PHYSICS FORM FIVE-CIRCULAR MOTION

PHYSICS FORM FIVE-CIRCULAR MOTION

UNAWEZA JIPATIA NOTES ZETU KWA KUCHANGIA KIASI KIDOGO KABISA:PIGA SIMU/WHATSAPP: 0787237719

ALSO READ;

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  2. A’ Level Study Notes All Subjects
  3. Pats Papers




PHYSICS FORM FIVE-CIRCULAR MOTION

CIRCULAR MOTION

Is the motion of the body around the circular track

There are two types of circular motion

(i)  Uniform circular motion

(ii)  Non-uniform circular motion

(i)UNIFORM CIRCULAR MOTION

This refers to the motion of a particle in circular path moves with a uniform speed.The word “uniform” refers to the constant speed.It means that in uniform circular motion,the object covers equal distances along the circumference in equal   intervals of time i.e. speed is constant.Although the magnitude of velocity (speed) remains constant, the direction of the velocity is changing continuously.Therefore, the object is undergoing acceleration. This is called centripetal acceleration and is directed radically towards the center of the circle as shown shown in the figure 1;

C:\thlb\cr\tz\__i__images__i__\motion.png

                                   Figure .1

PHYSICS FORM FIVE-CIRCULAR MOTION

(ii)NON UNIFORM CIRCULAR MOTION

This refers to the motion of the particle in circular path moves with a non uniform speed.

The speed of the particle in circular motion is different at different points along the circular path.

RELATIONSHIP BETWEEN LINEAR DISPLACEMENT AND ANGULAR DISPLACEMENT

Consider the particles of mass m moving around the circular track with angular speed ( C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image002.gif

) as illustrated on the figure 2

C:\thlb\cr\tz\__i__images__i__\wwp.png

Figure.2




From the figure above

O    =   Origin of the circle (circular track)

Q    =    Angular Displacement

S    =     Linear Displacement

R   =     Radius of the circular track

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image002.gif

   =     Angular velocity

Consider the particles moving from Q to P from the figure above

C:\thlb\cr\tz\__i__images__i__\ABOVE.png

Figure.3
From the figure above

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image005.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image006.gif

If C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image007.gif

 is very small

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image008.gif

Therefore

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image009.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image010.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image011.gif

Note

The angle C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image012.gif

 in radian is given by

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image013.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image014.gif


When C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image015.gif
 the length of the arc which is the circumference of the circle is C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image016.gif




PHYSICS FORM FIVE-CIRCULAR MOTION

TERMS RELATING TO CIRCULAR MOTION

Angular Displacement (C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image012.gif

)

Is the angle in radian traced at the center of the circle by the body moving around the circular track.

Or

Is the angle turned by an object moving along a circular path in a given time. Consider an object moving along a circular path with  center O as shown in figure 4.

C:\thlb\cr\tz\__i__images__i__\TZ1.jpg

Figure .4
Let us consider O as the origin of our coordinate system.

Suppose initially at t = O, the object is at point P.

At time t1, the object is at point P1 and its angular position is C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image012.gif

1

At time t2, the object is at point P2 and its angular position is C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image012.gif

2

During the time interval C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image018.gif

 the angular displacement is C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image019.gif




Angular Velocity ( C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image002.gif

)

Is the angular displacement traced per unit time.

Or

Is the rate of change of angular displacement of a body moving along a circular path

If the angular displacement of an object is C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image021.gif

  during the time interval
C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image022.gif
 then the angular velocity of the object is

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image023.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image024.gif

SI unit of angular displacement is rad/s

The angular velocity of an object moving along the circular path at any instant of time is called instantaneous angular velocity

It is denoted by C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image002.gif

.  It is  given by the limit of C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image026.gif
 as C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image027.gif
 approaches zero

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image023.gif

C:\thlb\cr\tz\__i__images__i__\nit58.png

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image029.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image030.gif

Angular Acceleration C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image031.gif

.

Is the rate of change of angular velocity

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image032.gif

If change in the angular velocity of an object is C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image033.gif

 during time interval
C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image022.gif
, the average angular velocity is given by

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image034.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image035.gif

The SI unit of angular acceleration is rad/s2

The angular acceleration of an object moving along a circular path at any instant of time is called instantaneous angular acceleration.

It is denoted byC:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image031.gif

.  It is given by the limit of C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image037.gif
  as C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image027.gif
 approaches zero.

∝ =   Δω/Δt

lim Δt → 0

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image039.gif

Time Period (T)

Is the time taken by the body moving along a circular path to complete one revolution.

It is denoted by T and its unit is second

For example, if an object completes 120 revolutions in 30 seconds, its time period is given by

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image040.gif

It means that the object will complete one cycle in 0.25s

Frequency (ƒ)

Is the number of revolution completed by the object moving along a circular track in one second.

It is denoted by f and its SI unit is s–1 or hertz (Hz)

Thus in the above case, the object completes 120 revolutions in 30 seconds. Therefore, the frequency of the object is

 

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image041.gif

It means the object will complete 4 revolutions in one second.




PHYSICS FORM FIVE-CIRCULAR MOTION

Relation between T and f

Suppose an object executing circular motion has frequency ƒ.

C:\thlb\cr\tz\__i__images__i__\nit59.png


It means that the object completes f revolutions in 1 second.

-Therefore, the time taken to complete one revolution is 1/f.

Relation between C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image044.gif

, f and T

When an object executing circular motion completes one revolution, angular displacement θ and time taken is T.

From

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image045.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image046.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image047.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image048.gif

     

Relation between V and C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image044.gif

As an object moves along the circumference of the circle, it has linear velocity V which is always Tangent to the circular path at every instant as shown in figure 5.

C:\thlb\cr\tz\__i__images__i__\nit62.png

Figure . 5

The relation between the linear velocity V and angular velocity w of the object can be found as under

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image050.gif

But

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image051.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image052.gif


C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image053.gif

But

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image030.gif


C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image054.gif

This is an important relationship between the circular motion of an object and the linear motion that results from rotation.




Relation between linear acceleration and angular acceleration.

The relation between linear acceleration C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image012.gif

 and angular acceleration C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image031.gif
can be found as follows.

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image055.gif


C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image056.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image054.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image057.gif


C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image058.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image039.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image059.gif

PHYSICS FORM FIVE-CIRCULAR MOTION

EQUATIONS OF UNIFORM MOTION AS APPLIED TO CIRCULAR MOTION.

Consider the particle of mass m moving around the circular track with uniform angular acceleration.

1)  To Derive C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image060.gif

From

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image039.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image061.gif

At C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image062.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image063.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image064.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image065.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image066.gif

2)  To derive C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image067.gif

From    C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image030.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image068.gif

At C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image069.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image070.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image071.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image072.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image067.gif

 

3)  To derive C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image073.gif

From

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image030.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image039.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image074.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image075.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image076.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image077.gif

When C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image078.gif





Centripetal Acceleration (ac)

Is the acceleration possessed by the body which is moving around the circular track and always directed towards the center.

It is also called radial acceleration because it always acts radially towards the center of the circle.

This acceleration must be in the same direction as it passes e towards the center of the circle.

For a body which is moving with constant angular velocity ω along a circular path of radius r, the magnitude of the centripetal acceleration to be given by

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image079.gif

If the linear speed of the particle is V, then the centripetal acceleration is given by

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image080.gif


It’s SI unit is m/s2

For non uniform circular motion, acceleration has two components, centripetal component and tangential component (at).
The magnitude of the resultant acceleration is

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image081.gif


C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image082.gif

To show that the centripetal

Acceleration = C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image083.gif

Consider a particle moving with constant speed V along an arc NOP as in figure 6.

C:\thlb\cr\tz\__i__images__i__\PAPAA1.jpg

Figure. 6

The x – component of velocity of the particle has the same value at P as at N and therefore its x-component of acceleration ax is zero

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image085.gif

As the particle moves from N to P its y-component of velocity changes by 2Vsinθ.

If this takes place in a time interval t its y – component of acceleration, ay is given by

C:\thlb\cr\tz\__i__images__i__\one78.PNG

             

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image086.gif

The speed of the particle along the arc is V, and therefore

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image087.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image088.gif

Sub equation (ii) into equation (i)

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image089.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image090.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image091.gif


If N and P are now taken to be coincident at O, then C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image012.gif
approaches to zero, and C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image092.gif
 has its limiting value of 1 in this case.

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image093.gif


Thus at O, ax = O and ay = C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image094.gif
 and therefore the acceleration is directed along Oz i.e. towards the center of the circle.

PHYSICS FORM FIVE-CIRCULAR MOTION

Alternately

C:\thlb\cr\tz\__i__images__i__\nit64.png


Figure  7

From the figure 7 above, the distance of displacement S arc is given by

S = r C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image012.gif

Differentiate by product rule

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image011.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image098.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image099.gif

But per body moving in circular path of common radius (Since radius is constant)

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image100.gif

Then

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image101.gif

Make C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image102.gif

 the subject

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image103.gif


From

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image104.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image105.gif




Differentiate by product rule

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image106.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image107.gif

At,   C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image108.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image109.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image110.gif

Substitute equation (ii) into equation (i)

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image111.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image112.gif

Divide by t on the C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image113.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image114.gif

———————————————————————— C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image115.gif

Since

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image116.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image117.gif


Also from

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image118.gif

When the body is moving around the circular track, velocity remains constant

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image119.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image120.gif

Substitute equation C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image121.gif

 into equation C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image122.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image123.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image124.gif


Equate the equation C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image125.gif
 and equation C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image126.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image127.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image128.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image129.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image130.gif




PHYSICS FORM FIVE-CIRCULAR MOTION

Centripetal Force (Fc)

Is the force possessed by the body moving around circular path and always directs the body towards the centre

OR

Is the force acting on a body moving along a circular path with uniform speed and is directed towards the centre of the circle.

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image131.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image132.gif


From

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image133.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image134.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image135.gif


C:\thlb\cr\tz\__i__images__i__\joe.png

If

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image137.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image138.gif

Also

C:\thlb\cr\tz\__i__images__i__\nit69.png

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image140.gif

More over

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image132.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image141.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image142.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image143.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image144.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image145.gif

From

C:\thlb\cr\tz\__i__images__i__\nit70.png

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image146.gif

But

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image142.gif

C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image147.gif

No work is done by the centripetal Force

Work done C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image148.gif

 F.S Cos C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image012.gif

= C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image149.gif

C:\thlb\cr\tz\__i__images__i__\let91.jpg




Some Common Examples of Centripetal Force

(i)  In the case of planets orbiting  around the sun the centripetal force is provided by the gravitational force of attraction between the planets and the sun.

(ii)  In the case of an electron moving around the nucleus of the atom, the centripetal force is provided by the electrostatic force of attraction between the electron and proton.

(iii)  When a particle tied to a string and whirled in a horizontal circle then the tension in the string provides the centripetal  force.

(iv)  When charged particle describing a circular path in a magnetic field, then magnetic force exerted on a charged particle that set up  the centripetal force .

(v)    When a vehicle moves in a circular path on a level road the force of lateral friction between the wheels and the road provides the centripetal force.

Application of Centripetal force in Every Day Life

(a)   In separating honey from bees wax

(b)  In separating cream from milk

(c)   In separating sugar crystals from molasses

(d)  In spin drier machines, water particles fly off tangentially through holes in the wall of the machine.

PHYSICS FORM FIVE-CIRCULAR MOTION

Centrifugal Force (Fictitious Force)

Is the force which does not really act on a body but appears due to the acceleration of the frame

In order to move a body in a circular path, a centripetal force C:\thlb\cr\tz\UNIFORMCIRCULARMOTION_files\image156.gif

 is required.  This force acts along the radius towards the centre of the circle.

The reaction of this centripetal force is the centrifugal force.

Both these forces are equal in magnitude but opposite in direction and act on different bodies.

Force example, consider the case of a stone tied at one end and rotated in a circle as shown below.

C:\thlb\cr\tz\__i__images__i__\nit71.png

Figure 8

Note that the centripetal force F1 is applied on the stone by the hand and acts towards the center.

The centrifugal force F2 acts on hand and pulls it away from the center of the circle.

Centrifugal means center – Fleeing or away from the center.

Therefore, the outward force in circular motion is called centrifugal force.

The magnitude of centrifugal force is the same as that of the centripetal force and its direction is opposite to that of the centripetal force.

C:\thlb\cr\tz\__i__images__i__\angel1.jpg



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