NJC 2025 H2 Physics Quantities and Measurement Problem Set

National Junior College Science Department | Physics 1. Quantities & Measurement Problem Set PHYSICAL QUANTITIES AND SI UNITS Guided Exercises For this section of the problem set, you should try the exercises without looking at the guided solutions. The guided solutions are there if you get stuck and you should also use it to understand how to present your work. 1 Perform the following conversion of units: (i) 50 km h-1 to m s-1. [13.9 m s-1] (ii) 320 m s-1 to km h-1. [1152 km h-1] Step-by-step guide Convert km to m and h to min 50 km h-1 = 50 km 1 h = 50 000 𝑚 60 min = 50 000 𝑚 60 ×60 𝑠 = 13.9 m s-1 Convert min to s 320 m s-1 = 320 m 1 s = 320 1000 km 1 60×60 h = 1152 km h-1

National Junior College Science Department | Physics 2 Express the units of the following quantities in SI base units: (i) Gravitational constant G, where 𝐹 = 𝐺𝑀1𝑀2 𝑟2 , 𝐹 is force, 𝑀1 and 𝑀2 are masses and 𝑟 is separation distance between the masses. [m3 kg-1 s-2] (ii)  , if 𝑣 = √ 𝑇 𝜇 where 𝑣 is the speed of the wave on a string of length L held at a tension of T, a type of force. [kg m-1] Look at Examples 1.1 and 1.2 in your lecture notes for a step-by-step guide, it will be useful to rearrange the equations to write G and  in terms of other quantities before determining the units. Please also be mindful of the presentation of your working! (i) 𝐺 = 𝐹𝑟2 𝑀1𝑀2 and 𝐹 = 𝑚𝑎 Units of G = (kg m s−2)(m2) (kg)(kg) = m3 kg-1 s-2 (ii) 𝑣2 = 𝑇 𝜇  𝜇 = 𝑇 𝑣2 and 𝑇 = 𝑚𝑎 Units of 𝜇 = kg m s−2 (m s−1)2 = kg m s−2 m2 s−2 = kg m-1 Presentation style: Always write “units of” in front of the variable you are finding units for. E.g 𝜇 = kg m s−2 (m s−1)2 would be marked wrong for presentation.

National Junior College Science Department | Physics 3 The critical flow speed vc is related to the width a of the obstacle, the density 𝜌 liquid, and its viscosity 𝜂 by one of the equations: (i) vc = 𝐴𝜂𝑎 𝜌 (ii) vc = 𝐵𝜂 𝑎𝜌 (iii) vc = 𝐶𝜌𝑎 𝜂 A, B and C are constants with no units and unit of 𝜂 is Pa s. Determine the correct equation. You are given 3 equations. A dimensionally correct equation is one where the units of each term has the same base units. (pg 8 of your lecture notes) Note that a dimensionally correct equation need not be correct! Hence you can’t actually determine the correct equation, just the dimensionally correct one! Step-by-step guide using (i) Unit of vc = m s-1 Unit of 𝐴𝜂𝑎 𝜌 = 𝑃𝑎 𝑠 (𝑚) (𝑘𝑔 𝑚−3) = (𝑘𝑔𝑚𝑠−2 𝑚2 ) 𝑠 (𝑚) (𝑘𝑔 𝑚−3) = 𝑠−1 (𝑚−3) = 𝑚3𝑠−1 LHS ≠ RHS, (i) is not dimensionally correct Option (ii) is the dimensionally correct equation. 4 Bernoulli’s equation, which applies to fluid flow, states that 𝒑 + ℎ𝜌𝑔 + 1 2 𝜌𝑣2 = 𝑘 Where 𝒑 is a pressure, h a height, 𝜌 a density, g an acceleration, v a velocity and k a constant. Show that the equation is homogenous and state an SI base unit for k. [kg m-1 s-2] Note: An equation where each term has the same base units (dimensionally consistent) is known as a homogeneous equation. To determine whether an equation is homogenous, you will need to check whether all the terms in the equation have the same base units. Hence you will need to check whether units of p = units of ℎ𝜌𝑔 = units of 1 2 𝜌𝑣2 and the units for k will be the same. Units of pressure = N m-2 = kg m s-2 m-2 = kg m-1 s-2 Units of ℎ𝜌𝑔 = m kg m-3 m s-2 = kg m-1 s-2 Units of 1 2 𝜌𝑣2 = kg m-3 (m s-1)2 = kg m-1 s-2 Since units of 𝑝 = units of ℎ𝜌𝑔 = units of 1 2 𝜌𝑣2 equation is homogenous if units of 𝑘 = kg m-1 s-2 To determine base unit of Pa Pa is the unit of pressure. We have learnt that pressure is Force Area and force is mass × acceleration Therefore, 𝑢𝑛𝑖𝑡𝑠 𝑜𝑓 𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 = 𝑢𝑛𝑖𝑡𝑠 𝑜𝑓 𝑚𝑎𝑠𝑠 × 𝑢𝑛𝑖𝑡𝑠 𝑜𝑓 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑢𝑛𝑖𝑡𝑠 𝑜𝑓 𝑎𝑟𝑒𝑎 𝑃𝑎 = 𝑘𝑔𝑚𝑠−2 𝑚2

National Junior College Science Department | Physics 5 The activity A of a radioactive sample is given by the equation, 𝐴 = 𝐴𝑜𝑒−𝜆𝑡 where Ao is the initial activity of the sample, 𝜆 is the decay constant and t is the time taken for the sample to decay. Determine the unit of 𝜆. Hint: you do not need to know the units of A. [s-1] All exponents are dimensionless. The unit of 𝐴 = unit of 𝐴𝑜and −𝜆𝑡 has no units. Units of (−𝜆𝑡) = units of 𝜆 × units of t 1 = units of 𝜆 × s Units of 𝜆 = 𝑠−1 6 Express the following quantities using the required prefixes: (i) 800,000 W with prefix kW, MW and TW; [800 kW, 0.8 MW, 8 x 10-7 TW] (ii) 0.0000325 m with prefix mm and μm; [ 0.0325 mm, 32.5 μm] (iii) 2.65 x 10-10 s with prefix ps [265 ps] Common prefixes required for A levels can be found in Pg 7 of your lecture notes. (i) 800,000 W = 800 × (103 W) =800 kW 800,000 W = 800,000 × 1×106 1×106 W = 800,000 1×106 × (1 × 106 W) = 0.8 MW 800,000 W = 800,000 × 1×1012 1×1012 = 800,000 1×1012 × (1 × 1012 W) = 8 × 10−7 TW (ii) 0.0000325 m = 0.0000325 x 1 m = 0.0000325 x 1000 mm = 0.0325 mm 0.0000325 m = 0.0000325 x 1 m = 0.0000325 x 1 x 106 m = 32.5 μm (iii) 2.65 x 10-10 x 1 second = 2.65 x 10-10 x 1 x 1012 ps = 265 ps

National Junior College Science Department | Physics Problems P1 The behaviour of many real gases can be represented quite closely over certain ranges of temperature and pressure by an equation of the form (𝑝 + 𝑎 𝑉2𝑚 )(𝑉𝑚 − 𝑏) = 𝑅𝑇 in which the values of a and b are characteristics constants of the particular gas. p is the pressure of gas, Vm is the volume of gas per mole and T is the thermodynamic temperature of gas. Express the units of a, b and R in SI base units. P2 Express the unit of 𝜀𝑜the permittivity of vacuum, in terms of base units from the equation 𝐹 = 𝑄1𝑄2 4𝜋𝜀𝑜𝑟2 where Q1 and Q2 are the charge of two particles separated by a distance r. The unit of𝜇𝑜, the permeability of a vacuum, is kg m s-2 A-2. Use the unit of 𝜇𝑜, and unit of 𝜀𝑜, to decide which one of the following relations between𝜀𝑜, 𝜇𝑜 and c, the speed of light in a vacuum, is possible. 𝜀𝑜𝜇𝑜 = 𝑐2 ; 𝜀𝑜𝜇𝑜 = 𝑐 ; 𝜀𝑜𝜇𝑜 = 𝑐−1 ; 𝜀𝑜𝜇𝑜 = 𝑐−2 P3 Suggest how to check whether a hypothetical equation is possible and explain why your suggested method of checking does not give confirmation that an equation is physically correct. P4 N13/1/2 What is the order-of-magnitude of the mass of a single sheet of A4 paper? A 0.01 g B 0.3 g C 1 g D 30 g P5 N08/1/2 Which estimate is realistic? A The kinetic energy of a bus travelling on an expressway is 30 000 J. B The power of a domestic light is 300 W C The temperature of a hot oven is 300 K. D The volume of air in a car tyre is 0.03 m3.

National Junior College Science Department | Physics Errors and uncertainties Exercises 1 N represents the number of data collected, x represents the possible values of the results and xo represents the actual value. Comment on the accuracy and precision of the data shown on each graph. Refer to Pg 14 of your lecture notes Accurate and precise Accurate but not precise Not accurate but precise Not accurate and not precise 2 Four students each made a series of measurements of the acceleration of free fall g, where g = 9.81 m s-2. The table shows the results obtained. Which student obtained a set of results that could be described as precise but not accurate? Student Results, g / m s-2 Average Spread (max – min) Precise Accurate A 9.81 9.79 9.84 9.83 9.82 B 9.81 10.12 9.89 8.94 9.69 C 9.45 9.21 8.99 8.76 9.10 D 8.45 8.46 8.5