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The angle of incidence is measured between the incident ray and the normal, so a ray travelling along the normal has an angle of incidence of 0°. By the law of reflection the angle of reflection equals the angle of incidence, hence it is also 0°, and the ray retraces its path back along the normal.
A series circuit provides only one continuous path, so the same charge passes any cross-section in a given time and the current is identical through every element. The potential difference divides in proportion to the resistances, while resistance and power dissipated may differ from one resistor to another.
The ciliary muscles change the curvature, and therefore the focal length, of the eye lens so that nearby and distant objects are both focused on the retina. This ability is termed the power of accommodation, and it weakens with age, producing presbyopia.
The refractive index is the ratio n = c/v, where c is the speed in vacuum or air and v the speed in the medium. Rearranging gives v = c/n = (3 × 10^8 m/s) ÷ 1.5 = 2.0 × 10^8 m/s. Light always slows down on entering an optically denser medium.
Cutting the wire into three equal lengths gives each piece one-third of the original resistance, that is 4 Ω. Three 4 Ω resistors in parallel obey 1/R = 1/4 + 1/4 + 1/4 = 3/4, so R = 4/3 Ω ≈ 1.33 Ω. A parallel combination is always smaller than the smallest individual resistance.
If the right thumb points along the current, the curled fingers give the direction of the magnetic field. For a straight conductor the field lines are closed concentric circles lying in planes perpendicular to the wire, and the field strength decreases as the distance from the wire increases.
Ohm's law states that I = V/R. Substituting V = 220 V and R = 44 Ω gives I = 220 ÷ 44 = 5 A. At this current the iron consumes VI = 220 × 5 = 1100 W of power.
Moving air possesses kinetic energy, which turns the blades of the windmill and drives a generator. The generator then converts this mechanical motion into electrical energy. Since no fuel is burnt, wind is a renewable and non-polluting source of energy.
Using 1/v − 1/u = 1/f with u = −30 cm and f = +20 cm gives 1/v = 1/20 − 1/30 = 1/60, so v = +60 cm. A positive image distance means a real image formed on the far side of the lens. The magnification is m = v/u = 60/(−30) = −2, so the image is inverted and twice the size of the object.
In a series connection the same current flows through both bulbs, so the power actually dissipated in each is I²R and the bulb of larger resistance glows brighter. The rated power gives R = V²/P, so the 60 W bulb has the greater resistance and therefore glows brighter, even though its rating is lower.
Pressure is defined as force acting normally per unit area, so its SI unit is the newton per square metre, which is given the name pascal. The newton is the unit of force, the joule is the unit of energy and the watt is the unit of power.
Acceleration is the rate of change of velocity. For motion along a straight line with constant speed, the direction also remains unchanged, so velocity is constant and acceleration is zero.
A convex mirror diverges rays, and for any real object placed in front of it the reflected rays appear to come from a point behind the mirror. Therefore the image is always virtual, erect and diminished, whatever the object distance.
The heat required is Q = mcΔT. Here m = 2 kg, c = 4200 J kg⁻¹ °C⁻¹ and ΔT = 30 − 20 = 10 °C, so Q = 2 × 4200 × 10 = 84,000 J = 8.4 × 10^4 J.
For a wave, speed = frequency × wavelength, so wavelength = speed ÷ frequency. Substituting 330 m/s and 500 Hz gives λ = 330/500 = 0.66 m.
The parallel equivalent resistance is given by 1/R = 1/6 + 1/3 = 1/2, so R = 2 Ω. The total current is I = V/R = 6/2 = 3.0 A. Equivalently, the branch currents are 1 A and 2 A, giving 3 A.
Inside a long current-carrying solenoid the magnetic field is uniform and directed along its axis. The field lines there are therefore parallel straight lines, while outside they close through the ends like those of a bar magnet.
Useful power equals work done per unit time. The pump lifts 200 kg through 10 m in 60 s, so work = mgh = 200 × 10 × 10 = 20,000 J and power = 20,000/60 ≈ 333 W.
Using the mirror formula with sign convention, f = −10 cm and u = −15 cm. Then 1/v = 1/f − 1/u = −1/10 + 1/15 = −1/30, so v = −30 cm. A negative image distance for a concave mirror means a real, inverted image 30 cm in front of the mirror.
The acceleration due to gravity at height h is g' = g(R/(R+h))². Setting g'/g = 1/4 gives R/(R+h) = 1/2, so R+h = 2R and h = R.
Electric charge is measured in coulombs, where one coulomb is the charge that flows when a steady current of one ampere passes for one second (Q = It). The ampere, volt and ohm are the SI units of current, potential difference and resistance, so only charge fits the unit given.
Sound is a mechanical wave, so it can travel only through a material medium in which particles can vibrate and pass the disturbance on. When the air is removed, the number of particles available to carry the vibrations falls, and in a vacuum no sound is heard at all.
A freely suspended bar magnet behaves like a compass needle because the Earth itself acts as a large magnet with poles near the geographic poles. The magnet therefore settles along the magnetic meridian, with its north pole pointing towards geographic north.
Resistance is given by R = ρL/A. Stretching keeps the volume L × A constant, so when the length doubles the area of cross-section becomes half, and R becomes ρ(2L)/(A/2) = 4R. With R = 10 Ω, the new resistance is 40 Ω.
The power of the heater is P = VI = 220 V × 5 A = 1100 W, which equals 1.1 kW. Energy in kilowatt-hours is power in kW multiplied by time in hours, so 1.1 × 2 = 2.2 kWh.
At the first face the ray bends towards the normal on entering the denser glass, and at the second face it bends away from the normal by an equal amount because the two faces are parallel. The emergent ray is therefore parallel to the incident ray, shifted sideways by a small lateral displacement.
Difficulty in seeing distant objects clearly while near vision remains good is myopia, or short-sightedness, in which the eye lens converges light too strongly and the image forms in front of the retina. A concave, diverging lens of suitable power spreads the rays slightly so that the image is pushed back onto the retina.
The sound has to travel to the cliff and back, so the distance covered in 0.6 s is twice the distance to the cliff. Thus 2d = 340 × 0.6 = 204 m, and the distance d is 204/2 = 102 m.
The parallel pair has resistance (3 × 6)/(3 + 6) = 2 Ω, so the total resistance of the circuit is 2 + 4 = 6 Ω and the battery current is 12/6 = 2 A. This current produces a potential difference of 2 A × 2 Ω = 4 V across the parallel section, so the current through the 6 Ω resistor is 4/6 ≈ 0.67 A.
For thin lenses in contact, the powers add: P = (+5 D) + (−2 D) = +3 D. The focal length is f = 1/P = 1/3 m ≈ 33.3 cm, and since the net power is positive, the combination behaves as a converging lens.
Resistance is the opposition offered by a conductor to the flow of electric current. The SI unit of resistance is the ohm, which is defined as the resistance of a conductor when a potential difference of one volt produces a current of one ampere.
The least distance of distinct vision is the closest distance at which a normal eye can see an object clearly without undue strain. For a normal adult human eye, this distance is about 25 cm.
Pitch is the sensation that distinguishes a shrill sound from a grave sound, and it is determined by frequency. A higher frequency gives a higher pitch, whereas amplitude mainly controls the loudness of the note.
Using s = ut + ½at² with u = 0, a = 2 m/s² and t = 5 s gives s = ½ × 2 × 25 = 25 m. The final speed of the car would be 10 m/s, but the question asks for the distance travelled.
Weight is the gravitational force acting on a body and is calculated as W = mg. On the Moon, W = 10 kg × 1.6 m/s² = 16 N, whereas the same mass would weigh about 98 N on the Earth.
Kinetic energy is given by KE = ½mv². Substituting m = 2 kg and v = 3 m/s gives KE = ½ × 2 × 9 = 9 J.
During melting, the temperature remains constant at 0°C, so the heat absorbed is Q = mL. For 1 kg of ice, Q = 1 kg × 336 kJ/kg = 336 kJ.
In series, the total resistance is 4 Ω + 8 Ω = 12 Ω, so the current is I = V/R = 12/12 = 1 A. The potential difference across the 4 Ω resistor is V = IR = 1 × 4 = 4 V.
For a concave lens both the focal length and the virtual image distance are taken as negative in the lens formula. Using 1/v − 1/u = 1/f with v = −10 cm and f = −15 cm gives u = −30 cm, so the object is 30 cm from the lens.
The magnetic force on a current-carrying straight conductor is F = BIL sin θ. Here θ = 90°, so F = 0.5 T × 2 A × 0.10 m = 0.1 N.
A convex mirror always forms a diminished, erect virtual image and has a wider field of view, so it shows more traffic behind the vehicle. A plane mirror gives an image of the same size but a narrower field. Concave mirrors can form magnified or inverted images depending on object distance and are not used for this purpose.
At the highest point the velocity becomes zero, but the acceleration due to gravity continues to act on the ball. Its magnitude is about 9.8 m/s² and its direction is vertically downward throughout the flight. The vanishing of velocity does not make the acceleration zero.
The wave equation v = fλ gives λ = v/f. Substituting v = 340 m/s and f = 200 Hz gives λ = 340/200 = 1.70 m. Frequency and speed are independent properties here, so the wavelength is fixed by their ratio.
When the object lies within the focal length of a concave mirror, the reflected rays diverge and meet only when produced behind the mirror. The image is therefore virtual, erect and larger than the object, as in a shaving mirror. Beyond the focus, the image would be real and inverted.
For two resistors in parallel, 1/R = 1/R1 + 1/R2 = 1/4 + 1/12 = 3/12 + 1/12 = 4/12, so R = 3 Ω. The equivalent resistance of a parallel combination is always smaller than the smallest individual resistance.
The rated power P and voltage V are related to resistance by P = V²/R, so R = V²/P. Substituting gives R = (220)²/100 = 48400/100 = 484 Ω. This is the resistance when the bulb glows at its normal brightness.
The gain in gravitational potential energy is ΔPE = mgh. Substituting m = 2 kg, g = 10 m/s² and h = 5 m gives ΔPE = 2 × 10 × 5 = 100 J. The result is independent of the path taken if the body moves slowly.
The conversion formula is F = (9/5)C + 32. Setting F = C gives C = (9/5)C + 32, so −(4/5)C = 32 and C = −40. Thus −40 °C equals −40 °F.
For a real image, magnification m = v/u = 3, so v = 3u. Using the lens formula 1/f = 1/v + 1/u gives 1/20 = 1/(3u) + 1/u = 4/(3u), hence u = 80/3 ≈ 26.7 cm. The image distance is then 80 cm on the other side.
Cutting the wire into three equal parts makes each piece have resistance R/3, since resistance is proportional to length. Three such resistors in parallel give 1/Req = 3/(R/3) = 9/R, so Req = R/9. The parallel combination is therefore much smaller than any single piece.