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Category: Physics

  • Upthrust in Gases

    Upthrust in Gases

    Just like liquids, gases exerts upthrust on objects that are in them.Air is the most common gas whose upthrust maybe of interest because we float objects on air and sometimes we use parachutes to float in air as human beings.

    The upthrust in air is small because air has lower density compared to most of substances.

    The density of air is about 1.3kgm-3 or 0.0013gcm-3.

    If we trap a gas that has a lower density than air in a balloon, then that balloon can float in air. For example hydrogen has a density of 0.09kgm-3 whereas helium has a density of 0.18kgm-3. Therefore a balloon filled with helium or hydrogen will rise on air provided density of the balloon fabric and air will be less than density of air.

    Consider the figure below that illustrates a balloon filled with air .

    If we consider the balloon filled with air to a certain volume, the weight of air in the balloon plus it’s fabric is greater than the weight of air displaced by the balloon, since the volume of air in the balloon is nearly equal to the volume of air displaced.

    The upthrust force on the balloon due to the air is thus less than the weight. The balloon therefore stay grounded because the it’s weight is less than the upthrust force that could set it up to float on air.

    That is W-U > resultant downward forces.

    If the balloon is filled with a gas which is has lower density compared to that of air, the weight of the gas plus the balloon fabric is less than the weight of the air displaced by the balloon, hence the upthrust force U exerted by the air on the balloon is greater than the weight W of the inflated balloon.the resultant upward force is greater than W-U and hence the balloon set to accelerate upward.see the illustrations below.

    Example Question

    A meteorological balloon has a volume of of 55m3 and is filled with a helium gas of density 0.18kgm-3. If the weight of the balloon fabric is 170N, calculate the maximum load the balloon can lift given that density of air is 1.3kgm-3

    solution

    Volume of air displaced by the balloon is 55m3 which is volume occupied by the balloon.

    mass of air displaced by the balloon = 55m3 x 1.3kgm-3 = 71.5kg

    weight of air displaced =71.5kg x 10 Nkg-1 = 715N

    mass of helium in the balloon = 55m3 x 0.18kgm-3 = 9.9Kg

    weight of helium in the ballon = 9.9kg x 10 Nkg-1 = 99N

    Total weight of the inflated balloon = 170N + 99N = 269 N

    From the law of floatation, upthrust is the weight of air displaced which should also be weight of the balloon plus the weight of load it will carry. hence,

    upthrust = weight of the balloon + load in the ballon = weight of air displaced

    269N + load in the ballon = 715N

    load in the balloon = (715-269)N=446N

    So the total mass of the goods to be included in the balloon should never exceeded 44.6kg.

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  • Density of mixtures

    Density of mixtures

    A mixture is obtained by putting together two or more substances such that they do not react with one another.

    Density of a mixture lies between the densities of the constituent substances and depends on their proportions.

    Volume of the mixture is obtained by summing up the masses of the individual constituents that makes the mixture and dividing it with the sum of their individual volumes. I.e

    Density of the mixture=mass of the mixture /volume of the mixture

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    References

    • Secondary Physics. 2nd ed., Kenya Literature Bureau, 2011, https://doi.org/KLB10524 20m 2012. pp. 8-48.

  • The Archimedes’s Principle

    It states that:

    When a body is partially or totally immersed in a fluid, it experiences an upthrust equal to the weight of the fluid displaced.

    The law of flotation

    It is a special case of the Archimedes’s Principle which states that:

    A floating object displaces it’s own weight of the fluid in which it is floating.

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  • Describing materials

    Describing materials

    A material is selected for a particular use depending on its ability to withstand forces it may be subject to. A given material has some attributes that defines it’s characteristic. The characteristics of a given material determines its suitability for a certain work.

    Some of the important physical characteristics of materials includes:

    Strength

    This is the ability of a material to resist breakage when under stretching, compression or shearing force. A material with much strength can be able to withstand a large force without breaking.

    Stiffness

    This is the resistance a material offers to forces which tend to change its shape or size or both. Stiff materials are not flexible and resist bending.

    Ductility

    It is characteristic of a material that give it a structure that can lead to permanent change of size and shape.

    Ductile materials are material which will elongate considerably under stretching forces and undergo plastic deformation until they break .

    Examples of ductile materials may include:

    • Lead
    • Copper
    • Wrote iron
    • Plasticine

    Ductile materials can be rolled into sheets, drawn into wires or worked into other useful shapes without breaking. They are usually very useful in making things like staples, rivets and paper clips

    Brittleness

    It is the characteristics of a material that makes it tend to break just after the elastic limit is reached .

    Brittle materials are fragile and do not undergoes any noticeable extension on stretching but snap suddenly without warning.

    brittle materials can only absorb a limited amount of energy before breaking.

    Examples of brittle materials includes:

    • Bricks
    • Glass
    • cast iron board
    • Dry biscuits
    • ceramic
    • graphite
    • diamond
    • crystal
    • Porcelain
    • Tin-rich Bronze
    • Sodium Chloride
    Elasticity

    Elasticity is the ability of a material to recover it’s original shape and size after the force causing it’s deformation is removed. Materials that regain their shape after deformation under force are said to be elastic.

    Elastic materials recovers to their previous shape after enduring deformations like compression and expansion

    A material that does not recover but is permanently deformed is said to be plastic.examples of elastic materials includes:

    • Rubber
    • springs
    • wires
    • Trampoline
    • Rubber Bands
    • Elastin
    • Nylon
    • Lycra
    • Gum
    • Wool
    • Silicon
    • Polyester
    • Balloons

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  • Cause of Upthrust

    Cause of Upthrust

    Consider a cylindrical solid of cross-section area A which is totally immersed in a fluid of density ρ as shown.

    The pressure due to liquid column is usually given by P=ρgh.

    Pressure at the top of the solid will be given by, PT = h1ρg.

    Where h1 is the height of the liquid column above the top of the object.

    Pressure at the lower end of the object will be given by

    Pb=h2ρg where h2 is the height of the liquid above the lover surface of the cylinder .

    The pressure at the top of the cylinder will provide downward force exerted by the liquid up on the object.

    From the pressure laws, F=pressure P x Area A.

    i.e F=PA.

    Taking the area of the cylinder at the top, the force from the liquid acting on that surface is Given by F=PT x A=h1ρgA.

    Similarly, pressure at the bottom is given as F=PB x A=h2ρgA.

    The total resultant upwardward force F is this given as

    F=F2-F1

    Hence F=h2ρgA-h1ρgA

    Factoring out the common factors: F=ρgA (h2-h1)

    Let h be the difference between liquid column on top and the one at bottom h2 such that h=h2-h1

    Hence F=ρgAh

    But Volume is always given by V=Ah

    The resultant force F is the upthrust force U and will thus be expressed as.

    F=U=Aρpg=pgV

    where V is the volume of the liquid displaced.

    Mass of the liquid is usually given by density x volume. Hence mass m of liquid displaced will be given by m=Ahρ

    Weight is usually given as Weight W=mg

    Hence weight of liquid displaced will be W=U=Ahρg which represents the upthrust force we calculated earlier. This confirms the archimedes principle that upthrust force is equal to the weight of the fluid it displaces.

    From our mathematical arguments, it should be easy to see that Magnitude of the upthrust force is equal a function of volume of the object and density of the liquid considering that gravitational pull g is a constant.

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  • Effect of Magnetic field on current

    Effect of Magnetic field on current

    Professor Hans Oersted discovered that a conductor carrying current has magnetic field associated with it in 1820.

    This discovery has led scientists to always interfere magnetic field with electric fields causing development of many devices that has become very instrumental in human life. These devices includes electric motors, loud speakers, coil meter, circuit break, magnetic tapes, electric motors etc.

    To investigate magnetic effect on electric current, we consider two compasses , one placed on top of a current carrying wire and the one one placed under it as shown below.

    The compass is held on current conducting wire with the compass needle in line with the wire. And when the switch is closed , the compass north pole deflects as shown relative to direction of current for compass under wire.

    The deflection will be as shown when current is over the wire.

    When current is increased, by reduced resistance through a rheostat, the compass needle deflection increases.

    Suppose now we change the direction of the current by reversing the batter polarity, the direction of the deflection is observed to change to the opposite direction as shown

    When circuit is broken, the compass needle returns to it’s original position such that the north is in parallel with the wire showing no deflection.

    see the figure below:

    Magnetic needle position on wire with no current

    The above illustrations shows that magnetic deflection occurs only when there is current that is passing through the conductor.

    This simple experiments confirms that there is association between magnetic field and electric current and that the association is dependent on number of factors like.

    • Direction of current
    • Direction of magnetic field
    • Magnitude of the current flowing
    • Strength of the magnetic field

    Explanations

    A flow of current is basically a flow of charge. Flowing charges has magnetic fields associated with them which can be observed when a magnetic compass is made to interact with current conduting wire.

    Strength of magnetic field created by flowing current increases with increase of the flowing current which can be shown by greater deflection on a magnetic compass when current in wire is increased.

    Somebody called Ampere is the one that devised a rule called Ampere’s swimming rule that physicists can use to predict direction of compass deflection when it encounters a current flowing in a wire.

    The Ampere’s rule states that:

    If one imagines to be swimming along a wire in the direction of the current and facing the compass needle, then the north of the needle will be deflected towards the swimmers left hand.

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  • Phase and Phase Difference

    Phase and Phase Difference

    The word phase in normal usage means any stage in a series of events or in a process of development.

    cambridge University dictionary defines phase as one of the stages or points in a repeating process measured from a specific starting point.

    Two Waves can be of the same amplitude but with the different frequencies as shown in figure below.

    The wave profile P makes it’s one complete oscillations before wave Q. Wave P has shorter wavelength compared to Q and hence P has higher frequency.

    We can also see that P has smaller period as waves with shorter wavelength has smaller period.

    Wave P completes it’s first cycle at A while Q finishes it’s first oscillation at B. We can say that P is leading Q. The maximum displacement of the two waves is the same hence they are operating at the same amplitude but different frequency. The two waves are said to be out of phase.Think about two radio receivers tuned to two different stations but with equal volume.

    waves can also be of the same frequency but different amplitudes. Think of when we tune in our two radio receivers to the same station and then set them at different volumes

    The figure below illustrates two waves operating at same frequency but at different amplitudes.

    As can be seen from the diagram, one wave is having more displacement than the other one but they are arriving at the same horizontal position simultaneously. We say they are in phase.

    Pendulum bobs in phase

    To further illustrate the concept of phase and out of phase oscillations, consider two identical pendulums with bobs P and Q below.

    The two masses, P and Q are set in oscillation by giving them some displacement on the left and then releasing them simultaneously. Because they have equal displacement and released at the same time , they will pass through the lowest point Y at the same time as they move on to the opposite direction.. They attain displacement together at Z and swing back together to complete the oscillation at x.

    At any particular moment, the two masses will be moving in the same direction and at the same level of displacement in their oscillations.The masses are said to be oscillating in phase.

    Particles in a wave motion which happens to be oscillating in the same direction and at the same level of displacement in their oscillation are said to be in phase.

    The diagram below have highlighted two positions of particles A and B. The particles are in the same displacement level from the reference line and they are both facing the same direction as indicated by the arrows. The particles A and B are said to be in phase and their distance apart is the wavelength λ of the wave motion whereas time taken to move from A to B is the periodic time T.

    Particles in a wave motion can be in phase even if they have different amplitude.

    In our previous pendulum oscillation of mass P and Q ; If P is Initially given a larger displacement than Q, the two will oscillate i n phase even though P will always be at a larger magnitude of displacement than Q.

    A typical displacement time graph for two wave motions in phase with different amplitudes is shown below.

    two waves in phase at different amplitude.

    Oscillations out of phase

    Consider two masses P and Q displaced from opposite directions from each other as in figure below.

    When released simultaneously, they pass through the rest position at the same time as they move in the opposite direction and they reach a point of maximum displacement at the same time but their maximum displacement is in opposite direction to each other.

    180o out of phase

    The two objects above are always at the opposite levels of displacement and their oscillations opposite direction to each other and they are said to be in opposite phase.

    Wave motions that have same displacement and makes complete oscillations at the same time with their maximum displacements in exact opposite to each other are said to be in 180o phase difference (180o out of phase).

    The figure below shows two wave motions at 180o phase difference.

    90o out of phase

    suppose in our pendulum oscillations we displaces the objects P and Q to X ; we release Q before P and then we release p when Q is exactly at Y. The angle of oscillation between P and Q will be 90o in difference and the resulting oscillation will be 90o out of phase.

    The displacement time graph for waves 90o out of phase is illustrated below.

    two waves can be out of phase at any angle. We are likely to see that in our future lessons

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  • Calculating the wave speed

    From topics on linear motion, speed v is usually given as rate of change of distance with time. i.e. speed (v) =distance(s)/time(t)

    Speed of a wave is the distance covered by a wave in one second.

    The SI unit of a wave speed is metres per second.

    Consider a particle on a wave motion that gets displaced from point A to B such that distance AB makes one wavelength as shown

    The time taken to complete one oscillation is called the periodic time (T) and so the speed of the particle can be expressed in terms of wavelength (λ) and periodic time T as in the equation below.

    For the particle to cover a distance of exactly one wavelength, it means it has completed exactly one oscillation.

    The time taken to complete one oscillation is called the periodic time (T) and so the speed of the particle can be expressed in terms of wavelength (λ) and periodic time T as in the equation below.

    λ is a greek letter known as lambda which means wavelength in english.

    but T = 1/f. hence

    Multiplying both numerator and denominator by f, we get v=fλ.

    v=fλ is a relationship that helps determine speed of a wave given it’s frequency and wavelength.

    In a given media, the speed of a wave motion is constant and so frequency is inversely proportional to the the wavelength such that:

    The above equation shows that an increase in frequency results in decrease of wavelength.

    The relationship between frequency and wavelength is represented graphically as shown in figure below.

    If a wavelength covers λ distance, then the total distance covered by the wave profile can be determined as λ x number of oscillations and the total time taken as periodic time x number of oscillations..

    Example Question

    The figure below shows a wave form in a string.The wave is moving at 30.0ms-1 and the grid is in centimeters.

    Detrmine:

    (a) Amplitude

    (b) Wavelength

    (c) frequency

    (d) Period

    solution

    (a) amplitude is the largest displacement of the wave profile. It is the crest in the wave motion. In our diagram, the largest displacement is ±5. hence the amplitude of the wave is 5.0 cm

    (b) we can determine wavelength by determining distances between two points that are in phase. (40-0)cm or (50-10)cm or (60-20)cm. Hence the wavelength is 40cm.

    (c) v=fλ

    hence

    v= 30.0ms-1 and λ = 40 cm = 0.4m

    hence

    (d) Period T = 1/f

    Practice Question 1

    Waves on a spring are produced at the rate of 200 wavelengths every 10 seconds.

    (a) Find the frequency of the wave motion. (Answer: 20 Hz)

    (b) If the wavelength of the waves is 0.15m, find the speed of the waves (Answer: 3ms-1)

    (c) Find the period of the waves (Answer: 0.05s)

    Practice Question 2

    A water wave travels at 24m in 3 s. If source of the water waves vibrates at 5Hz, determine:

    (a) The speed of the wave (Answer: 8 ms-1)

    (b) The wavelength (Answer: 1.6m)

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  • Density of some substances

    Density of some substances

    The table below shows density of some common substances.

    Please note that density of water being 1.0gcm-3 can be used as a relative density to compare densities of other substances. For example , density of gold is 19.3 times that of water.

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  • Explaining upthrust force

    Explaining upthrust force

    Objects will weigh less in water than in air.

    Take a spring balance and hang some mass on it. Determine the weight of the mass and then push the mass up slightly with your hand. What have u observed?

    When you place some upward force on a mass hanging on the spring, it’s weight is seemed to reduce as observed by lesser leading of the spring balance.

    When you apply a force upward on the object hanging on a spring balance, you are providing some force that is acting opposite to the weight of the object. Weight is always acting downward on a straight line that is directed towards the center of the earth.

    When you push the object upward, you are reduce the overall resultant downward force by providing some force acting opposite to the weight.

    From the law of addition of forces, when two forces are acting in opposite direction on the same object, then one force is considered positive force and the other one taken as negative force . The total resultant force acting on the object is the algebraic sum of the forces acting on that object Consider the setup below that shows some weight acting on an object hanging freely on air.

    Spring balance measuring some weight

    We consider the force acting on the object which is it’s weight as W and any force applied upward as U as shown.

    Illustrating forces acting on an object hanging on air

    The resultant force will be given as W’=W-U. Where W’ represents the reduced weight.

    If U is greater than W’, then the object will accelerates upward, otherwise it will accelerates downward with reduced force.

    The downward acceleration force is balancing with tensional forces on the spring causing some extension, hence the object remains on the spring balance but causing it to extend in length.

    Upthrust Force

    When an object is immersed in a fluid, the upward forces on the object are provided by pressure in the fluid. That is why objects weighs less in water because some weight of the object is being cancelled out by the upward forces in water. This upward forces produced by fluid on an object is known as the upthrust force. It is the same force that causes object to float in water.

    Illustrating upthrust force,

    However, it is important to note that, for heavier objects falling in air, the upthrust by air is soo small such that it cannot be notices. We say that upthrust of air on an object is negligible.

    Paper and Stone

    If you release a piece of paper and a stone from some distance above the ground, you will notice that the stone reaches the ground faster than the paper. This is because upthrust force on paper is comparable to that of paper, because a piece of paper has very small weight. However, the stone weight is much more than the upthrust that can be provided by paper hence the total resultant downward forces is larger than that of paper hence causing more acceleration downward.

    Later on, we will see that upthrust fall is a characteristic of both volume of the object and density of the fluid.

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