Word equation Symbol equation Units 1 2 kinetic energy, Ek ... · A heavy-duty transformer delivers...

Word equation Symbol equation Units

𝑘𝑖𝑛𝑒𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 = 0.5 × 𝑚𝑎𝑠𝑠 × 𝑠𝑝𝑒𝑒𝑑2

𝐸𝐾 =1

2𝑚𝑣2

or 𝐸𝐾 =𝑚𝑣2

2

kinetic energy, Ek, in joules, J mass, m, in kilograms, kg speed, v, in metres per second, m/s

Example 1 What is the kinetic energy of a car of mass 1500 kg travelling at 31 m/s?

Example 2 A BB gun is a type of air gun that fires plastic pellets of mass 0.25 g with a kinetic energy of 1.3 J. What is the velocity of the pellet as it leaves the gun?

Practice 1 In an experiment in the laboratory, a trolley of mass 1.4 kg is given a push and then goes through a light gate which records a speed of 2.3 m/s. What is the kinetic energy of the trolley?

Practice 2 A model electric car has a kinetic energy of 27 J when travelling at a speed of 3.2 m/s. What speed will the same car need to travel at for it to have a kinetic energy of 100 J?


𝑒𝑙𝑎𝑠𝑡𝑖𝑐 𝑒𝑛𝑒𝑟𝑔𝑦 = 0.5× 𝑠𝑝𝑟𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 × 𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛2

𝐸𝑒 =1

2𝑘𝑒2

elastic potential energy, Ee, in joules, J spring constant, k, in newtons per metre, N/m extension, e, in metres, m

Example 1 A spring has an extension of 13 cm when stretched by a force of 9.0 N. What will be the elastic energy stored in the spring when stretched to an extension of 18 cm?

Example 2 The spring in a new toy car track system is required to store an energy of 5.0 J. The spring has a spring constant of 180 N/m. By how much will the spring need to be compressed?

Practice 1 What is the increase in the elastic energy stored in a spring of spring constant 150 N/m when the extension increases from 15 cm to 25 cm (provided the elastic limit has not been exceeded)?

Practice 2 The spring in a toy is required to launch a small ball of mass 28 g to a height of 1.0 m. The spring will be compressed by 3.0 cm. What spring constant must the spring have? (the value of g is 10 m/s2)


𝑔𝑝𝑒= 𝑚𝑎𝑠𝑠 × 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑖𝑒𝑙𝑑 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ× ℎ𝑒𝑖𝑔ℎ𝑡

𝐸𝑝 = 𝑚𝑔ℎ

gravitational potential energy, Ep, in joules, J mass, m, in kilograms, kg gravitational field strength, g, in newtons per kilogram, N/kg height, h, in metres, m

Example 1 What is the increase in the gravitational potential energy of a 90 kg astronaut when they climb 3.7 m up the ladder into their moon lander? (The value of g on the Moon is 1.6 N/kg)

Example 2 The mass in a grandfather clock can be lifted 1.1 m and must provide energy to the clock mechanism of at least 35 J. What is the minimum value of mass needed?

Practice 1 A climber of mass 78 kg is carrying 8 kg of food, water and equipment. What will be his increase in gravitational potential energy when he climbs from Fort William the 1345 m to the summit of Ben Nevis? (In Scotland the gravitational field strength is 9.8 N/kg)

Practice 2 A ski-jumper of mass 72 kg wants to have a take-off speed of 28 m/s. Assuming that air resistance and friction are negligible, calculate the height of his starting point above the take-off. (g = 9.8 N/kg).


Change in thermal energy = mass × specific heat capacity × temperature

change ∆𝐸 = 𝑚𝑐∆𝜃

change in thermal energy, ΔE, in joules, J mass, m, in kilograms, kg specific heat capacity, c, in joules per kilogram per degree Celsius,J/kg °C temperature change, Δθ, in degrees Celsius, °C

Example 1

What is the energy required to raise the temperature of 2.5 kg of iron from 20 C to the melting point?

(Data for iron: specific heat capacity 440 J/kg C, melting point 1540 C)

Example 2

An electric kettle of power 2700 W is switched on for 60 s. The kettle is filled with 1.2 kg of water at 18 C.

(c = 4200 J/kg C) What will be the temperature of the water after the 60 s?

Practice 1

What is the energy required to raise the temperature of 5.2 kg of aluminium from 20 C to the melting point?

(Data for aluminium: specific heat capacity 900 J/kg C, melting point 660 C)

Practice 2 In an experiment to determine the specific heat capacity of copper the following measurements were made. What value of specific heat capacity does this data produce? Mass of copper: 1.05 kg

Start temperature: 18.5 C

Finish temperature: 42.0 C Energy supplied: 10.5 kJ


𝑝𝑜𝑤𝑒𝑟 =𝑒𝑛𝑒𝑟𝑔𝑦 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟𝑟𝑒𝑑

𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑃 =

∆𝐸

𝑡

power, P, in watts, W energy transferred, ΔE, in joules, J time, t, in seconds, s

Example 1 What is the power output of an electric motor that can provide 250 kJ in 2 minutes?

Example 2 The electric motor in a model crane has a power output of 480 mW. How long will it take to lift a load of 450 g through a height of 90 cm? (g = 10 N/kg)

Practice 1 What is the power output of a boy of mass 52 kg who can run up a flight of stairs of height 3.7 m in 2.4 s? (g = 10 N/kg)

Practice 2 An electric kettle has a power output of 2.7 kW. How long will it take to raise the temperature of 1.3 kg of

water from 20 C to 100 C?

(specific heat capacity of water is 4200 J/kgC)


𝑝𝑜𝑤𝑒𝑟 =𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒

𝑡𝑖𝑚𝑒 𝑡𝑎𝑘𝑒𝑛 𝑃 =

𝑊

𝑡

power, P, in watts, W time, t, in seconds, s work done, W, in joules, J

Example 1 It takes 1.5 hours for a machine to transfer 4.2 MJ of energy. What is the power output of the machine?

Example 2 A car engine has an output power of 28 kW when the car is being driven at a constant 29 m/s. What is the total work done against friction and drag forces when the car is driven 250 km?

Practice 1 The tension in a winch cable is 4500 N when it drags a car through the mud at a steady 0.6 m/s?

Practice 2 Mains electricity is supplied at a cost of 25 p per unit (1 kWh or 3.6 MJ). How much will it cost to run a 3.0 W LED lamp for 4 hours a day for a year?


𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =𝑢𝑠𝑒𝑓𝑢𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟

𝑡𝑜𝑡𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦 𝑖𝑛𝑝𝑢𝑡 𝜀 =

∆𝐸𝑜𝑢𝑡

∆𝐸𝑖𝑛

Energy transfer, ΔE, in joules, J Efficiency, ε, as decimal or percentage

Example 1 A capacitor is a device that can store energy electrically. When charged by connecting to a battery the maximum energy that can be delivered by the capacitor is half that transferred from the battery. What is the efficiency of the capacitor?

Example 2 The turbine and generator system in a power station has an efficiency of 40%. In a particular period the power station produces a useful electrical energy transfer of 70 MWh, with 5 MWh wasted as a thermal energy transfer to the surroundings in the transformers and cables. 10 MWh of the stored energy in the fuel is not transferred to the turbine system due to thermal transfer to the atmosphere. Sketch a Sankey diagram for these energy transfers and hence calculate the total energy input to the power station from the chemical store in the fuel.

Practice 1 The Sankey diagram shown is for a particular type of wave energy generator. What is the efficiency of this system?

Practice 2 What is the energy wasted by a lamp with an efficiency of 35% when the total energy input is 750 J?


𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =𝑢𝑠𝑒𝑓𝑢𝑙 𝑝𝑜𝑤𝑒𝑟 𝑜𝑢𝑡𝑝𝑢𝑡

𝑡𝑜𝑡𝑎𝑙 𝑝𝑜𝑤𝑒𝑟 𝑖𝑛𝑝𝑢𝑡 𝜀 =

𝑃𝑜𝑢𝑡

𝑃𝑖𝑛

Power, P, in watts, W Efficiency, ε, as decimal or percentage

Example 1 The label shown to the right is fixed to a microwave oven. Assuming that the data shown is accurate, what is the efficiency of the oven at producing microwaves?

Example 2 A petrol engine provides an output power of 55 kW and has an efficiency of 38%. Petrol has an energy content of 46.4 MJ/kg. What will be the fuel consumption of this engine in kg/hour?

Practice 1 A heavy-duty transformer delivers an output current of 50 A at a potential difference of 12 V when the input potential difference and current are 230 V and 2.5 A respectively. What is the efficiency of the transformer?

Practice 2 The label shown is for a boiler with an efficiency of 88%. What is the useful output power in watts when operating on natural gas?


𝐶ℎ𝑎𝑟𝑔𝑒 𝑚𝑜𝑣𝑒𝑑 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 × 𝑡𝑖𝑚𝑒 𝑄 = 𝐼𝑡 charge flow, Q, in coulombs, C current, I, in amperes (amp), A time, t, in seconds, s

Example 1 A 1.2 V rechargeable cell can deliver a constant current of 0.50 A for 5 hours. What is the charge moved around the circuit in this time?

Example 2 Electrons each have a charge of 1.6 × 10-19 C. What is the current in a wire when a mole of electrons (6.02 × 1023 electrons) moves past a point in the wire in one hour?

Practice 1 The lights on a car have a total current of 11 A. What is the total charge moved by the car battery when the lights are left on for 8 hours?

Practice 2 Electron-beam deposition is a method of coating surfaces with thin metal layers. A beam of electrons strikes an anode causing the metal contained in it to be vaporised. How many electrons hit the anode in each 20 s operating cycle when the beam current is 2.5 A? (Electrons each have a charge of 1.6 × 10-19 C)


𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒= 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 × 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑉 = 𝐼𝑅 potential difference, V, in volts, V current, I, in amperes, A resistance, R, in ohms, Ω

Example 1 What is the potential difference across a 33 Ω resistor if the current through it is 0.45 A?

Example 2 The insulation around the electrical wiring is tested by applying a potential difference of 500 V across it. The current through the material is measured to be 0.25 mA. What is the resistance of the insulator?

Practice 1 A resistor has a stated value of 10 Ω ± 10%. What could be the maximum current if this resistor was connected across a 12 V battery?

Practice 2 In an experiment to determine the resistance of a resistor the following data was obtained:

Potential Difference (V) Current (A) 3.80 0.240 7.52 0.512 9.31 0.605

Use this data to calculate a mean value for the resistance. Give your answer to 3 significant figures.


𝑇𝑜𝑡𝑎𝑙 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒= 𝑠𝑢𝑚 𝑜𝑓 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒𝑠 𝑖𝑛 𝑠𝑒𝑟𝑖𝑒𝑠

𝑅𝑡𝑜𝑡𝑎𝑙 = 𝑅1 + 𝑅2 resistance, R, in ohms, Ω

Example 1 The resistance of a human body between hands and feet is 5.0 kΩ. Shoes have a resistance of 20 kΩ. What is the total resistance between the hands and the floor?

Example 2 Two resistors of value 22 Ω are connected in series with a third resistor. If the total resistance is 59 Ω, what is the value of the third resistor?

Practice 1 An engineer has the following resistors in their box of spare resistors. What is the maximum resistance they can make by connecting any two of these in series? 68 Ω, 12 kΩ, 33 kΩ, 470 Ω

Practice 2 A student has a number of 330 Ω resistors and a 12 V power supply. How many of these resistors, connected in series will give a current closest to 10 mA? Show how you obtained your answer.


𝑃𝑜𝑤𝑒𝑟 = 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒× 𝑐𝑢𝑟𝑟𝑒𝑛𝑡

𝑃 = 𝑉𝐼 power, P, in watts, W potential difference, V, in volts, V current, I, in amperes, A

Example 1 What is the total power output of a bulb rated at 12 V, 0.30 A?

Example 2 What is the current through a 2.5 kW kettle when connected to UK mains electrical supply?

Practice 1 What is the maximum power device that can be connected to a car power socket that has a potential difference from the battery of 12 V and is fitted with a 15 A fuse?

Practice 2 The output power of a power station is 400 MW at 25 kV. What is the current in the cables to the National Grid connection?


𝑃𝑜𝑤𝑒𝑟 = 𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 × 𝑟𝑒𝑠𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑃 = 𝐼2𝑅 power, P, in watts, W current, I, in amperes, A resistance, R, in ohms, Ω

Example 1 What is the power transferred thermally to the surroundings from a cable in the National Grid that has a resistance of 0.12 Ω and is carrying a current of 230 A?

Example 2 A heating element has a constant resistance. When the current through it is 1.2 A the power transferred is 48 W. What will be the power transferred when the current is increased to 5.0 A?

Practice 1 A resistor has a resistance of 15 Ω and a maximum power rating of 25 W. What is the maximum current that this resistor can carry?

Practice 2 What is the resistance of a heater rated at 2.0 kW when connected to UK mains?


𝐸𝑛𝑒𝑟𝑔𝑦 𝑡𝑟𝑎𝑛𝑠𝑓𝑒𝑟= 𝑐ℎ𝑎𝑟𝑔𝑒 𝑚𝑜𝑣𝑒𝑑× 𝑝𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒

∆𝐸 = 𝑄𝑉 energy transferred, ΔE, in joules, J charge flow, Q, in coulombs, C potential difference, V, in volts, V

Example 1 What is the energy transferred to an electron (Q = 1.6 × 10-19 C) when it is accelerated by a potential difference of 120 kV?

Example 2 A Van de Graaff generator produces a spark of energy 50 J when the potential difference reaches 100 000 V. What is the charge moved by the spark?

Practice 1 What is the energy transferred by a potential difference of 120 V when it moves a charge of 45 mC?

Practice 2 What is the energy transferred by an electron beam produced by a potential difference of 4800 V with a beam current of 12 mA when it is switched on for 5.0 minutes?


𝑑𝑒𝑛𝑠𝑖𝑡𝑦 =𝑚𝑎𝑠𝑠

𝑣𝑜𝑙𝑢𝑚𝑒 𝜌 =

𝑚

𝑉

density, ρ, in kilograms per metre cubed, kg/m3

mass, m, in kilograms, kg volume, V, in metres cubed, m3

Example 1 A measuring cylinder is filled with water to the 20 cm3 mark. When a rock of mass 130 g is lower into the measuring cylinder the water level rises to the 70 cm3 mark. What is the density of the rock?

Example 2 A manufacturer plans to make a set of masses out of aluminium where each mass is a cylinder of thickness 4.0 cm. What will need to be the diameter of the 500 g mass? (Density of aluminium = 2710 kg/m3)

Practice 1 A cuboid metal block has dimensions 1.0 cm by 2.0 cm by 4.0 cm and a mass of 150 g. What is the density of the metal used to make the block?

Practice 2 A crane manufacturer wants to use concrete to make the balancing weights for a mobile crane. The weights must each have a mass of 4.5 tonnes and are made from concrete of density 2400 kg/m3. The base dimensions of a block will be 1.0 m by 2.5 m. What will be the height of each block?


𝑒𝑛𝑒𝑟𝑔𝑦 (𝑓𝑜𝑟 𝑎 𝑐ℎ𝑎𝑛𝑔𝑒 𝑜𝑓 𝑠𝑡𝑎𝑡𝑒)= 𝑚𝑎𝑠𝑠× 𝑠𝑝𝑒𝑐𝑖𝑓𝑖𝑐 𝑙𝑎𝑡𝑒𝑛𝑡 ℎ𝑒𝑎𝑡

∆𝐸 = 𝑚𝐿

energy, ΔE, in joules, J mass, m, in kilograms, kg specific latent heat, L, in joules per kilogram, J/kg

Example 1 What is the energy required to boil away 150 g of ethanol? (LV for ethanol = 8.5 ×105 J/kg)

Example 2 In an experiment the specific latent heat of vaporisation for water an electric kettle of power 2.7 kW was modified so that it does not turn off when the water boils. The mass of the kettle and water was determined using an electronic balance. A timer was started when the water started to boil and the kettle was turned off when the timer reached 2 min 30 s. The kettle was then re-weighed. If the mass of the kettle and water reduced from 1.262 kg to 1.096 kg, what value for the specific latent heat of vaporisation of water does this experiment give?

Practice 1 Water can be purified using a simple distillation process. Water is boiled and the vapour is condensed and collected. What will be the minimum daily energy input to this system to produce a daily water requirement of 3.0 kg? (Specific latent heat of vaporisation for water = 2.3 ×106 J/kg)

Practice 2 A container of water is just at the boiling point of water when an iron block of mass 1.5 kg at a

temperature of750 Cis plunged into the water. What mass of water boils away as a result of this?

(Specific heat capacity of iron = 480 J/kgC, specific latent heat of vaporisation of water = 2.3 ×106 J/kg)

Word equation – Physics only Symbol equation Units

𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 × 𝑣𝑜𝑙𝑢𝑚𝑒 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑝𝑉 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

𝑝1𝑉1 = 𝑝2𝑉2

pressure, p, in pascals, Pa volume, V, in metres cubed, m3

Example 1 The volume of a sealed helium balloon at sea-level (p = 1.01 ×105 Pa) is 1.0 m3. What will be the volume of the balloon at a height of 10 km where the atmospheric pressure has fallen to 2.6 ×104 Pa (The pressure exerted by the balloon fabric is negligible)

Example 2 A pump consists of a cylinder of diameter 2.6 cm with a moveable piston enclosing a length of air of 0.35 m. The outlet of the pump is sealed. By what distance must the piston be pushed inwards in order to quadruple the pressure in the cylinder?

Practice 1 A pump is used to pump up a bicycle tyre. The pressure inside the tyre and pump is initially 1.01 ×105 Pa with a combined volume of 0.0028 m3. When the pump handle is pushed in the volume of air in the tyre and pump reduces by 0.0002 m3. What will be the new pressure in the tyre?

Practice 2 A new cylinder of “helium” gas for party balloons contains 50 dm3 of a mixture of helium and nitrogen at a pressure of 2.5 ×107 Pa. Each balloon filled has a volume of 0.025 m3 and initially contains this “helium” gas at a pressure of 1.2 ×105 Pa. How many balloons can be filled from this cylinder?


𝑓𝑟𝑎𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑐𝑡𝑖𝑣𝑖𝑡𝑦 = (1

2)

𝑛

𝐴

𝐴0= 0.5𝑛

activity, A, in Becquerel, Bq initial activity, A0, in Becquerel, Bq number of half-lives, n

Example 1 What will be the percentage of activity of a particular radioactive source remaining after 8 half-lives?

Example 2 A sample of neon-24 emits beta particles at a rate of 650 Bq. The half-life of neon-24 is 3.4 minutes. What will be the activity of the source 10 minutes later?

Practice 1 A source has an initial activity of 200 Bq. What will be the activity of this source 5 half-lives later?

Practice 2 A radioactive source is considered safe to dispose of in normal refuse if the count rate from it is less than half that due to the background radiation. A scientist measures the background count and the count with the source, which has a half-life of one week. Will the source be safe to dispose of four weeks after the initial measurement? Show how you made your decision. Background count rate = 1.5 Count/s, With source count rate = 9.5 Count/s


𝑤𝑒𝑖𝑔ℎ𝑡= 𝑚𝑎𝑠𝑠 × 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑖𝑒𝑙𝑑 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ 𝑊 = 𝑚𝑔

weight, W, in newtons, N mass, m, in kilograms, kg gravitational field strength, g, in newtons per kilogram, N/kg

Example 1 What is the weight of a 95 kg astronaut on the Moon (where g = 1.6 N/kg)?

Example 2 A very sensitive spring balance can be used to find differences in gravitational field strength by measuring the force on a known mass of 5.000 kg. At most points across a particular building site the force measured is 49.050 N but at one measuring point the force drops by 6 × 10-6 N. What is the decrease in gravitational field strength at this point (due to a large cave just below the surface!)?

Practice 1 What is the weight of a 1300 kg electric sports-car in Earth orbit where the gravitational field strength is 8.7 N/kg?

Practice 2 The thrust from the engines of a particular rocket is 1.44 ×107 N. If the mass of the rocket at launch is 1.42 ×106 kg, what is the maximum payload (mass of the satellite to be put into orbit)? (At the Earth’s surface, g = 9.81 N/kg)


𝑤𝑜𝑟𝑘 𝑑𝑜𝑛𝑒 = 𝑓𝑜𝑟𝑐𝑒 × 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑚𝑜𝑣𝑒𝑑 𝑊 = 𝐹𝑠

work done, W, in joules, J force, F, in newtons, N distance, s, in metres, m (distance and force must be in the same direction)

Example 1 What is the work done by the engine of a car that drives the car at a constant velocity a distance of 1.5 km against drag and resistive forces that total 3500 N?

Example 2

A rope acting at an angle of 40 above the horizontal pulls a block 5.4 m along the floor at a steady speed when the tension in the rope is 57 N. What is the work done against friction by the rope?

Practice 1 What is the value of the energy transferred by a force of 28 N moving at 5.9 m/s for 5.0 minutes?

Practice 2 The work done by a force dragging a sinking boat 2.0 km to shore was 2.2 MJ. What was the magnitude of the force?


𝑡𝑒𝑛𝑠𝑖𝑙𝑒 𝑓𝑜𝑟𝑐𝑒 = 𝑠𝑝𝑟𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡× 𝑒𝑥𝑡𝑒𝑛𝑠𝑖𝑜𝑛

𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑣𝑒 𝑓𝑜𝑟𝑐𝑒

= 𝑠𝑝𝑟𝑖𝑛𝑔 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡× 𝑐𝑜𝑚𝑝𝑟𝑒𝑠𝑠𝑖𝑜𝑛

𝐹 = 𝑘𝑒

force, F, in newtons, N spring constant, k, in newtons per metre, N/m extension, e, in metres, m

Example 1 A particular spring has a spring constant of 250 N/m. What tension will be needed to extend this spring by 15 cm?

Example 2 A copper wire has a length of 1.250 m when loaded with a force of 15 N. The length increases to 1.255 m when the tension is increased by 75 N. What is the spring constant of this wire?

Practice 1 What is the spring constant for a spring that extends by 2.5 cm for every 100 g that is suspended from it? (g = 9.8 N/kg)

Practice 2 An empty lorry of mass 8.0 tonnes drives on to a weighbridge and the platform depresses by 2.0 cm. The same lorry depresses the platform by 4.8 cm after loading. Was the lorry overloaded? Show how you worked out your answer. (The maximum mass for a lorry of this type is 18 tonnes)

Word equation - Physics only Symbol equation Units

𝑚𝑜𝑚𝑒𝑛𝑡= 𝑓𝑜𝑟𝑐𝑒 × 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑀 = 𝐹𝑑

moment of a force, M, in newton-metres, Nm force, F, in newtons, N distance, d, is the perpendicular distance from the pivot to the line of action of the force, in metres, m.

Example 1 What is the moment of a 660 N force acting 2.5 m from a pivot?

Example 2 A see-saw of total length 6.4 m, with its pivot at the centre, is balanced when Mum of weight 550 N is at one end and Dad is 1.0 m away from the other end. What is Dad’s weight?

Practice 1 A metal bar of length 1.2 m is pivoted at one end and a force of 20 N pulls at angle

of 60 to the bar as shown. What is the moment of this force about the pivot?

Practice 2 A uniform metre ruler has a mass of 85 g and has a mass of 150 g suspended from it at the 10 cm mark. From what point should the ruler be suspended so that it balances horizontally?

60


𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 =𝑓𝑜𝑟𝑐𝑒

𝑎𝑟𝑒𝑎 𝑝 =

𝐹

𝐴

pressure, p, in pascals, Pa force, F, in newtons, N area, A, in metres squared, m2

Example 1 What is the maximum pressure that a brick of weight 24 N and dimensions 8.0 cm by 12 cm by 24 cm can exert through its own weight acting on a horizontal surface?

Example 2 A double glazing unit consists of two panes of glass 80 cm by 60 cm separated by a gap of 22 mm that is filled with nitrogen gas at atmospheric pressure. Atmospheric pressure is 1.01 × 105 Pa. What would be the inwards force on one pane of glass if the gap was not filled with nitrogen gas but was evacuated (ie there is a vacuum between the panes of glass)?

Practice 1 What is the pressure exerted on the floor by a single stiletto heel of contact area 0.50 cm2 worn by someone of weight 640 N?

Practice 2 In a hydraulic system the oil pressure is the same at each point. There are two pistons in the system with diameters of 2.5 cm and 8.5 cm. If the force exerted on the 2.5 cm piston is 200 N, what is the force exerted by the 8.5 cm piston?


𝑝𝑟𝑒𝑠𝑠𝑢𝑟𝑒 𝑖𝑛 𝑎 𝑙𝑖𝑞𝑢𝑖𝑑= 𝑑𝑒𝑝𝑡ℎ × 𝑑𝑒𝑛𝑠𝑖𝑡𝑦× 𝑔𝑟𝑎𝑣𝑖𝑡𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑓𝑖𝑒𝑙𝑑 𝑠𝑡𝑟𝑒𝑛𝑔𝑡ℎ

𝑝 = ℎ𝜌𝑔

pressure, p, in pascals, Pa height of the column, h, in metres, m density, ρ, in kilograms per metre cubed, kg/m3

gravitational field strength, g, in newtons per kilogram, N/kg

Example 1 What is the water pressure above atmospheric at the bottom of a dive pool of depth 5.0 m? (Water density is 1000 kg/m3, g = 9.8 N/kg)

Example 2 Atmospheric pressure at sea level is 1.01 ×105 Pa where the density of air is 1.2 kg/m3. If we assume that the density of the air does not change with height, what is the height of the atmosphere?

Practice 1 A submarine in the Atlantic ocean dives from a depth of 65 m to a depth of 320 m. What is the increase in pressure acting on the hull? (Density of sea water is 1025 kg/m3, g = 9.8 N/kg)

Practice 2 A one-man submarine has a circular window of diameter 45 cm. The air pressure inside the submarine is 1 atm (the same as at sea level). What will be the inwards force on this window when the submarine dives to a depth of 23 m? (Density of sea water is 1025 kg/m3, g = 9.8 N/kg)


𝑠𝑝𝑒𝑒𝑑 =𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑡𝑖𝑚𝑒 𝑣 =

𝑠

𝑡

distance, s, in metres, m speed, v, in metres per second, m/s time, t, in seconds, s

Example 1 What is the speed in m/s of a car travelling at a steady 56 kph?

Example 2 How long will it take a radio signal to travel from Earth to Neptune (a distance of 4.5 × 1012 m)? (Speed of light = 3.00 × 108 m/s)

Practice 1 In 1993, Qu Yunxia ran 1500 m in a time of 3 min 50.46 s. What was her average speed for the race?

Practice 2 The speed of sound in air is 340 m/s. A man is standing some distance away due south from a high cliff and is watching a cannon being fired at a point due south from his location. He notices two bangs, one 1.4 s after the flash from the cannon and one 1.9 s after the flash. How far was the man from the cliff?


𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑠𝑝𝑒𝑒𝑑 × 𝑡𝑖𝑚𝑒 𝑠 =(𝑣 + 𝑢)

2𝑡

distance, s, in metres, m final speed, v, in metres per second, m/s initial speed, u, in metres per second, m/s time, t, in seconds, s

Example 1 A car has a 0 to 60 mph (27 m/s) time of 7.2 s. Calculate the distance that the car travels whilst accelerating.

Example 2 A car is being driven towards a wall at a speed of 18 m/s. The driver brakes when 12 m from the wall but hits the wall 1.1 s after applying the brakes. What was the impact speed with the wall?

Practice 1 It takes 2.8 s for a car to accelerate from 13 m/s to 30 m/s along the slip-road of a motorway. How far did the car travel along the slip-road?

Practice 2 An experimental rocket-sled can only fire its rocket whilst inside a sound-absorbing tunnel of length 85 m. If the rocket-sled is travelling at 15 m/s when it enters the tunnel and the rocket fires for 2.3 s, what is the value of the final speed as the sled exits the tunnel?


𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 =𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦

𝑡𝑖𝑚𝑒

𝑎 =∆𝑣

𝑡

or:

𝑎 =(𝑣 − 𝑢)

𝑡

acceleration, a, in metres per second squared, m/s2

change in velocity, Δv (= v-u), in metres per second, m/s time, t, in seconds, s

Example 1 Calculate the acceleration of a racing car that has a zero to 60 mph (27 m/s) of 3.2 s.

Example 2 A train can accelerate at 1.1 m/s2. It passes through a station at a speed of 12 m/s and then accelerates for 4.2 s. Calculate the speed of the train after this acceleration.

Practice 1 A rollercoaster car travelling at 27 m/s brakes for 2.1 s and enters the loading area at a speed of 5.0 m/s. Calculate the deceleration of the rollercoaster car.

Practice 2 In an experiment using a linear air track and a pair of light gates the following data was obtained for a glider with a 55 mm interrupt card. Use the data to calculate the acceleration of the glider.

Event Time (s)

Beam A broken 0.000

Beam A resume 0.012

Beam B broken 0.648

Beam B resume 0.653


𝑓𝑖𝑛𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦2 − 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦2

= 2 × 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛× 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒

𝑣2 − 𝑢2 = 2𝑎𝑠

final velocity, v, in metres per second, m/s initial velocity, u, in metres per second, m/s acceleration, a, in metres per second squared, m/s2 distance, s, in metres, m

Example 1 A stone of mass 250 g is dropped from a point 7.5 m above the ground. What will be the impact speed of the stone with the ground? (Use the value of g = 10 m/s2)

Example 2 A car joining the motorway will have an acceleration of 4.2 m/s2. What will be the length of slip-road needed if the car is to accelerate from 13 m/s to 31 m/s on the slip road before joining traffic on the motorway?

Practice 1 A car, initially travelling at 28 m/s can decelerate at 7.5 m/s2. The driver applies the brakes 30 m from a stationary vehicle. Calculate the speed at which the car will collide with the stationary vehicle.

Practice 2 A tram accelerates uniformly from 5.6 m/s to 9.3 m/s in 4.8 s. How far does the tram travel in this time?


𝑟𝑒𝑠𝑢𝑙𝑡𝑎𝑛𝑡 𝑓𝑜𝑟𝑐𝑒 = 𝑚𝑎𝑠𝑠 × 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝐹 = 𝑚𝑎

resultant force, F, in newtons, N mass, m, in kilograms, kg acceleration, a, in metres per second squared, m/s2

Example 1 A trolley is pulled along a horizontal surface by a string. The mass of the trolley is 1.4 kg and it accelerates at 4.3 m/s2. There is a frictional force of 2.6 N acting on the trolley. Calculate the tension in the string.

Example 2 A small rocket can provide a thrust of 5.6 N. The rocket of mass 125 g is fixed to a trolley of mass 780 g with free-running wheels. The rocket fires for 2.7 s. Calculate the maximum speed reached by the trolley.

Practice 1 A trailer of mass 550 kg is pulled by a force of 320 N at a constant speed. The pulling force is suddenly doubled. What will be the acceleration of the trailer at this instant?

Practice 2 The new McClaren sportscar has a mass of 1200 kg and an acceleration given by the 0 - 62 mph (28 m/s) time of 2.8 s. Calculate the resultant force producing this acceleration.


𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 = 𝑚𝑎𝑠𝑠 × 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑝 = 𝑚𝑣

momentum, p, in kilograms metre per second, kg m/s mass, m, in kilograms, kg velocity, v, in metres per second, m/s

Example 1 What is the momentum of a 25 g bullet travelling at 300 m/s?

Example 2 A cannon and the cannonball fired from it have momenta that are equal in value but opposite in direction. If the cannonball has a mass of 15 kg and a speed of 110 m/s and the cannon has a mass of 1.0 tonne, determine the recoil speed of the cannon.

Practice 1 What is the momentum of a 1800 kg car travelling at 31 m/s?

Practice 2 A softwood block of mass 750 g is mounted on a stationary, friction-free trolley on a horizontal track. An air-gun pellet of mass 5 g is fired in line with the track into the block which then moves at a speed of 0.50 m/s. Calculate the speed of the pellet just before impact.


𝑓𝑜𝑟𝑐𝑒 × 𝑡𝑖𝑚𝑒 = 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑚𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝐹∆𝑡 = 𝑚∆𝑣

Force, F, in newtons, N time force acts for, Δt, in seconds, s mass, m, in kilograms, kg change in velocity, Δv, in m/s (m Δv = change in momentum)

Example 1 When a car moving at 28 m/s crashes into a wall the driver, who has a mass of 75 kg, is brought to rest by the seat belt in a time of 0.53 s. Determine the force that the seat belt exerts on the driver.

Example 2 A football (mass 430 g) is travelling towards a player at a speed of 11 m/s. The player kicks the ball causing it to travel back along its path at 22 m/s. If the impact time of the ball with the player’s boot was 0.3 s, determine the average force that the player’s boot exerted on the ball.

Practice 1 The maximum force that a rope can exert on a climber is 16 kN. A climber of mass 70 kg falls and reaches a speed of 32 m/s when the rope becomes tight. What must be the time that the rope stretches for whilst bringing the climber to a stop?

Practice 2 A small hovercraft of mass 2400 kg is moving at 4.0 m/s due north. A motor provides a horizontal thrust of magnitude 2200 N causing the hovercraft to travel at 3.0 m/s due east. Calculate the time that the motor was providing thrust on the hovercraft.


𝑝𝑒𝑟𝑖𝑜𝑑 =1

𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑇 =

1

𝑓

period, T, in seconds, s frequency, f, in hertz, Hz

Example 1 What is the duration of one cycle of the oscillation of a loudspeaker cone when it is producing a sound of frequency 440 Hz?

Example 2 How far will light travel in the time it takes for one cycle of a radio wave of frequency 98.8 MHz? (Speed of light = 3.00 × 108 m/s)

Practice 1 It takes 2.5 s for the wave generator in a leisure pool to complete one cycle of its oscillation. What is the frequency of the water waves produced?

Practice 2 How far will light travel in the time it takes for one cycle of an electromagnetic wave of frequency 9.7 GHz? (Speed of light = 3.00 × 108 m/s)


𝑤𝑎𝑣𝑒 𝑠𝑝𝑒𝑒𝑑 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦× 𝑤𝑎𝑣𝑒𝑙𝑒𝑛𝑔𝑡ℎ 𝑣 = 𝑓𝜆

wave speed, v, in metres per second, m/s frequency, f, in hertz, Hz wavelength, λ, in metres, m

Example 1 A sound wave has a frequency of 13 kHz and a wavelength of 2.6 cm. Calculate the speed of this sound wave.

Example 2 An X-ray machine has an operating potential difference of 120 kV and produces X-rays of wavelength 1.04 × 10-11 m. What is the frequency of these X-rays? (Speed of light = 3.00 × 108 m/s)

Practice 1 The wave generator in a leisure pool produces waves of wavelength 7.5 m that travel at 2.4 m/s. What is the frequency of the wave generator?

Practice 2 Ultrasound waves travel at 1500 m/s through water. An echo-sounder produces pulses of ultrasound with a pulse frequency of 75 Hz. The time delay between emitting a pulse and receiving reflected ultrasound is 4.1 ms. In order for the echo-sounder to detect fish the wavelength of the ultrasound must be no more than 5.0 cm. Calculate the minimum frequency of the ultrasound.


𝑚𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛 =𝑖𝑚𝑎𝑔𝑒 ℎ𝑒𝑖𝑔ℎ𝑡

𝑜𝑏𝑗𝑒𝑐𝑡 ℎ𝑒𝑖𝑔ℎ𝑡 𝑀 =

ℎ𝐼

ℎ𝑂

heights, h, in m or (cm) magnification, M has no units

Example 1 Calculate the magnification produced by a lens in a camera that gives an image 4.1 mm high of a 1.83 m tall man.

Example 2 A film projector is set up to give a magnification of 130. The image on the screen is 4.5 m tall. What is the height of the film frame in the projector?

Practice 1 A ripple tank uses a projection system to show the wave patterns on a screen. On the screen the wavelength was measured to be 8.7 cm when the wavelength of the waves on the water surface was 2.2 cm. Calculate the magnification of this projection system.

Practice 2 A visualiser (document camera) projects an A4 page in landscape orientation (210 mm high) onto a screen that has a height of 1.2 m. If I want any letters that I write on the A4 page to have a height of at least 2.5 cm on the screen, what is the minimum height that I must write the letters on the page?


𝑓𝑜𝑟𝑐𝑒 = 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐 𝑓𝑙𝑢𝑥 𝑑𝑒𝑛𝑠𝑖𝑡𝑦× 𝑐𝑢𝑟𝑟𝑒𝑛𝑡× 𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑜𝑟

𝐹 = 𝐵𝐼𝑙

force, F, in newtons, N magnetic flux density, B, in tesla, T current, I, in amperes, A length, l, in metres, m

Example 1 The magnetic flux density between the poles of a horseshoe magnet is 75 mT. A wire placed in this field has a length of 5.0 cm and carries a current of 13 A. Determine the size of the force produced on the wire.

Example 2 At a point near the equator the magnetic flux density due to the Earth’s magnetic field is 41 μT due north. An aluminium wire has a weight per unit length of 0.083 N/m and, if placed in an east-west direction might levitate – the magnetic force might be enough the support the weight of the wire. Calculate the minimum current required for this wire to levitate in the Earth’s magnetic field.

Practice 1 The magnetic flux density inside a magnetic resonance imaging machine is 1.7 T. What will be the force acting on the 1.1 cm long tungsten filament of a lamp with an operating current of 1.8 A when it is placed in the magnetic field?

Practice 2 The current in a wire produces a magnetic field. The magnetic flux density depends on the size of the current and the distance from the wire. At a distance of 15 cm from a conductor carrying a current of 1500 A the flux density is 2.0 mT. What will be the force between two 3.0 m long, parallel conductors 15 cm apart each carrying a current of 1500 A?


𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑣𝑜𝑙𝑡𝑎𝑔𝑒𝑠 = 𝑟𝑎𝑡𝑖𝑜 𝑜𝑓 𝑡𝑢𝑟𝑛𝑠 𝑉𝑃

𝑉𝑆=

𝑁𝑃

𝑁𝑆

potential difference, Vp and Vs in volts, V number of turns on primary, NP number of turns on secondary, NS

Example 1 The transformer used in a battery charger needs to have a peak output potential difference of 15 V when the input potential difference has a peak value of 325 V. The secondary coil has 230 turns. Calculate the number of turns required on the primary coil.

Example 2 A teacher connects a transformer designed to step down 230 V mains to 12 V the wrong way round (they connect the 12 V coil to the mains). What will be the output potential difference produced across the other coil (assuming that the transformer is not damaged)?

Practice 1 The primary coil in a transformer designed to step up the output voltage of a power station from 25 000 V to 440 kV has 550 turns on the primary coil. Calculate the number of turns on the output coil.

Practice 2 A transformer has two coils, one with 350 turns and the other with 2500 turns. Use the equation to calculate the two possible output voltages when this transformer is connected to a 12 V ac power supply.


For a transformer: 𝑜𝑢𝑡𝑝𝑢𝑡 𝑝𝑜𝑤𝑒𝑟 = 𝑖𝑛𝑝𝑢𝑡 𝑝𝑜𝑤𝑒𝑟

𝑉𝑆𝐼𝑆 = 𝑉𝑃𝐼𝑃

Vs × Is is the power output (secondary coil) Vp × Ip is the power input (primary coil). power input and output, in watts, W

Example 1 An industrial heater is designed to operate on a building site at a potential difference of 110 V and is connected to the mains supply by means of a transformer. The operating current for the heater is 14 A. What will be the current input to the transformer from the mains?

Example 2 A set of Christmas tree lights has 100 LEDs each of which operates with a potential difference of 1.9 V and a current of 450 mA. These are connected to the mains (230 V ac) using a transformer. What will be the input current to the transformer from the mains supply?

Practice 1 A battery charger includes a transformer that produces an output current of 5.5 A at a potential difference of 14 V. Calculate the input current when this transformer is connected to mains voltage (230 V).

Practice 2 A filament lamp is rated at 12 V, 55 W. If this lamp is operated from the mains by using a transformer, what will be the value of the input current to the transformer from the mains?

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