UNIT – II- KINEMATICS�CHAPTER – 4�MOTION IN A PLANE

PREPARED

BY

K.SANKAR (PGT PHYSICS)

JNV KARAIKAL

TOPIC TO BE DISCUSSED

• Introduction
• Scalar and vector quantities
• Vectors and their notations; equality of vectors, multiplication of vectors by a real number; addition and subtraction of vectors, relative velocity, Unit vector;
• Resolution of a vector in a plane-rectangular components,
• Scalar and Vector product of vectors.
• Motion in a plane, cases of uniform velocity and uniform acceleration
• Projectile motion
• Uniform circular motion.

INTRODUCTION

• To describe motion of an object in two dimensions (a plane) or three dimensions (space), we need to use vectors.
• Then we discuss motion of an object in a plane. As a simple case of motion in a plane, we shall discuss motion with constant acceleration and treat it in detail the projectile motion.
• Circular motion which is a familiar class of motion that has a special significance in daily-life situations.

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SCALARS AND VECTORS QUANTITIES

• Scalar Quantities: The physical quantities which are completely specified by their magnitude or size alone are called scalar quantities.�Examples: Length, mass, density, speed, work, etc.

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• Vector Quantities: The physical quantities which are characterised by both magnitude and direction.�Examples: Velocity, displacement, acceleration, force, momentum, torque etc.�

CHARACTERISTICS OF VECTORS

1. It possess both magnitude and direction.
2. It does not obey the ordinary laws of Algebra.
3. It changes either magnitude or direction or both magnitude and direction.
4. The vectors are represented by bold letters or letters having arrow over them.�

POSITION AND DISPLACEMENT VECTORS

(a) Position (OP or OP’) and displacement (PP’)vectors.

(b) Displacement vector PQ and different courses of motion. The displacement vector is the same PQ for different paths of journey, say PABCQ, PDQ, and PBEFQ.

EQUALITY OF VECTORS

• Two vectors A and B are said to be equal if, they have the same magnitude and the same direction.

1. Two equal vectors A and B.
2. Two vectors A′ and B′ are unequal though they are of the same length.

MULTIPLICATION OF VECTORS BY REAL NUMBERS

• Multiplying a vector A with a positive number λ gives a vector whose magnitude is changed by the factor λ but the direction is same as that of A.
• Note:

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ADDITION OF VECTORS — GRAPHICAL METHOD

1. Vectors A and B. (b) Vectors A and B added graphically.

(c) Vectors B and A added graphically.

In this procedure of vector addition, vectors are arranged head to tail, this graphical method is called the head-to-tail method.

PROPERTIES OF VECTOR ADDITION

• vector addition obey Commutative law.

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• vector addition also obeys Associative

law.

TYPES OF VECTOR ADDITION

• Triangle Law

If two sides of a triangle completely represent two vectors both in magnitude and direction taken in same order, then the third side taken in opposite order represents the resultant of the two vectors both in magnitude and direction.

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TYPES OF VECTOR ADDITION

• Parallelogram Law

If two vectors are represented both in magnitude and direction by the adjacent sides of a parallelogram drawn from a point, then the resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram passing through the same point.

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TYPES OF VECTOR ADDITION

• Polygon Law

If a number of vectors are represented both in magnitude and direction by the sides of a polygon taken in the same order, then the resultant vector is represented both in magnitude and direction by closing side of the polygon taken in the opposite order.

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SUBTRACTION OF VECTORS

• The difference of two vectors A and B as the sum of two vectors A and –B :

1. Two vectors A and B, – B is also shown.
2. Subtracting vector B from vector A – the result is R2. For comparison, addition of vectors A and B, i.e. R1 is also shown.

ZERO OR NULL VECTOR

• If the magnitude of the two vectors are same, but opposite in direction, then the resultant vector has zero magnitude wthout any specific directionand is called a null vector or a zero vector.

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• The main properties of 0 are :

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TRIANGLE LAW OF VECTOR ADDITION

• If two sides of a triangle completely represent two vectors both in magnitude and direction taken in same order, then the third side taken in opposite order represents the resultant of the two vectors both in magnitude and direction.

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In ∆ O C M,

O C ² = O M² + C M²

O C ² = (OA + A M)² + C M²

OC ² = OA² + 2 OA × AM + AM² + C M²

OC² = OA² + 2OA × AM + AC²

In ∆CAM,

cos θ = AM/AC

AM = AC cos θ

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OC² = OA² + 2OA × AC cos θ+ AC²

R² = P² + 2P × Q cos θ + Q²

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R² =P² + 2 PQ cos θ + Q²

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R=(P² + 2PQ cos θ + Q²) ^{1 /2}

OC² = OA² + 2OA × AM + AC²

DIRECTION OF RESULTANT VECTOR

In triangle CAM;

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sin θ = C M /AC

cos θ = AM / AC

In triangle OCM;

tan α = C M/ OM

= C M/ (OA + AM)

= AC sin θ / (P + A C c o s θ)

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tan α = Q sin θ / (P + Q cos θ)

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PARALLELOGRAM LAW OF VECTOR ADDITION

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If two vectors are represented in direction and magnitude by two adjacent sides of parallelogram then the resultant vector is given in magnitude and direction by the diagonal of the parallelogram starting from the common point of the adjacent sides.

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R = A + B

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The diagram above shows two vectors A and B with angle p between them.

R is the resultant of A and B

R = A + B

This is the resultant in vector

R is the magnitude of vector R

Similarly A and B are the magnitudes of vectors A and B

R = √(A² + B² +2ABCos p) or

[A² + B² +2ABCos p]^1/2

To give the direction of R we find the angle q that R makes with B

Tan q = (A Sin p)/(B + A Cos q)

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UNIT VECTORS

• A unit vector is a vector of unit magnitude and points in a particular direction.
• It has no dimension and unit.
• It is used to specify a direction only.

Unit vectors along the x-, y-andz-axes of a rectangular coordinate system are denoted by iˆ , jˆ and kˆ respectively

In general, a vector A can be written as

where n is a unit vector along A.

RESOLUTION OF A VECTOR IN A PLANE�(Rectangular Components of vectors)

• Consider a vector A that lies in x-y plane as shown in Fig.
• From fig, we have
• where Ax and Ay are real numbers. Thus
• The quantities Ax and Ay are called x-, and y-

components of the vector A.

• Using simple trigonometry, we can express Ax and Ay in terms of the magnitude of A and the angle θ it makes with the x-axis :

• its magnitude A and the direction θ it makes with the x-axis;

RESOLUTION OF A VECTOR IN �3-DIMENSION

• If α, β, and γ are the angles between A

and the x-, y-, and z-axes, respectively,

we have

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• where x, y, and z are the components of r along x-, y-, z-axes, respectively.

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SCALAR OR DOT PRODUCT OF VECTORS

• Scalar product of two vectors is defined as the product of the magnitude of two vectors with cosine of smaller angle between them.

VECTOR OR CROSS PRODUCT OF VECTORS

Vector product of two vectors is defined as the product of the magnitude of two vectors with sine of smaller angle between them with unit vector perpendicular to those two vectors.

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• θ is angle between A and B taken in anti-clockwise direction.
• is unit vector in the direction perpendicular to the plane containing A and B.

Note: In Chapter-7, vector product will be discussed in detail.

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POSITION VECTOR AND DISPLACEMENT

• The position vector r of a particle P located in a plane with reference to the origin of an x-y reference frame is given by
• where x and y are components of r along x-, and y- axes or simply they are the coordinates of the object.

• Suppose a particle moves along the curve shown by the thick line and is at P at time t and P′ at time t′ . Then, the displacement is :

and is directed from P to P′ .

VELOCITY

• The average velocity (v) of an object is the ratio of the displacement and the corresponding time interval :

The velocity (instantaneous velocity) is given by the limiting value of the average velocity as the time interval approaches zero :

As the time interval Δt approaches zero, the average velocity approaches the velocity v. The direction

of v is parallel to the line tangent to the path.

ACCELERATION

• The average acceleration a of an object for a time interval Δt moving in x-y plane is the change in velocity divided by the time interval :

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• The acceleration (instantaneous acceleration) is the limiting value of the average acceleration as the time interval approaches zero :

The average acceleration for three time intervals (a) Δt1, (b) Δt2, and (c) Δt3, (Δt1> Δt2> Δt3). (d) In the

limit Δt 0, the average acceleration becomes the acceleration.

MOTION IN A PLANE WITH CONSTANT�ACCELERATION

Expression for velocity in a plane

• Suppose that an object is moving in x-y plane

and its acceleration a is constant. Now, let the velocity of the object be v0 at time t = 0 and v at time t. Then, by definition

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• In terms of components :

Expression for Displacement in a plane

• Let ro and r be the position vectors of the particle at time 0 and t and let the velocities at these instants be vo and v. The displacement is the average velocity multiplied by the time interval :

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• In component form as,

Note: Motion in a plane (two-dimensions) can be treated as two separate simultaneous

one-dimensional motions with constant acceleration along two perpendicular

directions.

RELATIVE VELOCITY IN TWO�DIMENSIONS

• Suppose that two objects A and B are moving with velocities vA and vB (each with respect to some common frame of reference, say ground.). Then, velocity of object A relative to that of B is :

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and similarly, the velocity of object B relative to that of A is :

PROBLEM ON RELATIVE VELOCITY IN A PLANE

Rain is falling vertically with a speed of 35 m/s. A woman rides

a bicycle with a speed of 12 m/s in east to west direction.

What is the direction in which she should hold her umbrella ?

• Since the woman is riding a bicycle, the velocity of rain as experienced by her is the velocity of rain relative to the velocity of the bicycle she is riding. That is vrb = vr – vb
• This relative velocity vector as shown in Fig. makes an angle θ with the vertical. It is given by

Therefore, the woman should hold her umbrella at an angle of about 19° with the vertical towards the west.

PROJECTILE MOTION

• Projectile is a body thrown with an initial velocity in the vertical plane and then it moves in two dimensions under the action of gravity alone without being propelled by any engine or fuel. Its motion is called projectile motion. The path of a projectile is called its trajectory.
• Projectile motion is a case of two-dimensional motion .

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A body can be projected in two ways :

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Horizontal projection- When the body is given an initial velocity in the horizontal direction only.

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Angular projection- When the body is thrown with an initial velocity at an angle to the horizontal direction.

• ASSUMPTIONS:

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• There is no resistance due to air
• Rotational motion of the earth is absent
• Acceleration due to gravity is constant both in magnitude and direction

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HORIZONTAL PROJECTION

A body is thrown with an initial velocity u along the horizontal

direction. We will study the motion along x and y axis separately by

taking the starting point to be the origin.

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Along x-axis

1. u_x=u (Component along x-axis)

2. Acceleration along x-axis a_x=0 (F =0)

3. Velocity in the horizontal direction � (here a=0),

4. The displacement along x-axis at any instant t,�

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Along y-axis

1. Component of initial velocity along y-axis. u_y=0

2. Component of velocity along the y-axis at any instant t.�

3. The displacement along y-axis at any instant t�

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EQUATION OF A TRAJECTORY�(PATH OF A PROJECTILE)�

• We know at any instant x = ut �t=x/u

Also, y= (1/2)gt²

Substituting for t we get

y= (1/2)g(x/u)²

y= (1/2)(g/u²)x²

  y= kx²  where k= g/(2u² )

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• Thus, the path of projectile, projected horizontally from a height above the ground is a parabola.

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TIME OF FLIGHT (T):

• It is the total time taken by a body to completes its projectile motion.

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• To find T , we will find the time for vertical fall 

�From y= u_yt + (1/2) gt²

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Put y= h and time t = T

h= 0 + (1/2) gT²

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T= (2h/g)^1/2

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^{RANGE (R) :}

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• ^{It is the horizontal distance covered during the time of flight T.}
• ^{From x= ut �When t=T , x=R}

R=uT But T= (2h/g)^1/2

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 R=u(2h/g)^1/2

R

Net velocity & direction at any instant of time t

• At any instant t

We know that

v_x= u and v_y= gt

v= (v_x² + v_y²)^1/2 

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v= [u² + (gt)²]^1/2

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• ^{Direction of v with the horizontal at any instant :}

θ = tan^-1 (v_y/v_x)= tan^-1 (gt/u)

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PROJECTILE FIRED AT AN ANGLE WITH THE HORIZONTAL

Along X axis

Along Y axis

Consider an object is projected with an initial velocity u at an angle Φ to the horizontal direction.We will study the motion along x and y axis separately by taking the starting point to be the origin.

Equation of Trajectory �(Path of projectile)

This equation is of the form y= a x + bx² where 'a' and 'b are constants. This is the equation of a parabola. Thus, the path of a projectile is a parabola .

Time of flight (T)

• It is a time taken by a body to complete its projectile motion.

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• Time taken by projectile to reach the maximum height, T_a (time of ascent)=T/2

Ta=Td (time of ascent =time of descent)

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Also v_y= usinΦ – gt

At t=T/2 , v_y= 0 , So

0= u sinΦ – g (T/2)

T= (2usinΦ)/g

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Maximum height (H)

MAXIMUM HEIGHT AT 90⁰

Hmax= (u²)/2g

Horizontal Range (R) :

Range is the total horizontal distance covered during the time of flight.

MAXIMUM RANGE AT 45⁰

Rmax= u²/g

Net velocity of the body at any instant of time t:

v_x=ucosΦ

v_y=usinΦ – gt

v= (v_x² + v_y²)^1/2

Φ= tan^-1(v_y/v_x) 

Where Φ is the angle that the resultant velocity(v) makes with the horizontal at any instant .

Note: projectile-motion_en.html

CIRCULAR MOTION 

It is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation.

ANGULAR DISPLACEMENT

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The angular displacement is defined as the angle through which an object moves on a circular path.

It is the angle, in radians,

between the initial and final positions.

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(θ₂- θ₁) = θ angular displacement

θ = s/r

θ = angular displacement through which movement has occurred

s = distance travelled

r = radius of the circle

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ANGULAR VELOCITY

It is a quantitative expression of the amount of rotation that a spinning object undergoes per unit time. It is a vector quantity, consisting of an angular speed component and either of two defined directions or senses.

TIME PERIOD:

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Time taken by a particle to complete one rotation is called time period. It is denoted by T.

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FREQUENCY:

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Number of rotations completed by the particle in one unit second is called frequency. It is denoted by v.

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RELATION BETWEEN ANGULAR VELOCITY and TIME

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Let dθ be the angular displacement made by the particle in time dt , then the angular velocity of the particle is dθ /dt

ω = dθ /dt .

Its unit is rad s-1

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For one complete revolution, the angle swept by the radius vector is 360⁰ or 2π radians.

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then the angular velocity of the particle is

ω= θ/t = 2 π/T .

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If the particle makes n revolutions per second,

then ω=2π(1/T) = 2πn

where n = 1/T  is the frequency of revolution.

RELATION BETWEEN ANGULAR VELOCITY AND TIME

Let PQ = ds, be the arc length covered by the particle moving along the circle, then the angular displacement d θ is expressed as dθ = ds/r.

But ds=vdt.

UNIFORM CIRCULAR MOTION

Uniform circular motion can be described as the motion of an object in a circle at a constant speed. As the object moves in a circular path, its direction changes constantly. At all instances, the direction of the object is tangent to the circle.

ANGULAR ACCELERATION

Angular acceleration is the rate of change of angular velocity.

In SI units, it is measured in radians per second squared (rad/s²)

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It is denoted by the Greek letter alpha (α).

RELATION BETWEEN ANGULAR ACCELERATION AND LINEAR ACCELERATION

Linear acceleration = Rate of change of linear velocity

a= dv/dt

But v=r ω

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Therefore a= d(r ω)/ dt = r dω/ dt

Or a= r α

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In vector form;

CENTRIPETAL ACCELERATION

When an object follows a circular path at a constant speed, the motion of the object is called uniform circular motion. From fig

The acceleration is always directed towards the centre. This is why this acceleration is called centripetal acceleration.

ACKNOWLEDGEMENT

• First of all, I take this opportunity to thank NVS for selecting me as one of the member to prepare e-content for Class XI, Physics Subject,Chapter – 4 – Motion in Plane.

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• At same time , I thank the Chairman of this task Mr.B.GovindaRao, Principal, JNV Srikakulam who guided us to complete the task well in time by conducting video conference along with Co-ordinator and Member secretary.

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• I thank Mrs. Helen Mary, Principal, JNV Karaikal for her valuable guidance in preparing the e-content and all the team members.

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• I also express my sincere thanks to Mr.S.Subramanian (Art Teacher), JNV Karaikal for his technical support and the Lab attendant Mr.Veerapandian for capturing activity based videos.

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• Finally I thank almighty to provide strength and confidence to complete the work with full spirit.

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K.Sankar (PGT Physics)

JNV Karaikal

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