Chapter 07 Motion in a Circle

CAMBRIDGE A – LEVEL

PHYSICS

MOTION IN A

CIRCLE

LEARNING OUTCOMES

No. LEARNING OUTCOME

i Understand the concept of circular motion

ii Learn the kinematics of circular motion

iii Interpret situations where the circular motion is

classified as uniform

iv Apply Newton’s 2nd Law to understand the dynamics

of uniform circular motion

v Understand the significance of the centripetal force to

uniform circular motion

C O N C E P T O F C I R C U L A R

M O T I O N

C O N C E P T O F C I R C U L A R

M O T I O N

• An object is in circular motion when it is

rotating about a stationary point

(centre) at a fixed distance (radius)from

it.

K I N E M AT I C S O F C I R C U L A R

M O T I O N


M O T I O N

• The diagram shows an

object, P undergoing

circular motion.

• The point O is the

centre, the fixed point.

• The angle through

which it has rotated =

�.

Figure 9.9, page 286: Chapter 9: ROTATION OF RIGID BODIES; SEARS AND ZEMANSKY’S

UNIVERSITY PHYSICS WITH MODERN PHYSICS; Young, Hugh D. and Freedman, Roger A.,

Addison Wesley, San Francisco, 2012.

• An object undergoing circular motion

will have two displacements:

1. �, the angular displacement , that is

measured in radians, (could be clockwise

or anticlockwise), and

2. �, the arc length, that has the same units

as distance (m, km, etc.)


M O T I O N : D I S P L A C E M E N T



� � �

• Geometrically, the angular displacement,

�, the arc length, � and the radius, � of

the circular path are related by the

equation � �

�





Figure 17.3 (a), page 260, Chapter 17: CIRCULAR

MOTION; Cambridge International AS and A Level

Physics Coursebook, Sang, Jones, Chadha and

Woodside, 2nd edition, Cambridge University Press,

Cambridge, UK,2014.

• Definition: “1 radian is the angle

subtended at the centre of the circle by

an arc equal in length to the radius of

the circle.”

Figures 17.4 , page 260, Chapter 17: CIRCULAR

MOTION; Cambridge International AS and A Level

Physics Coursebook, Sang, Jones, Chadha and

Woodside, 2nd edition, Cambridge University Press,

Cambridge, UK,2014.





• We also need to measure how fast the

�

• We also need to measure how fast theobject is moving.

• An object that is undergoing circularmotion has two speeds.

• The two speeds are:1. a linear speed, (in m s-1), that measures

how fast the object is moving, and

2. an angular speed, � (in rad s-1) thatmeasures how fast the object is rotating.


M O T I O N : V E L O C I T Y







Questions 5 and 6, page 261, Chapter 17: CIRCULAR MOTION; Cambridge

International AS and A Level Physics Coursebook, Sang, Jones, Chadha and

Woodside, 2nd edition, Cambridge University Press, Cambridge, UK,2014.

�

� ��

• The linear speed, and the angular

speed, � can be related mathematically.

• Recall that � � ��.

• By differentiating both sides with

respect to � , we obtain��

��

��

��or

simply � ��.






M O T I O N


M O T I O NQuestions 7,8,

and 9, page 262,

Chapter 17:

CIRCULAR

MOTION;

Cambridge

International AS

and A Level

Physics

Coursebook,

Sang, Jones,

Chadha and

Woodside, 2nd

edition,

Cambridge

University Press,

Cambridge,

UK,2014.

•What about the acceleration?

• An object in circular motion

has two accelerations:

1. angular acceleration, , and

2. linear acceleration,


M O T I O N : A C C E L E R AT I O N



• The angular acceleration,

measures the rate at which the

angular speed, is changing.

• The angular acceleration,

in this syllabus. Hence, is

constant.





• The linear acceleration, is made

�

• The linear acceleration, is madeup of two mutuallyperpendicular components:

1. the tangential acceleration,

��, and

2. The centripetal acceleration,

�.





• The tangential acceleration, ��

��

• The tangential acceleration, ��determines the change in themagnitude of the linear velocity:i. If �� > 0, the linear speed

increases.

ii. If �� < 0, the linear speeddecreases.

iii. If �� = 0, the linear speed isconstant.





�

• The centripetal acceleration,

� changes the direction of

the linear velocity vector.

• For circular motion to occur,

� must exist!





• How do we calculate the values• How do we calculate the valuesof the tangential and centripetalaccelerations?

• At A Level, �� . This meansthat the linear speed of theobject in circular motion isconstant. This is called uniformcircular motion.





For uniform circular motion, the linear

� ��

� ��

�

• For uniform circular motion, the linear speed, is constant.

• We introduce a new quantity called the period, �, which is the time taken to make one complete revolution.

• We now have � ��

�since

� ��

�

U N I F O R M C I R C U L A R

M O T I O N

U N I F O R M C I R C U L A R

M O T I O N

��• What about ��?

• We will work on deriving an expression for it

now by using the diagram below.





r

r

Figure 3.28, page 86: Chapter 3:

MOTION IN TWO OR THREE

DIMENSIONS; SEARS AND

ZEMANSKY’S UNIVERSITY PHYSICS

WITH MODERN PHYSICS; Young,

Hugh D. and Freedman, Roger A.,

Addison Wesley, San Francisco,

2012.

∆� ∆�

∆� ∆� �

• Since both triangles are similar, ∆�

�

∆�

�.

• Dividing both sides of the equation

by , we obtain �

�

∆�

∆�

�

�

∆�

∆�. Note

that ∆�

∆�and

∆�

∆� �.





�

�

�

� � � �

�• We now get

�

�

�

� � or � �

�

• Since , we can also obtain

� �

�

��

�

�

• The two equations we have derived for � calculates its magnitude. The direction will always be towards the centre of the circular path.





DY N A M I C S O F C I R C U L A R

M OT I O N

DY N A M I C S O F C I R C U L A R

M OT I O N

Recalling Newton’s 2nd Law, the• Recalling Newton’s 2nd Law, theexistence and direction of thecentripetal acceleration tells us:

1. That there is a resultant forcetowards the centre of the circularpath, and

2. How to calculate the magnitude ofthis resultant force.

CENTRIPETAL FORCE

�

• This resultant force towards the

centre of the circular path is known

as the centripetal force.

• Its value can be calculated by using

the equation � � �

��

CENTRIPETAL FORCE

• What we need to understand is that

the centripetal force is a resultant

force, not an actual physical force.

• It is a good idea for us to identify the

forces that provide the centripetal

force.

EXAMPLESEXAMPLESQuestions 10 and

11, page 263,

Chapter 17:

CIRCULAR MOTION;

Cambridge

International AS and

A Level Physics

Coursebook, Sang,

Jones, Chadha and

Woodside, 2nd

edition, Cambridge

University Press,

Cambridge,

UK,2014.

EXAMPLESEXAMPLESQuestions 14 and

15, page 265,

Chapter 17:

CIRCULAR MOTION;

Cambridge

International AS and

A Level Physics

Coursebook, Sang,

Jones, Chadha and

Woodside, 2nd

edition, Cambridge

University Press,

Cambridge,

UK,2014.

EXAMPLESEXAMPLESMay/Jun 2008, Paper 4, question 1.

EXAMPLESEXAMPLESMay/Jun 2008, Paper 4, question 1 (cont’d).




H O M E W O R K

1. May/Jun 2010, Paper 41, question 1.

Documents

Chapter 07 Motion in a Circle