes level with that bottom point, read the corresponding mark on the scale, and record the volume in mL.

Q3. An irregular outline is traced onto graph paper whose small squares each represent one square unit. Explain how to count complete and partly covered squares to estimate its area. [4 marks]

1. Count each complete square inside the outline as one square unit, establishing the contribution from the fully covered part of the region.
2. Ignore covered portions smaller than half a square when applying this estimation convention.
3. Count each covered portion larger than half a square as one square unit.
4. Count each exactly half-covered square as half a square unit, add all contributions, and state the result as an estimated area in square units.

Q4. A cuboid-shaped notebook is 25 cm long, 18 cm wide and 2 cm high. Calculate its volume, explaining the choice of formula and unit. [3 marks]

1. A cuboid has three perpendicular dimensions, so use volume = length × width × height rather than multiplying only two lengths.
2. Substitute the supplied measurements: volume = 25 × 18 × 2 = 900.
3. All three lengths are in centimetres. Their product therefore has the cubic unit cm³, giving a notebook volume of 900 cm³.

Q5. A stone has mass 16.400 g. It does not dissolve in or absorb water. Complete immersion without trapped bubbles or water loss changes a cylinder reading from 50 mL to 55 mL. Calculate its density and explain why the final reading alone cannot be used as its volume. Use 1 mL = 1 cm³. [5 marks]

1. The solid's volume is found from the increase in the cylinder reading, so subtract the initial reading from the final reading.
2. The displaced volume is 55 − 50 = 5 mL. Using the supplied conversion, the stone's volume is 5 cm³.
3. Density means mass per unit volume, so the required relationship is density = mass ÷ volume.
4. Substitute the stone's mass and calculated volume: density = 16.400 ÷ 5 = 3.28 g/cm³.
5. The final reading includes the original water as well as the rise caused by the stone. Using 55 mL would therefore not use the volume of the stone alone.

Q6. Object A has mass 200 g and volume 40 cm³. Object B has mass 240 g and volume 60 cm³. Calculate both densities, identify the denser object and explain why mass alone gives the wrong comparison. [4 marks]

1. For each object, calculate density as its own mass divided by its own volume. Object A has density 200 ÷ 40 = 5 g/cm³.
2. Object B has density 240 ÷ 60 = 4 g/cm³.
3. Object A is denser because 5 g/cm³ is greater than 4 g/cm³.
4. Object B has greater mass but also occupies more space. Density compares mass per unit volume, so greater mass alone does not establish greater density.

Q7. Swati cycles 3.6 km to school in 15 min. Calculate her average speed in m/s and explain what this answer tells us. Use 1 km = 1000 m and 1 min = 60 s. [4 marks]

1. Convert the distance to metres using the given relationship: 3.6 × 1000 = 3600 m.
2. Convert the journey time to seconds: 15 × 60 = 900 s.
3. Average speed = total distance ÷ total time = 3600 ÷ 900 = 4 m/s.
4. This is the average for the whole journey. It does not establish that Swati travelled at 4 m/s at every instant; her speed might have varied.

Q8. A train travels 180 km in 3 h. Find its average speed in km/h, then calculate the distance it will cover during a further 4 h if it maintains that speed throughout. Explain why the stated condition matters. [5 marks]

1. The average speed for the first journey is calculated by dividing the total distance covered by the total time taken.
2. Substitution gives average speed = 180 km ÷ 3 h = 60 km/h.
3. For the further journey at the stated maintained speed, use distance = speed × time, with time measured in hours.
4. The predicted distance is therefore 60 × 4 = 240 km during the further 4 h.
5. This result depends on maintaining 60 km/h throughout that interval. The earlier average alone does not guarantee the same distance if the train's later speed changes or it stops.

Key takeaways

• A measurement needs a numerical value and a unit; check the units before comparing quantities or substituting into a formula.
• Volume is occupied space and uses cubic units; one millilitre represents the same volume as one cubic centimetre.
• Read the bottom of the water meniscus at eye level, after checking the value of the cylinder's scale divisions.
• Find a cuboid's volume by multiplying length, width and height; find a suitable irregular solid's volume by water displacement.
• Graph-paper counting estimates irregular areas by combining complete squares with agreed rules for partly covered boundary squares.
• Density is mass divided by volume; a heavier object need not be denser if it also occupies more space.
• Average speed is total distance divided by total time; its value need not describe every instant of the journey.
• Use compatible units when calculating speed, and retain the stated speed condition when predicting later distance or time.

Test yourself

How does a cubic unit differ from a square unit?

A cubic unit measures occupied space, or volume. A square unit measures the size of a flat region, or area.

What does the tare control on a balance do?

It resets the reading to zero after a container is placed on the pan, allowing the added object's mass to be read.

Where should your eyes be when reading water volume?

Keep the eyes at the same level as the bottom of the water meniscus and read the corresponding scale mark.

Why is an irregular solid's volume found using two cylinder readings?

The final reading includes the original water. Subtracting the initial reading gives the volume increase caused by the submerged solid.

How is an exactly half-covered graph square counted?

It contributes half a square unit when each complete small square represents one square unit.

What measurements are needed to find density?

Measure the object's mass and volume, then divide mass by volume using compatible units.

For equal travel times, which of two objects has greater average speed?

The object covering the greater distance has the greater average speed over that time interval.

A bus travels at 50 km/h for 2 h. What distance does it cover?

Distance = speed × time = 50 × 2 = 100 km, using the speed stated for that journey.