Uniform Circular Motion

When the velocity of an object changes, we say that the object is accelerating. The change in the velocity could be due to change in its magnitude or the direction of the motion or both. Can you think of an example when an object does not change its magnitude of velocity but only its direction of motion?

(a) Rectangular track                   

 (b) Hexagonal track

(c) Octagonal shaped track                   

(d) A circular track

Fig.9 The motion of an athlete along closed tracks of different shapes.

Let us consider an example of the motion of a body along a closed path. Fig 9 (a) shows the path of an athlete along a rectangular track ABCD. Let us assume that the athlete runs at a uniform speed on the straight parts AB, BC, CD and DA of the track. In order to keep himself on track, he quickly changes his speed at the corners. How many times will the athlete have to change his direction of motion, while he completes one round? It is clear that to move in a rectangular track once, he has to change his direction of motion four times.

Now, suppose instead of a rectangular track, the athlete is running along a hexagonal shaped path ABCDEF, as shown in Fig. 9(b). In this situation, the athlete will have to change his direction six times while he completes one round. What if the track was not a hexagon but a regular octagon, with eight equal sides as shown by ABCDEFGH in Fig. 9(c)? It is observed that as the number of sides of the track increases the athelete has to take turns more and more often. What would happen to the shape of the track as we go on increasing the number of sides indefinitely? If you do this you will notice that the shape of the track approaches the shape of a circle and the length of each of the sides will decrease to a point. If the athlete moves with a velocity of constant magnitude along the circular path, the only change in his velocity is due to the change in the direction of motion. The motion of the athlete moving along a circular path is, therefore, an example of an accelerated motion.

We know that the circumference of a circle of radius r is given by 2πr . If the athlete takes t seconds to go once around the circular path of radius r, the velocity v is given by

v = 2πr ⁄ t

When an object moves in a circular path with uniform speed, its motion is called uniform circular motion.

Graphical Representation of Motion

Distance-Time Graphs

The change in the position of an object with time can be represented on the  distance-time graph adopting convenient scale of choice. In this graph fig2 ,time is taken along the x-axis and distance is taken along the y-axis. Distance-time graphs can be employed under various conditions where objects move with uniform  speed , non-uniform speed, remain at rest etc.

We know that when an object travels equal distance in equal interval of time, it moves with uniform speed. This shows that the distance travelled by the objects is directly proportional to the time taken. Thus , for uniform speed, a graph of di

stance travelled against time is a straight line , as shown in fig2. The portion OB of the graph shows that the distance is increasing at a uniform rate.

We can use the distance-time graph to determine the speed of an object. To do so, consider a small part AB of the distance –time graph shown in fig 2. Draw a line parallel to the x-axis from point A and another line parallel to the y –axis from the point B. These two lines meets each other at point C to form a triangle ABC.Now,on the graph,AC denotes the time interval(t₂-t₁) while BC corresponds to the distance(s2-s1). We can see from the graph that as the object moves from the point A to B, it cover a distance(s2-s1) in the (t2-t1). The speed,v of the object, therefore  can represented as

We can also plot the distance-time graph for accelerated motion.

The nature of this graph shows non-linear variation of the distance travelled by the car with time.

Velocity-Time Graphs

The  variation in velocity with  time for an object moving  can be represented by a velocity-time graph.

When an object is moving with a constant velocity, the line on the graph is horizontal. When an object is moving with a steadily increasing velocity, or a steadily decreasing velocity, the line on the graph is straight, but sloped. The diagram shows some typical lines on a velocity-time graph.

Motion

                                                  

To describe the position of an object we need to specify a reference point called the origin.

For example ,Let us assume that a hospital in a village is 2 km north of the railway station.  We 

have specified the position of hospital with respect to the railway station. In this example , the railway station is  the reference point(origin). We could have also chosen other reference points according to our convenience.

Difference Between Distance and Displacement.

Distance : Suppose a car start its journey from O which is treated as its reference point(origin).         

                                                       Fig 1

From O  it goes to B and back to C.

Total Distance cover by car =OB+BC

=35+10=45KM

So What is Displacement.

Displacement = OC

=25KM

Displacement is the shortest distance measured from the initial position (O) to the final position(C).      

The numerical value of a physical quantity is its magnitude.

Can the magnitude of the displacement be equal to the distance travelled by an object?

Consider the example given in fig 1. If car move from O to C. Then distance cover is 25KM and

Displacement is also 25KM.Durning it motion from O to B and B to C,

The distance cover is OB+BC=35+10=45KM.While displacement is 25KM.

Thus displacement is not equal to path length in this case.

Further ,we will notice that the magnitude of the displacement for a couse of motion may be zero

But the corresponding distance covered is not zero. If our car move from O to A and come back to O.

Than distance travelled is OA+AO=60+60=120KM.But displacement is 0 because distance between final and initial position is 0.

Uniform Motion And Non Uniform Motion

Uniform Motion:-  When an object cover equal distance in equal interval of time .it called to have uniform motion. For example , if a car cover 40KM in1st Hours ,40KM in 2^nd hour and so on .It said to have uniform motion.

Non uniform Motion:-  If the car travelled 40KM in 1^st  Hours ,60 Km in 2^nd  Hours ,70KM in 3^rd Hour so on. Then it said to have non uniform motion.

Speed:- 

It is distance travelled by an object in unit time. To specify the speed of an object, we require only its magnitude. The speed of an object need not be constant. In most cases ,objects will be in non uniform motion. Therefore , we describe the rate of motion of such object in term of their average speed. The average speed of an object is obtained by dividing the total distance travelled by the total time taken. That is,

Average speed  v =d/t

Where  d= Total distance Travelled

t = Total time taken

The SI unit of speed is metre per second. This is represented by the symbol  ms-1 or m/s.

The other units of speed include centimetre per second(cm/s) and kilometre per hour(km/h).

Velocity

Velocity is the speed of an object moving in a definite direction. The velocity of an object can be uniform or variable.it can be changed by changing the object’s speed, direction of motion or both.

When an object is moving along a straight line at a variable speed, we can express the magnitude of its rate of motion in term of average velocity.It is calculated in the same way as we calculate average speed.

In case the velocity of the object is changing at a uniform rate ,then average velocity is given by the arithmetic mean of initial velocity and final velocity for a given period of time.

Where v= average velocity

             U= initial velocity,

              f= final velocity

Also

ACCELERATION :-RATE OF CHANGE OF VELOCITY

Aceelaration is a measure of the change in the velocity of an object per unit time. That is

               

Acceleration =  

                         

If the velocity of an object changes from an initial value u to the value v in time t,

The acceleration a is

The acceleration is taken to be positive if it is in the direction of velocity and negative when it is  opposite  to the direction of velocity. The SI unit of acceleration is ms^-2.

  If an object travel in a straight line and its velocity increases or decreases by equal amounts in equal internals of time ,then the acceleration of the object is said to be uniform. the  motion of a freely falling body is an example of uniformly accelerated motion. On the other hand, an object can travel with non-uniform acceleration if its velocity changes  at a  non-uniform rate. For example, if a car travelling along a straight road increases its speed by unequal amounts in equal interval of time,then the car is said to be moving with non-uniform acceleration.