Physics Notes Form 2 – Free Download

In a convex mirror the ray is reflected and appears to originate from the point. F is virtual for a convex mirror while it is real for a concave mirror.

4. Radius of curvature (r) – this is the distance from the pole to the centre of curvature. The distance from the pole to the principal focus is called the focal length (f).

Parabolic mirrors.

They produce a wide parallel beam or converge a large beam of light to a point. They are widely used in making car headlights or in spotlights.

Images formed by spherical mirrors.

Location of images using ray diagrams.

When drawing ray diagrams the following symbols are used to represent the mirrors.

The image is located by drawing any two of the following rays:

i) A ray parallel to the principal axis which is reflected through the principal focus.

ii) A ray through the centre of curvature which is reflected along its own path since it hits the mirror normally.

iii) A ray through the principal focus which is reflected parallel to the principal axis.

Virtual images are formed when rays diverge and as such the rays are extended backwards using dotted line till they meet.

The image formed is also dotted since it is not formed by an intersection of real rays.

A real image is formed by intersection of real rays.

Concave mirror.

a) Object at infinity: image is formed at F. It is real, inverted and diminished.

Convex mirror.

g) Image is always formed behind the mirror. It is virtual, erect and always diminished.

Applications of curved reflectors.

a) They are used in satellite dishes.

b) They are used in making shaving mirrors.

c) They are used in telescopes.

d) They are used in driving mirrors.

Magnification.

Magnification is the ratio of the image size to the object size.

Magnification (M) = height of the image / height of the object.

When the ratio is greater than one we say the image is magnified and when less than one we say it is diminished.

Also magnification = image distance from the mirror / object distance from the mirror.

Examples

1. Determine the size, position and nature of the image of an object 5.0 cm tall, placed on the principal axis of a concave mirror of focal length 15 cm, at a distance 35 cm from the mirror.

Solution

Let 1 cm represent 5 cm. Then the focal length is 3 cm.

Object distance = 7 cm, object height = 1 cm.

From the scale drawing,

Image position = 5.4 cm × 5 = 27 cm in front of the mirror.

Image size = 0.75 cm × 5 = 3.75 cm. Image is real and inverted.

2. A vertical object 5 cm high is placed 10 cm in front of a convex mirror of focal length 15 cm.

find the position, size and nature of image formed. Determine the magnification of the image.

Solution

Let 1 cm represent 5 cm, then the focal length = 3 cm, object size = 1 cm
Object distance = 2 cm.

From the scale drawing,

Image position = 1.2 cm × 5 = 6.0 cm behind the mirror. Image size = 0.6 cm × 5 = 3.0 cm.

The image is virtual and erect.

Magnification = image dist. / object dist. Hence 6 /10 = 0.6 (diminished).

Chapter Six

Magnetic Effect of an Electric Current.

Introduction: Oersted’s discovery.

Hans Christian Oersted discovered the magnetic effect of a current in 1819. The direction of the field is dependent on the direction of the current.

This discovery brought about the development of electric bells, electric motors, telephone receivers and radios.

Determining the direction of the lines of force.

The direction of the lines of force can be determined using a simple rule called the right-hand screw rule.

This rule states that “if a right-hand screw advances in the direction of the current , then the rotation of the screw is in the direction of the field”.

Another rule is the right-hand grip rule which states that “if the wire carrying a current is gripped with the right hand, using the thumb along the conductor and pointing in the direction of the current, then the direction of curled fingers is in the direction of the lines of force”.

Magnetic field due to a solenoid. The rule for polarity.

A solenoid is a cylindrical coil of wire acting as a magnet when carrying electric current.

The direction of the field can be determined using a simple rule stated as follows “if the coil (solenoid) is viewed from one end and the current flows in an anticlockwise direction at that end, then that end is the North Pole.

If the current flows in a clockwise direction, then that end is the South Pole”.

Electromagnets.

An electromagnet is a soft metal core made into a magnet by passing an electric current
through a coil surrounding it .

They only maintain their magnetism if current continues to flow, if switched off they lose their magnetism.

Factors affecting the strength of an electromagnet.

1. Increasing current through the coil.

2. Increasing the number of turns of the coil.

3. Using iron of C- core shape which brings both magnetic poles together.

Some applications of electromagnets.

a) Electric bell

When the switch is closed the current passing through the solenoids magnetizes them and
they pull the soft iron armature which makes the hammer hit the gong therefore producing sound. When the hammer hits the gong the contact between the spring and the screw is broken and then stops the current from flowing.

The soft iron core loses its magnetism and releases the armature which is then pulled back by the screw.

The contact between the spring and the screw is regained and the process repeats itself again and again therefore
the gong is struck continuously.

b) Telephone receiver.

It consists of a u-magnet made by attaching two soft-iron bars to the end of a short
permanent magnet.

The solenoids are wound in opposite directions around the bars. When the phone is lifted the current flows through the solenoids depending on the microphone
on the other end of the line.

These varying current spasms induce magnetism of varying strengths in the iron bars which in turn causes the magnetic alloy diaphragm to vibrate differently producing sound.

Force on a current-carrying conductor in a magnetic field.

When a conductor carries a current in a magnetic field a force acts on it.

The direction of the force depends on the directions of the field and current.

The factors affecting the magnitude of the force are;

a) The current flowing in the conductor

b) The strength of the magnet

c) The length of the conductor in the magnetic field.

The relationship between the directions of the current, field and force are mutually
perpendicular.

They are summarized in a law called Fleming’s right-hand rule or the motor rule.

This rule states that “if you hold the first finger, the second finger and the thumb of your left hand mutually perpendicular to each other, so that the first finger points in the direction of the magnetic field and the second finger points in the direction of the current in the conductor, then the thumb points in the direction of the force acting on the conductor”.

Applications of the force on a conductor.

Simple D.C motor.

Consists of a rectangular coil of wire mounted on an axle which can rotate between the poles of a magnet.

For the rotation to be continuous the ends of the coil is connected to half -rings called the split-ring commutators.

The battery terminals are attached to brushes which slide on these half-rings. D.C motors are useful as car starter motors, hand drills, machine motors, fans etc.

Chapter Seven

Hooke’s Law.

Hooke’s law states that “the extension of a spring is proportional to the applied force,
provided that the force is not large enough to deform the spring permanently”.

Mathematically expressed as Force α extension.

Spring constant

Since Force α extension then Force / Extension = constant (k). The constant of proportionality (k) is called the spring constant. F / e = k or Force (N) = k e.
The spring constant is a measure of the stiffness of a spring.

The greater the constant the stiffer the spring.

The spring constant varies with the following;-

a) Material – identical springs mad of different materials will have different constants i.e.steel and copper.

b) Diameter – the stiffness decreases with the increase in diameter.

c) Thickness of the wire – a spring made of a thicker wire is stiffer than the one made of thin wire of the same material.

d) Length of spring – a short spring is stiffer than a longer one.

e) Number of turns per unit length – a spring with higher number of turns per unit length is
less stiff than the one with fewer turns per unit length.

Example

1. If the springs shown below are similar and the constant of proportionality (k) is
100 Nm^-1, determine total extension in each arrangement.

The spring balance

It is made up of a spring mounted in a metal or plastic casing.

The spring is fitted with a pointer which moves along a calibrated scale divided into ten equal parts.

Examples

1. A load of 4 N causes a certain copper wire to extend by 1.0 mm. Find the load that will
cause a 3.2 mm extension on the same wire. (Assume Hooke’s law is obeyed).

Solution

F α e also F₁ / F₂ = e₁ / e₂ = F₂ = (4 × 3.2) / 1.0 = 12.8 N.
2. A body of 200 g was hung from the lower end of a spring which obeys Hooke’s law.
Given that the spring extended by 100 mm, what is the spring constant for this spring? Solution
F = α e, F = k e. F = 200 × 10-3 kg × 10 N /kg = 2 N. Extension = 100 × 10³ m = 0.1 m.

Spring constant (k) = 2 / 0.1 = 20 N/m.

3. Two identical springs, whose spring constant is 6.0 N/cm, are used to support a load of
60 N as shown below. Determine the extension of each spring.

Solution

Since the springs are parallel their spring constant equals 2k. Therefore extension = Force / k = 2 F / k = 60 / 2 × 6 = 5 cm. Each spring will extend by 5 cm.

Chapter Eight

Waves I

A wave is simply a disturbance that moves through a medium . Other waves do not require a medium to travel i.e. they can travel in a vacuum, are known as electromagnetic waves e.g.

radio, X-rays, gamma rays UV rays etc. Other waves require a material medium to be transferred and are called mechanical waves i.e. water, sound waves etc.

Transverse and longitudinal pulses and waves.

1. Transverse waves – they consist of a crest and a trough. In this case the displacement of the medium caused by these pulses are perpendicular to the direction in which the wave (disturbance) travels.

A pulse is a single non-repeated disturbance.

If the pulses are repeated periodically (regularly) they produce a series of waves called periodic transverse wave train.

They can be produced as shown below. Examples are water waves, light and radio waves.

2. Longitudinal waves – these are waves whereby the particles of the medium vibrate parallel to the direction of movement of the disturbance .

When several turns of a spring are pulled together (compression) and then released they tend to spread out to their original position.

When pulled apart (rarefaction) they also turn to their original
position.

In this case the displacement of the spring is parallel to the motion of the wave and this is known as longitudinal. Examples are the sound waves.

Characteristics of waves

1. All waves have speed which depends on the nature of disturbance.

2. All waves have wavelength (distance between two successive points in a wave ).

Represented by the symbol λ and is measured in metres.

3. All waves have frequency ‘f’ which is the number of waves passing a point in one second. It is measured in cycles per second or hertz (Hz). The period of a wave is the time required for a complete wave to pass a given point.

Therefore T = 1 / f or f = 1 / T (period is measured in seconds).

The speed ‘v’ is given as; v = λ / T, since f = 1 / T then
v = (1 / T) × λ = f λ or v = f λ. This is the wave equation.

4. All waves have amplitude which is the maximum displacement of the particles of the
medium as the wave passes.

Examples

1. A rope is displaced at a frequency of 3 Hz. If the distance between two successive crests of the wave train is 0.8 m, calculate the speed of the waves along the rope.

Solution

v = f λ = 3 × 0.8 = 2.4 m Hz = 2.4 m/s.

2. The figure below illustrates part of the displacement-time graph of a wave travelling across water at a particular place with a velocity of 2 ms^-1. Calculate the wave’s;

a) Amplitude

b) Frequency (f)

c) Wavelength (λ)

Solution

a) From the graph, maximum displacement (a) = 0.4 cm

b) From the graph, period T = time for one cycle = 0.20 seconds

So f = 1 / T = 1 / 0.20 = 5 Hz.

c) Velocity = f λ hence λ = 2 / 5 = 0.4 m.

Chapter Nine

Sound.

Sound is an important example of a longitudinal wave which is composed of both compressions and rarefactions. The frequency range in which compressional disturbances occur is called the sonic spectrum and is very large.

Sound is a range of compressional wave frequencies sensitive to human ear and is known as audio range which ranges from 20 Hz to 20,000 Hz.

Compressional waves with frequencies above and below the audio range are called ultrasonic and infrasonic frequencies respectively.

Nature and transmission of sound waves.

Sound waves require a material medium for transmission from one point to another. Sound waves therefore cannot be transmitted in a vacuum.

The rate of transmission diminishes as you move from solids, to liquids then to gases.

Characteristics of sound waves

1. Intensity and loudness – intensity of sound refers to the rate of flow of energy through an area.

The loudness of a sound generally varies with the intensity of sound.

The frequency of sound waves determines their intensity while the amplitude determines their loudness.

2. Frequency and pitch – pitch refers to the sharpness of a sound and is determined by its frequency.

3. Fundamental tones and harmonics – fundamental frequency is the vibration in a wire
which has the lowest possible frequency.

The fundamental frequency and the tones with the frequencies that are whole multiples of the fundamental are called harmonics.

The fundamental frequency is also the first harmonic.

The number of segments vibrating in a string depends on the point at which the string is plucked.

4. Quality of sound or timbre – quality of sound can be improved by adding the second harmonic to the fundamental frequency hence the quality depends on the number of harmonics produced simultaneously and their relative intensities.

Echo

Echoes are produced by reflection of sound waves from hard surfaces such as a wall or a cliff.

To hear an echo, the sound waves travels double the distance between the source and the reflector. So to determine the velocity of sound ‘v’ between two successive claps is given as
v = distance from the reflecting surface / half the time taken between two successive claps
Hence; speed ‘v’= distance / time = m/s.

Example

A boy strikes a railway line (steel) with a hammer.

If the speed of sound in steel is 5,200 m/s, determine the time taken for the sound to reach another boy 2.3 km down the railway line with his ear on the rail.

Solution

Time taken = distance / speed = 2300 / 5200 = 0.44 seconds.

Factors affecting the speed of sound

a) Temperature – speed increases with increase in temperature especially gases since the change in liquids and solids is small it can be neglected.

b) Nature of transmitting medium – different substances transmit sound waves at various speeds as shown below

(i) Air – 346 m/s

(ii) Hydrogen – 1339 m/s

(iii) Water – 1498 m/s

(iv) Aluminium – 5,000 m/s

(v) Iron – 5,200 m/s

(vi) Glass – 4,540 m/s

Example

A girl claps her hands once at a distance 250 m from a vertical cliff.

If the temperature in the surrounding is 50^oC, how long does it take for her to hear the echo?

Solution

Distance travelled = 250 × 2 = 500 m

Speed of sound = (331.5 m/s at 00) the speed in air increases at about 0.6 m/s per 0C.

Therefore speed at 50^oC = 331.5 + (0.6 × 5) = 334.5 m/s

Time taken = distance / speed = 500 / 334.5 = 1.5 seconds.

Chapter Ten

Fluid Flow

Fluid in physics refers to liquids and gases.

To study fluid flow we have to make the following assumptions:

1. We consider fluids to be incompressible

2. We assume that they have little or no internal friction or viscosity.

Streamline and turbulent flow.

The path followed by a small element of a moving fluid is called a line of flow . A streamline is a curve whose tangent at any point is in the direction of the fluid velocity at that point.

A streamline flow occurs when all elements of a fluid passing a particular point follow the same path or line of flow as the elements that passed through that point previously.

A streamline flow is achieved only when the speed is low.

If the speed increases it is characterized by whirls and eddies then it becomes a turbulent flow. Turbulent flow generally occurs when the speed is high and where there are sharp bends along the path of the fluid.

Equation of continuity

Consider a fluid flowing (streamline flow) through a horizontal pipe with different cross –
sectional areas as shown.

Let the cross -sectional area in both sections be A₁ and A₂ and the corresponding speeds of the fluid be V₁ and V₁ respectively.

The volume of fluid flowing per second in each section is given by; V = A L = A v t = A v. Where L, v t and v is the distance moved in one second.

Since the volumes in each section is the same, then
A₁ V₁ = A₂ V₂ , hence A v = constant.

The above equation is known as the equation of continuity. Since A₁ A₂ , then V₂ V₁ . i.e. the speed increases when a tube narrows.

The quantity (A v) is called volume efflux i.e. volume flowing per second.

Example.

A horizontal pipe of cross-sectional area 50 cm² carries water at the rate of 0.20 litres per second. Determine the speed;

a) Of the speed of water in the pipe.

b) When the tube narrows to 20 cm² at another point.

Solution

a) Volume efflux = o.20 l per second = A v
From V (volume) = A v, then v = V / A = 0.20 × 10-3 / 50 × 10-4 = 0.04 m/s b) Since A₁ v₁ = A₂ v₂ then v₁ = (0.05 × 0.04) / 0.02 = 0.1 m/s

Bernoulli’s principle

Daniel Bernoulli (1700 – 1782) explained the variation of pressure exerted by a moving fluid when its speed is changed. The pressure is lower where the speed is higher.

Bernoulli’s principle states that “For a fluid flowing through a tube, the sum of the pressure, the kinetic energy per unit volume and the potential energy per unit volume of the fluid is a constant”. Mathematically expressed as;

P + ½ ρ v₂ + ρ g h = constant. Where P – pressure, ρ – density, v= velocity,
g – acceleration due to gravity and h – height.

Bernoulli’s effect

When air is blown through the tunnel formed, the area marked ‘T’ collapses inwards showing that pressure outside is more than the one inside the tunnel.

The pressure inside the tunnel decreases as the air through it increases in speed.

Applications of Bernoulli’s principle.

1. Car carburetor– inside the carburetor the air passage is partially constricted at the point
where petrol mixes with air hence air intake increases the speed of air while decreasing the pressure inside for petrol to vaporize quickly before it gets to the cylinder where combustion occurs.

2. Horizontal pipe – for a streamline flow through a pipe the term ρ g h is eliminated from the Bernoulli’s equation leaving P + ½ ρ v₂ = constant, indicating that pressure in liquid is greatest when speed is least.

When this is combined with the equation of continuity, the pressure is then greatest when the pipe is widest hence the following observation.

3. Dynamic lift – when air is blown at the top a flat sheet of paper the ends of the paper moves upward and this because the speed of air on top of the paper is greater than below and according to Bernoulli’s principle the pressure on top lowers and the pressure below becomes sufficient enough to produce a force which moves the paper
upwards.

This is what is referred to as the dynamic lift since it is caused by motion.

The upward force is equal to the product of the pressure difference and the area of the surface lifted.

It is applied in the taking off of air-planes, the trajectory of a spinning ball, paint sprayer and Bunsen burner among others.

KCSE Revision Notes Form 1 – Form 4 All Subjects

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