Physical Science Classes VI-X for School Assistants and TET Paper 2A · Chapter 1

Measurement, Units, Motion, Scalars and Vectors, Graphs, Circular Motion

What to remember

The SI system has seven base units (metre, kilogram, second, ampere, kelvin, mole, candela); every other unit is derived from them.
Velocity = displacement ÷ time (a vector); speed = distance ÷ time (a scalar). The three equations of uniform acceleration are v = u + at, s = ut + ½at², v² = u² + 2as.
In uniform circular motion speed is constant but velocity changes, so there is a centripetal acceleration a = v²/r directed to the centre.

Measurement and units

To measure is to compare a quantity with a standard of the same kind. A physical quantity is written as a number times a unit. Fundamental (base) quantities are independent; derived quantities are formed from them.

Base quantity	SI unit	Symbol
Length	metre	m
Mass	kilogram	kg
Time	second	s
Electric current	ampere	A
Temperature	kelvin	K
Amount of substance	mole	mol
Luminous intensity	candela	cd

Common derived units: area m²; volume m³; speed m/s; acceleration m/s²; force newton (N = kg m/s²); pressure pascal (Pa = N/m²); work and energy joule (J = N m); power watt (W = J/s); frequency hertz (Hz = 1/s); density kg/m³.

Prefixes: kilo 10³, centi 10⁻², milli 10⁻³, micro 10⁻⁶, nano 10⁻⁹, mega 10⁶, giga 10⁹. Examples: 1 km = 1000 m; 1 cm = 0.01 m; 1 quintal = 100 kg; 1 tonne = 1000 kg; 1 litre = 10⁻³ m³ = 1000 cm³.

Other units: astronomical unit (distance of the Earth from the Sun, about 1.5 × 10¹¹ m), light year (distance light travels in one year, about 9.46 × 10¹⁵ m), parsec (larger than a light year). The light year is a unit of distance, not time. Time measured by clocks, pendulum, stop watch; length by metre scale, vernier callipers and screw gauge.

Dimensions: length [L], mass [M], time [T]. Velocity [LT⁻¹], acceleration [LT⁻²], force [MLT⁻²]. Quantities can be added or subtracted only if they have the same dimensions.

Measuring instruments, least count and errors

Least count (LC) is the smallest value an instrument can measure.
Vernier callipers: LC = value of one main scale division − value of one vernier division = 1 MSD ÷ number of vernier divisions. If 1 MSD = 1 mm and there are 10 vernier divisions, LC = 0.1 mm = 0.01 cm. Used for the diameter of a sphere or inner and outer diameters.
Screw gauge: LC = pitch ÷ number of divisions on circular scale. Pitch is the distance moved per full rotation. For pitch 1 mm and 100 divisions, LC = 0.01 mm. Used for thin wires and sheets.
Zero error is the reading when jaws are closed; the correct reading = observed reading − zero error.
Significant figures: all non-zero digits, zeros between them and zeros after the decimal at the end are significant. 0.0045 has two; 4.500 has four. A result cannot be more accurate than the least precise measurement.
Accuracy is closeness to the true value; precision is closeness of repeated readings. Errors may be systematic, random or personal (human).
Mean value = sum of readings ÷ number of readings. Absolute error = |measured − true|; percentage error = (absolute error ÷ true value) × 100.

Distance and displacement; speed and velocity

Distance is the length of the actual path (scalar, always positive).
Displacement is the shortest straight line from the starting point to the final point, with direction (vector). It can be zero when the object returns to the start.
Speed = distance ÷ time. Average speed = total distance ÷ total time.
Velocity = displacement ÷ time. SI unit m/s. Convert: 1 km/h = 5/18 m/s; 36 km/h = 10 m/s.
Uniform motion: equal distances in equal times. Non-uniform motion: unequal distances in equal times.
Average speed when equal distance d is travelled at speeds v₁ and v₂: 2v₁v₂ ÷ (v₁ + v₂). For 30 km/h and 60 km/h: 2 × 30 × 60 ÷ 90 = 40 km/h (not 45).

Acceleration and equations of motion

Acceleration a = (v − u) ÷ t (rate of change of velocity). Unit m/s². Negative acceleration is called retardation (deceleration).

Equations (uniform acceleration along a straight line):

1. v = u + at
2. s = ut + ½at²
3. v² = u² + 2as
4. Distance in the nth second: s(n) = u + a(2n − 1)/2

Worked example 1: A car starts from rest and accelerates at 2 m/s² for 5 s. v = 0 + 2×5 = 10 m/s; s = ½×2×25 = 25 m.

Worked example 2: A car moving at 20 m/s stops with retardation 5 m/s². v² = u² + 2as gives 0 = 400 − 10s, so s = 40 m; time = 20 ÷ 5 = 4 s.

Free fall: a body falling under gravity alone has a = g = 9.8 m/s² (taken as 10 m/s² in simple problems), the same for all masses in the absence of air. Dropped from rest: v = gt, h = ½gt². A body thrown upward with speed u rises to height u²/2g in time u/g. Example: a stone dropped from rest falls for 2 s: v = 10 × 2 = 20 m/s; h = ½ × 10 × 4 = 20 m (with g = 10).

Scalars and vectors

Scalar (magnitude only)	Vector (magnitude and direction)
Distance, speed, mass, time, temperature, energy, work, density, pressure, power, electric current	Displacement, velocity, acceleration, force, momentum, weight, impulse

Vectors are added by the triangle law or parallelogram law: tail of one to the head of the other. Two vectors in the same direction add; in opposite directions subtract; at right angles, the resultant is √(A² + B²).
Example: forces of 3 N and 4 N at right angles: resultant = √(9+16) = 5 N.
The negative of a vector has the same size but opposite direction. A vector can be resolved into components (A cosθ, A sinθ).
Work, though made from two vectors, is a scalar.

Graphs of motion

Distance (or position)-time graph

Straight line through the origin means uniform speed; the slope = speed. A horizontal line means the body is at rest. A curve with increasing slope means increasing speed.

Velocity-time graph

Horizontal line: uniform velocity. Straight sloping line: uniform acceleration; slope = acceleration. Area under the graph = displacement (distance if no reversal).
Line sloping downward: retardation. Area of a trapezium or triangle gives distance. Example: velocity rises from 0 to 10 m/s in 5 s and then stays constant for 5 s: distance = ½×5×10 + 10×5 = 25 + 50 = 75 m.

Graph	Slope gives	Area gives
Position-time	Velocity	Nothing useful
Velocity-time	Acceleration	Displacement

Circular motion

Uniform circular motion: a body moves in a circle at constant speed. Velocity is tangent to the circle and changes in direction continuously, so there is acceleration even though speed is constant.
Time period T: time for one revolution; frequency f = 1/T (Hz); angular velocity ω = 2π/T = 2πf (rad/s); linear speed v = rω = 2πr/T.
Centripetal acceleration a = v²/r = ω²r, directed toward the centre. Centripetal force F = mv²/r, supplied by tension (stone on a string), gravity (satellites, planets), friction (car on a flat turn) or the normal reaction.
The centripetal force does no work because it is perpendicular to the motion. If the string breaks, the body moves along the tangent in a straight line (Newton's first law).
Centrifugal force is the apparent outward force felt in the rotating frame (it is not a real force acting in the ground frame). Applications of the centrifugal effect: washing machine dryer, cream separator. Banking of roads and turning of vehicles depend on centripetal force.
Example: a stone of mass 0.5 kg whirled in a circle of radius 2 m at speed 4 m/s: F = 0.5 × 16 ÷ 2 = 4 N.
Example: a wheel makes 50 revolutions in 10 s: f = 5 Hz, T = 0.2 s, ω = 10π rad/s.

Exam traps

Distance is a scalar and displacement is a vector; displacement can be zero, distance never.
Speed can be constant while velocity changes (circular motion).
Average speed is not the simple mean of two speeds when equal distances are covered.
A light year is a unit of distance, not time.
The slope of a velocity-time graph is acceleration; of a distance-time graph, speed.
Area under the v-t graph is displacement, not acceleration.
Centripetal acceleration points to the centre; centrifugal is apparent and outward.
Acceleration is zero in uniform straight-line motion but not in uniform circular motion.

One-liners

1. SI has seven base units.
2. The SI unit of force is newton: N = kg m/s².
3. 1 km/h = 5/18 m/s.
4. Acceleration due to gravity near Earth's surface is about 9.8 m/s².
5. Light year is a unit of distance.
6. Least count of a vernier = 1 MSD ÷ number of vernier divisions.
7. Least count of a screw gauge = pitch ÷ number of circular divisions.
8. Slope of a v-t graph gives acceleration.
9. Area under a v-t graph gives displacement.
10. Frequency = 1/time period.
11. Centripetal acceleration = v²/r.
12. In uniform circular motion, velocity changes but speed remains constant.

Practice questions

How many base units are there in the SI system?
1. 5
2. 6
3. 7
4. 9
Answer
C. 7
Metre, kilogram, second, ampere, kelvin, mole, candela.
The SI unit of electric current is
1. ohm
2. coulomb
3. volt
4. ampere
Answer
D. ampere
Ampere is the SI base unit of current.
The SI unit of temperature is
1. degree Celsius
2. kelvin
3. joule
4. degree Fahrenheit
Answer
B. kelvin
Kelvin is the base unit of temperature.
The SI unit of force, expressed in base units, is
1. kg m/s²
2. kg m²/s²
3. kg/m s²
4. kg m/s
Answer
A. kg m/s²
1 N = 1 kg m/s².
A light year is a unit of
1. speed
2. distance
3. mass
4. time
Answer
B. distance
It is the distance light travels in one year.
The prefix 'micro' stands for
1. 10⁻⁶
2. 10⁻³
3. 10⁶
4. 10⁻⁹
Answer
A. 10⁻⁶
Micro = 10⁻⁶; milli = 10⁻³; nano = 10⁻⁹.
A vernier callipers has 10 vernier divisions and 1 main scale division = 1 mm. Its least count is
1. 0.01 mm
2. 1 mm
3. 0.5 mm
4. 0.1 mm
Answer
D. 0.1 mm
LC = 1 MSD ÷ number of vernier divisions = 1 mm ÷ 10 = 0.1 mm.
A screw gauge has pitch 1 mm and 100 divisions on the circular scale. Its least count is
1. 1 mm
2. 0.1 mm
3. 0.01 mm
4. 0.001 mm
Answer
C. 0.01 mm
LC = pitch ÷ divisions = 1 ÷ 100 = 0.01 mm.
The number of significant figures in 0.0045 is
1. 2
2. 4
3. 3
4. 5
Answer
A. 2
Leading zeros are not significant; 4 and 5 are.
The correct reading of a vernier is obtained by
1. adding the zero error to every reading
2. multiplying by the zero error
3. subtracting the zero error from the observed reading
4. ignoring the zero error
Answer
C. subtracting the zero error from the observed reading
Correct reading = observed reading − zero error.
Which of the following is a vector quantity?
1. Mass
2. Displacement
3. Speed
4. Distance
Answer
B. Displacement
Displacement has both magnitude and direction.
Which of the following is a scalar quantity?
1. Momentum
2. Acceleration
3. Velocity
4. Speed
Answer
D. Speed
Speed has only magnitude.
A person walks once around a circular track and returns to the start. His displacement is
1. equal to the diameter
2. equal to the radius
3. zero
4. equal to the circumference
Answer
C. zero
Initial and final positions are the same.
Two forces of 3 N and 4 N act at right angles. The resultant is
1. 5 N
2. 12 N
3. 7 N
4. 1 N
Answer
A. 5 N
√(3² + 4²) = 5 N.
A car moves at 54 km/h. Its speed in m/s is
1. 54
2. 30
3. 5.4
4. 15
Answer
D. 15
54 × 5/18 = 15 m/s.
A train covers equal distances at 30 km/h and 60 km/h. The average speed for the whole journey is
1. 50 km/h
2. 40 km/h
3. 45 km/h
4. 30 km/h
Answer
B. 40 km/h
2v₁v₂/(v₁ + v₂) = 2 × 30 × 60 ÷ 90 = 40.
A car starting from rest accelerates at 2 m/s² for 5 s. Its speed is
1. 10 m/s
2. 25 m/s
3. 5 m/s
4. 2.5 m/s
Answer
A. 10 m/s
v = u + at = 0 + 2 × 5 = 10 m/s.
A car moving at 20 m/s is brought to rest with a retardation of 5 m/s². The stopping distance is
1. 80 m
2. 20 m
3. 4 m
4. 40 m
Answer
D. 40 m
v² = u² + 2as gives 0 = 400 − 10s, so s = 40 m.
A stone is dropped from rest. Taking g = 10 m/s², the distance fallen in 2 s is
1. 5 m
2. 10 m
3. 20 m
4. 40 m
Answer
C. 20 m
h = ½gt² = ½ × 10 × 4 = 20 m.
A ball is thrown upward at 20 m/s (g = 10 m/s²). The maximum height reached is
1. 200 m
2. 20 m
3. 40 m
4. 10 m
Answer
B. 20 m
h = u²/2g = 400 ÷ 20 = 20 m.
A body is dropped from a height of 45 m (g = 10 m/s²). The time taken to reach the ground is
1. 4.5 s
2. 9 s
3. 2 s
4. 3 s
Answer
D. 3 s
h = ½gt² gives 45 = 5t², so t = 3 s.
The slope of a velocity-time graph gives
1. displacement
2. speed
3. acceleration
4. force
Answer
C. acceleration
Slope = Δv/Δt = acceleration.
The area under a velocity-time graph gives
1. displacement
2. acceleration
3. force
4. time
Answer
A. displacement
Area = v × t = displacement.
A straight line through the origin on a distance-time graph indicates
1. retardation
2. uniform speed
3. rest
4. uniform acceleration
Answer
B. uniform speed
Equal distance in equal times; slope is constant speed.
A horizontal line on a distance-time graph shows that the body is
1. falling
2. at rest
3. moving with uniform speed
4. accelerating
Answer
B. at rest
Distance does not change with time.
A body accelerates uniformly from 0 to 10 m/s in 5 s and then moves at 10 m/s for 5 s. The total distance is
1. 50 m
2. 25 m
3. 100 m
4. 75 m
Answer
D. 75 m
Area = ½ × 5 × 10 + 10 × 5 = 25 + 50 = 75 m.
In uniform circular motion, which quantity changes continuously?
1. Velocity
2. Speed
3. Radius
4. Time period
Answer
A. Velocity
Direction of velocity keeps changing.
The direction of centripetal acceleration is
1. along the tangent
2. away from the centre
3. towards the centre of the circle
4. opposite to the velocity
Answer
C. towards the centre of the circle
It always points to the centre.
The centripetal acceleration of a body moving with speed v in a circle of radius r is
1. v/r²
2. vr
3. v²/r
4. r/v²
Answer
C. v²/r
a = v²/r = ω²r.
A stone of mass 0.5 kg is whirled in a circle of radius 2 m at 4 m/s. The tension in the string is
1. 1 N
2. 4 N
3. 8 N
4. 16 N
Answer
B. 4 N
F = mv²/r = 0.5 × 16 ÷ 2 = 4 N.
A wheel makes 50 revolutions in 10 s. Its frequency is
1. 500 Hz
2. 0.2 Hz
3. 50 Hz
4. 5 Hz
Answer
D. 5 Hz
f = 50 ÷ 10 = 5 Hz.
The time period of a body in circular motion with frequency 5 Hz is
1. 0.2 s
2. 2 s
3. 25 s
4. 5 s
Answer
A. 0.2 s
T = 1/f = 0.2 s.
If the string breaks when a stone is whirled in a circle, the stone moves
1. along the tangent in a straight line
2. towards the centre
3. in a smaller circle
4. vertically down only
Answer
A. along the tangent in a straight line
It continues with its velocity at that instant (first law).
The work done by the centripetal force on a body in uniform circular motion is
1. maximum
2. zero
3. negative
4. equal to mv²
Answer
B. zero
The force is perpendicular to displacement.
Consider the statements on motion. 1. Displacement can be zero for a moving body. 2. Distance can be zero for a moving body.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
A body that returns to the start has zero displacement; distance covered by a moving body is never zero.
Consider the statements on graphs of motion. 1. Slope of a distance-time graph gives acceleration. 2. Slope of a velocity-time graph gives acceleration.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
The slope of a distance-time graph gives speed.
Consider the statements on circular motion. 1. Speed is constant in uniform circular motion. 2. Acceleration is zero in uniform circular motion.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
There is a centripetal acceleration; statement 2 is wrong.
Consider the statements on vectors. 1. Work is a vector quantity. 2. Force is a vector quantity.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Work is a scalar.
Consider the statements on units. 1. 1 newton = 1 kg m/s². 2. A light year is a unit of time.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
A light year is a distance.
Consider the statements on free fall. 1. The acceleration of a freely falling body depends on its mass. 2. In vacuum all bodies fall with the same acceleration.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
Gravitational acceleration is independent of mass.
Consider the statements on instruments. 1. A screw gauge is suitable for measuring the thickness of a thin wire. 2. The least count of an instrument is the largest value it can measure.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Least count is the smallest value that can be measured.
Consider the statements on equations of motion. 1. v = u + at is true only for retardation. 2. v² = u² + 2as is used when time is not given.
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
B. 2 only
The equation holds for any uniform acceleration.
Match the quantity with its SI unit. P. Force Q. Pressure R. Work S. Frequency
1. P-hertz, Q-joule, R-pascal, S-newton
2. P-joule, Q-hertz, R-newton, S-pascal
3. P-pascal, Q-newton, R-hertz, S-joule
4. P-newton, Q-pascal, R-joule, S-hertz
Answer
D. P-newton, Q-pascal, R-joule, S-hertz
Standard SI units.
Match the graph feature with its meaning. P. Slope of v-t graph Q. Area under v-t graph R. Slope of s-t graph
1. P-velocity, Q-acceleration, R-displacement
2. P-displacement, Q-acceleration, R-velocity
3. P-acceleration, Q-displacement, R-velocity
4. P-acceleration, Q-velocity, R-displacement
Answer
C. P-acceleration, Q-displacement, R-velocity
The standard interpretations of motion graphs.
Match the quantity with its type. P. Speed Q. Velocity R. Mass S. Acceleration
1. P-vector, Q-vector, R-scalar, S-scalar
2. P-vector, Q-scalar, R-vector, S-scalar
3. P-scalar, Q-scalar, R-vector, S-vector
4. P-scalar, Q-vector, R-scalar, S-vector
Answer
D. P-scalar, Q-vector, R-scalar, S-vector
Speed and mass are scalars; velocity and acceleration are vectors.

Page 1 of 1

Chapter quiz · 10 questions

0/10 answered

On Physics - Measurement & Units, from the live question bank. 🔒 Answers and solutions are shown to registered students (free).
अध्याय पढ़ने के बाद अभ्यास करें।

1.The boiling and freezing points of water are exactly _____ degrees apart on the Fahrenheit scale.
2.The center of gravity of a semi-circle lies at a distance of __________ from its base measured along the vertical radius.
3.The crest of a wave is the point where the disturbance is at:
4.The critical angle for diamond is approximately _______.
5.The coordination number of atoms in a body-centered cubic (BCC) lattice is:
6.The atomic radius of a hydrogen atom is:
7.The atomic number of sulphur is-
8.The atomic number of phosphorus atom is:
9.The average temperature of human body is ______.
10.The force experienced by a current carrying conductor in a magnetic field is maximum when the angle between the direction of current and the direction of magnetic field is

10 left. You can check any time.