Units, measurement and motion basics for NDA

Units and motion are the entry point to NDA physics, and they pay in two ways. First, there are direct questions: which unit measures what, which instrument measures what, which two quantities share the same dimensions. Second, every later chapter uses the same units and the three equations of motion, so fluency here speeds up everything that follows.

The direct questions are pure recall, and they are easy marks for anyone who has written the tables below once and revised them a few times. The motion questions are one-step numericals with g taken as 10 m/s² unless the question says otherwise.

Typical question types are:

• the SI unit of a quantity, or the quantity a unit measures;
• the dimensional formula of a quantity, or which pair shares the same dimensions;
• conversions such as km/h to m/s, kWh to joules, °C to °F;
• the instrument used to measure a quantity;
• scalars versus vectors, distance versus displacement;
• one-step problems with v = u + at, s = ut + ½at² or v² = u² + 2as;
• free fall and a ball thrown straight up.

SI base units

The SI system has seven base quantities. Every other unit is built from them.

The radian (plane angle) and the steradian (solid angle), once called supplementary units, are also asked.

Derived units and dimensions

A dimensional formula shows how a quantity depends on mass [M], length [L] and time [T]. Speed is length ÷ time, so its dimensions are [L T⁻¹].

Pairs with the same dimensions are a favourite: work and torque; momentum and impulse; pressure, stress and Young's modulus; Planck's constant and angular momentum; surface tension and spring constant. Strain, angle and refractive index have no dimensions at all.

What dimensions are good for

• Checking an equation. Every term in a correct equation must have the same dimensions.
• Deriving a relation. Suppose the period T of a pendulum depends on its length l and on g. Write T ∝ lᵃgᵇ. Matching dimensions, [T] = [L]ᵃ[L T⁻²]ᵇ, so a + b = 0 and −2b = 1. That gives b = −½ and a = ½, so T ∝ √(l/g).
• Converting units from one system to another.

Their limits: dimensions cannot give a pure-number constant (the 2π in T = 2π√(l/g)), and they cannot handle sums of different terms or functions like sin and log.

Non-SI units worth knowing

Temperature: K = °C + 273.15, and °F = (9/5)°C + 32. The Celsius and Fahrenheit readings are equal at −40°.

Instruments

• Barometer: atmospheric pressure. Manometer: pressure of a gas.
• Hydrometer: density (relative density) of liquids. Lactometer: purity of milk.
• Hygrometer: relative humidity. Anemometer: wind speed.
• Altimeter: altitude. Fathometer: depth of the sea.
• Pyrometer: very high temperatures. Sphygmomanometer: blood pressure.
• Ammeter: current (in series). Voltmeter: potential difference (in parallel).
• Odometer: distance travelled by a vehicle. Speedometer: its speed. Tachometer: revolutions per minute.
• Seismograph: earthquake waves.

The least count of a Vernier calliper is 1 main scale division − 1 Vernier scale division. For a screw gauge, it is the pitch ÷ the number of circular scale divisions.

Motion basics

• Scalars have magnitude only: mass, speed, distance, time, energy, work. Vectors have magnitude and direction: displacement, velocity, acceleration, force, momentum.
• Distance is the path length; displacement is the straight-line change in position. Going halfway round a circle of radius r covers a distance πr but a displacement of 2r. After a full round, the displacement is zero.
• Speed = distance/time; velocity = displacement/time; acceleration = change in velocity/time.
• Graphs: the slope of a position–time graph is velocity; the slope of a velocity–time graph is acceleration; the area under a velocity–time graph is displacement.

Equations of uniformly accelerated motion:

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

Free fall. Without air resistance, all bodies fall with the same acceleration g (about 9.8 m/s², often taken as 10). A body dropped from height h lands after √(2h/g) seconds at speed √(2gh). A ball thrown up at speed u rises to u²/(2g) in u/g seconds, and returns to the hand at the same speed.

Worked NDA-style MCQs

Q1. The dimensional formula of pressure is:
(a) [M L T⁻²] (b) [M L⁻² T⁻²] (c) [M L² T⁻²] (d) [M L⁻¹ T⁻²]

Pressure = force/area = [M L T⁻²]/[L²] = [M L⁻¹ T⁻²]. Answer: (d).

Q2. Which pair has the same dimensions?
(a) work and power (b) momentum and force (c) work and torque (d) stress and strain

Both work and torque are force × distance, [M L² T⁻²]. Strain is dimensionless, so (d) fails. Answer: (c).

Q3. A body starts from rest with an acceleration of 2 m/s². The distance it covers in the 5th second is:
(a) 9 m (b) 10 m (c) 25 m (d) 5 m

u + (a/2)(2n − 1) = 0 + 1 × 9 = 9 m. Option (c) is the total distance in 5 seconds, ½ × 2 × 5². Answer: (a).

Q4. A stone is dropped from a height of 20 m (g = 10 m/s²). It reaches the ground after:
(a) 1 s (b) √2 s (c) 2 s (d) 4 s

t = √(2h/g) = √(40/10) = 2 s. It lands at 20 m/s. Answer: (c).

Q5. A car moving at 72 km/h is brought to rest uniformly in 4 s. The distance it covers while braking is:
(a) 20 m (b) 40 m (c) 80 m (d) 144 m

72 km/h = 72 × 5/18 = 20 m/s. The deceleration is 20/4 = 5 m/s², so s = v²/(2a) = 400/10 = 40 m. Answer: (b).

Q6. A person walks halfway round a circular track of radius 7 m. Taking π = 22/7, the distance and displacement are:
(a) 22 m and 14 m (b) 44 m and 0 (c) 14 m and 22 m (d) 22 m and 7 m

Distance = πr = 22 m; displacement = the diameter = 14 m. Answer: (a).

Common mistakes

• Calling a light year a unit of time, or a kWh a unit of power.
• Forgetting to convert km/h to m/s (multiply by 5/18) before using the equations.
• Mixing up the distance in t seconds with the distance in the tth second.
• Treating displacement like distance. Displacement can be zero after a long walk.
• Assuming heavier bodies fall faster. In the absence of air resistance, they fall together.

Practice set

1. A light year is a unit of: (a) time (b) distance (c) speed (d) intensity of light
2. Planck's constant has the same dimensions as: (a) energy (b) force (c) power (d) angular momentum
3. 36 km/h expressed in m/s is: (a) 3.6 (b) 10 (c) 36 (d) 100
4. Atmospheric pressure is measured with a: (a) hygrometer (b) hydrometer (c) barometer (d) manometer
5. A ball is thrown straight up at 30 m/s (g = 10 m/s²). Its maximum height is: (a) 45 m (b) 30 m (c) 90 m (d) 60 m
6. A screw gauge has a pitch of 1 mm and 100 circular divisions. Its least count is: (a) 0.1 mm (b) 0.01 mm (c) 0.001 mm (d) 1 mm
7. The area under a velocity–time graph gives: (a) acceleration (b) force (c) speed (d) displacement
8. 1 kWh equals: (a) 3.6 × 10³ J (b) 3.6 × 10⁵ J (c) 3.6 × 10⁶ J (d) 1,000 J

Answers:

1. (b). About 9.46 × 10¹⁵ m, the distance light travels in a year.
2. (d). Both are [M L² T⁻¹].
3. (b). 36 × 5/18 = 10.
4. (c). A manometer measures the pressure of an enclosed gas.
5. (a). u²/(2g) = 900/20 = 45 m, reached after 3 s.
6. (b). 1 mm ÷ 100.
7. (d). The slope gives acceleration; the area gives displacement.
8. (c). 1,000 W × 3,600 s.

What to do next

• Write the derived-units table from memory with dimensions, then list the same-dimension pairs.
• Make a two-column instrument list (instrument, what it measures) and test yourself until it is automatic.
• Solve 20 one-step motion problems with g = 10, converting every speed to m/s first.
• Continue with laws of motion and momentum, and see the NDA physics plan for the order of chapters.