In the previous section, we completed the discussion on the motion of objects along a straight line. In this section we will discuss the motion of an object along a circular path.
We saw that velocity has both magnitude and direction. For example, if an object is moving with a velocity 12 m s-1, it must be along a straight line path. Because if there are any bends or curves in that path, the direction of motion will change. Then the velocity will also change.
So velocity can change because of any one of the following three reasons:
• Changing speed keeping direction the same
♦ An example: An object was travelling with a velocity of 20 m/s, at an angle of 25o with the x-axis. After some time, the speed changed to 15 m/s. In this case, the velocity has changed even if the object continues to travel in the same direction. The new velocity should be specified by the new speed and the same direction
• Changing direction keeping speed the same
♦ An example: A car was travelling with a velocity of 30 Kmph in the north east direction. After some time, the car changed course. In this case, the velocity has changed even if the car continues to travel with the same speed. The new velocity should be specified by the speed 30 kmph and new direction.
• Changing both speed and direction.
We saw the above details here. Now we are going to discuss about a situation in which changes in velocity occur due to the change in direction only.
1. Consider an athlete running along a closed path ABCD shown in fig.1.30(a).
2. The path is square in shape. He is running at constant speed. Let us check if he is running at constant velocity:
3. Consider his motion along AB. When he reaches B, he will have to change his direction. Otherwise, he will not be able to keep himself within the track.
4. Because of this change in direction, his velocity will change at B. This will happen at C, D and A also. So when he completes one lap, his direction will have changed 4 times.
5. Consider a path of another shape. This time, a hexagonal path shown in fig.b. Here also the there are changes in direction. The change happen 6 times. They are: at A, B, C, D, E and F
6. What if the path is octagonal as shown in the fig.c?
Ans: The change happens 8 times. They are: at A, B, C, D, E, F, G and H
7. So we see that, when the 'number of sides' n of the path increase, the 'number of times that the athlete has to change his direction' will also increase. This is shown in fig.1.31 below:
8. What if the number of sides of the path is very large?
Ans: Then the length of the sides will become very less. So less that the sides will be just ‘points’. And the path will appear as a circle. In maths we use this principle to obtain the general formula for the perimeter of a circle. Details here.
9. So, if the athlete is moving with constant speed along a circular path, his velocity is changing at every point of the track. That means, his velocity is changing continuously.
10. We saw such a 'continuous change in velocity' earlier when we discussed ‘motion along a straight line’. There, the velocity change was due to the change in speed. There was no change in direction. The change in speed was due to an application of acceleration. So we can write this:
• In straight line motion, if there is a change in magnitude of the velocity, that velocity has changed.
♦ The change occurred due to the acceleration. This acceleration causes a continuous change in velocity.
• In circular motion, even if the magnitude of the velocity remains the same, the direction continuously changes. So the velocity continuously changes. Thus 'motion along a circle' is also an accelerated motion.
11. We know that the perimeter of a circle is 2πr. Where r is the radius of the circle.
So, if the athlete requires t seconds to complete one path, then:
Speed of the athlete = distance⁄time = 2πr⁄t .
1. Consider a small stone tied to a string. Hold one end of the string and move the stone in a circular path. This is shown in fig.1.32 (a) below:
2. Let the stone move at a constant speed. After some time, let the stone go by releasing the string. In what direction will the stone move?
Ans: When the string is released the stone will move in a straight line path. This straight line path is tangential to the circle. The point of tangency is the position of the stone at the instant when the string is released.
3. Let us analyse the above answer:
■ In the fig.1.32 (b), the string is released when the stone reaches B
(i) Then, instead of the circular path, the stone will begin to travel in a straight line path.
(ii) We want to know this path. For that, draw a tangent line at B. In the fig., this tangent is named as AC
(iii) The movement of the stone after the release will be along the line AC. The direction is shown by the arrow at A
■ Another point of release is also shown in the fig.1.32 (b). It is at Q. If it is released at Q:
(i) Then, instead of the circular path, the stone will begin to travel in a straight line path.
(ii) We want to know this path. For that, draw a tangent line at Q. In the fig., this tangent is named as PQ
(iii) The movement of the stone after the release will be along the line PQ. The direction is shown by the arrow at P
• In higher classes we will learn to draw tangent at any given point on a circle.
So we have completed the discussion on the 'Motion of objects'. In the next chapter, we will see some more solved examples.
We saw that velocity has both magnitude and direction. For example, if an object is moving with a velocity 12 m s-1, it must be along a straight line path. Because if there are any bends or curves in that path, the direction of motion will change. Then the velocity will also change.
So velocity can change because of any one of the following three reasons:
• Changing speed keeping direction the same
♦ An example: An object was travelling with a velocity of 20 m/s, at an angle of 25o with the x-axis. After some time, the speed changed to 15 m/s. In this case, the velocity has changed even if the object continues to travel in the same direction. The new velocity should be specified by the new speed and the same direction
• Changing direction keeping speed the same
♦ An example: A car was travelling with a velocity of 30 Kmph in the north east direction. After some time, the car changed course. In this case, the velocity has changed even if the car continues to travel with the same speed. The new velocity should be specified by the speed 30 kmph and new direction.
• Changing both speed and direction.
We saw the above details here. Now we are going to discuss about a situation in which changes in velocity occur due to the change in direction only.
1. Consider an athlete running along a closed path ABCD shown in fig.1.30(a).
 |
| Fig.1.30 |
3. Consider his motion along AB. When he reaches B, he will have to change his direction. Otherwise, he will not be able to keep himself within the track.
4. Because of this change in direction, his velocity will change at B. This will happen at C, D and A also. So when he completes one lap, his direction will have changed 4 times.
5. Consider a path of another shape. This time, a hexagonal path shown in fig.b. Here also the there are changes in direction. The change happen 6 times. They are: at A, B, C, D, E and F
6. What if the path is octagonal as shown in the fig.c?
Ans: The change happens 8 times. They are: at A, B, C, D, E, F, G and H
7. So we see that, when the 'number of sides' n of the path increase, the 'number of times that the athlete has to change his direction' will also increase. This is shown in fig.1.31 below:
 |
| Fig.1.31 |
Ans: Then the length of the sides will become very less. So less that the sides will be just ‘points’. And the path will appear as a circle. In maths we use this principle to obtain the general formula for the perimeter of a circle. Details here.
9. So, if the athlete is moving with constant speed along a circular path, his velocity is changing at every point of the track. That means, his velocity is changing continuously.
10. We saw such a 'continuous change in velocity' earlier when we discussed ‘motion along a straight line’. There, the velocity change was due to the change in speed. There was no change in direction. The change in speed was due to an application of acceleration. So we can write this:
• In straight line motion, if there is a change in magnitude of the velocity, that velocity has changed.
♦ The change occurred due to the acceleration. This acceleration causes a continuous change in velocity.
• In circular motion, even if the magnitude of the velocity remains the same, the direction continuously changes. So the velocity continuously changes. Thus 'motion along a circle' is also an accelerated motion.
11. We know that the perimeter of a circle is 2πr. Where r is the radius of the circle.
So, if the athlete requires t seconds to complete one path, then:
Speed of the athlete = distance⁄time = 2πr⁄t .
1. Consider a small stone tied to a string. Hold one end of the string and move the stone in a circular path. This is shown in fig.1.32 (a) below:
 |
| Fig.1.32 |
Ans: When the string is released the stone will move in a straight line path. This straight line path is tangential to the circle. The point of tangency is the position of the stone at the instant when the string is released.
3. Let us analyse the above answer:
■ In the fig.1.32 (b), the string is released when the stone reaches B
(i) Then, instead of the circular path, the stone will begin to travel in a straight line path.
(ii) We want to know this path. For that, draw a tangent line at B. In the fig., this tangent is named as AC
(iii) The movement of the stone after the release will be along the line AC. The direction is shown by the arrow at A
■ Another point of release is also shown in the fig.1.32 (b). It is at Q. If it is released at Q:
(i) Then, instead of the circular path, the stone will begin to travel in a straight line path.
(ii) We want to know this path. For that, draw a tangent line at Q. In the fig., this tangent is named as PQ
(iii) The movement of the stone after the release will be along the line PQ. The direction is shown by the arrow at P
• In higher classes we will learn to draw tangent at any given point on a circle.
So we have completed the discussion on the 'Motion of objects'. In the next chapter, we will see some more solved examples.
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