In the previous section we completed a discussion on rectilinear motion. In this chapter we will see motion in a plane.
In the next section, we will see displacement vector.
We have already seen how to specify the position of an object which lies on a plane. (Details here)
For convenience, the fig.3.5 is shown again below:
1. Let in fig.a, a point object move on the xy plane
• To describe the motion of that object, we will need the help of two coordinates:
(i) the x coordinate
(ii) the y coordinate
2. Similarly, to describe the motion of an object on the xz plane, we will need the help of two coordinates:
(i) the x coordinate
(ii) the z coordinate
3. Similarly, to describe the motion of an object on the yz plane, we will need the help of two coordinates:
(i) the y coordinate
(ii) the z coordinate
■ Motion in a plane is called two dimensional motion
• In one dimentional motion, we learned about four items:
(i) Position (ii) Displacement (iii) Velocity (iv) Acceleration
• In two dimentional motion also we have to learn about the same four items. For that, first we have to learn about vectors.
• Such a quantity can be completely specified by a single number and a proper unit
• Speed, area, mass, volume, temperature etc., are scalars
■ If a physical quantity has both magnitude and direction, it is called a Vector quantity
• To specify such a quantity completely, we need the following two items:
(i) The magnitude
It is given by a number and a proper unit
(ii) The direction
It is given by statements like:
♦ North direction
♦ East west direction
♦ South west direction
♦ 30o east of north
♦ 25o with x axis
etc.,
• Velocity, force, acceleration, weight, momentum etc., are vectors
We can learn more details using an example:
1. Distance between two points A and B is 250 m
• Distance is a scalar quantity
• We use distance in those situations where direction is not required. Amount of fuel required for a journey will depend on the distance. Amount of fuel required will not depend on the direction
2. Displacement of an abject is 350 m in the north west direction
• Displacement is a vector quantity
• We use displacement in those situations where direction is also necessary. An example is shown in fig.4.1(a) below:
(i) An object is displaced from A to B
• AB makes an angle 30o with the horizontal
■ So we can write:
Magnitude of the displacement = Length AB
Direction of the displacement = A Direction which makes 30o with the horizontal
(ii) Consider another possibility:
• The displacement can be along AB' also
• Length AB = Length AB'
• But AB' makes an angle 60o with the horizontal
• If the object moves along AB, it's final position will be at B
• If the object moves along AB', it's final position will be at B'
(iii) Now, a stationary point P is also present in the problem
• Distance PB' > Distance PB
• So, if the displacement is along AB', the object will end up being at a greater distance from P
• If the displacement is along AB, the object will end up being at a lesser distance from P
• Note that, the distance travelled is the same in both cases
■ So this is a situation where direction also has to be considered
• Like distance, speed, area, mass, volume, temperature etc., are scalars
• Like displacement, velocity, force, acceleration, weight, momentum etc., are vectors
Let us learn more about vectors:
1. In print media, vectors are represented by boldface letters. For example:
• v represents the velocity vector
• a represents the acceleration vector
2. But when wriiten by hand, boldface is impractical. So in hand writting, we place an arrow above the letter. For example:
• $\vec{v}$ represents the velocity vector
• $\vec{a}$ represents the acceleration vector
3. Now we need a method to represent the magnitude of a vector. It can be explained with the help of an example:
• Given a force vector $\vec{F}$.
• Let it's magnitude be 15 N.
■Then we write:
|$\vec{F}$|= 15 N
• An object is moving on the xy plane
• The green curve is the path along which the object moves
2. At time t1, the object is at P1
• Join the origin O and P1 using a straight line. This is shown in fig.4.1(c)
• Mark an arrow on the line OP1
• Then $\mathbf\small{\vec{OP_1}}$ is the position vector of the object at time t1
[The direction of the arrow on the line OP1 in the fig.4.1(c) is important
Based on the direction of that arrow, we say these:
♦ Tail of the position vector $\mathbf\small{\vec{OP_1}}$ is at O
♦ Tip of the position vector $\mathbf\small{\vec{OP_1}}$ is at P1]
■ We can write the following 3 points:
(i) Position vector of the object at time t1 is $\mathbf\small{\vec{OP_1}}$
(ii) Magnitude of $\vec{OP_1}$ :
$\mathbf\small{|\vec{OP_1}|}$ = Length OP1
(iii) Direction of $\vec{OP_1}$ :
The vector $\mathbf\small{\vec{OP_1}}$ makes an angle $\theta_1$ with the x axis
3. At time t2, the object is at P2
• Join the origin O and P2.
• Then $\mathbf\small{\vec{OP_2}}$ is the position vector of the object at time t2
■ We can write the following 3 points:
(i) Position vector of the object at time t2 is $\mathbf\small{\vec{OP_2}}$
(ii) Magnitude of $\mathbf\small{\vec{OP_2}}$:
$\mathbf\small{|\vec{OP_2}|}$ = Length OP2
(iii) Direction of $\mathbf\small{\vec{OP_2}}$:
The vector $\mathbf\small{\vec{OP_2}}$ makes an angle $\theta_2$ with the x axis
What is the practical application of position vectors?
Consider a situation:
1. A person gives us the following information:
(i) $\vec{OP_1}$ is the position vector of an object at time t1
(ii) Magnitude of $\vec{OP_1}$ is 250 m. That is., |$\vec{OP_1}$|= 250 m
(iii) Direction of $\vec{OP_1}$ is 70o with the x axis
2. We can use the above information to represent the 'position of the object at time t1' on a sheet of paper
Let us see how it is done:
(i) On a fresh sheet of paper (preferably a graph paper), draw x and y axis and mark distances to a suitable scale. (A scale of 1 cm = 40 m is appropriate for this problem)
(ii) From the origin O, draw a line OP1 such that:
• Length of OP1 = 250 m
• Angle which OP1 makes with the x axis = 70o
This is shown in fig.4.2 below:
■ Thus P1 represents the position of the object at time t1
• In this way, if we have information about $\vec{OP_2}$, we can show the position P2 also
For convenience, the fig.3.5 is shown again below:
 |
| Fig.3.5 |
• To describe the motion of that object, we will need the help of two coordinates:
(i) the x coordinate
(ii) the y coordinate
2. Similarly, to describe the motion of an object on the xz plane, we will need the help of two coordinates:
(i) the x coordinate
(ii) the z coordinate
3. Similarly, to describe the motion of an object on the yz plane, we will need the help of two coordinates:
(i) the y coordinate
(ii) the z coordinate
■ Motion in a plane is called two dimensional motion
• In one dimentional motion, we learned about four items:
(i) Position (ii) Displacement (iii) Velocity (iv) Acceleration
• In two dimentional motion also we have to learn about the same four items. For that, first we have to learn about vectors.
Scalars and Vectors
■ If a physical quantity has magnitude but no direction, it is called a Scalar quantity• Such a quantity can be completely specified by a single number and a proper unit
• Speed, area, mass, volume, temperature etc., are scalars
■ If a physical quantity has both magnitude and direction, it is called a Vector quantity
• To specify such a quantity completely, we need the following two items:
(i) The magnitude
It is given by a number and a proper unit
(ii) The direction
It is given by statements like:
♦ North direction
♦ East west direction
♦ South west direction
♦ 30o east of north
♦ 25o with x axis
etc.,
• Velocity, force, acceleration, weight, momentum etc., are vectors
We can learn more details using an example:
1. Distance between two points A and B is 250 m
• Distance is a scalar quantity
• We use distance in those situations where direction is not required. Amount of fuel required for a journey will depend on the distance. Amount of fuel required will not depend on the direction
2. Displacement of an abject is 350 m in the north west direction
• Displacement is a vector quantity
• We use displacement in those situations where direction is also necessary. An example is shown in fig.4.1(a) below:
 |
| Fig.4.1 |
• AB makes an angle 30o with the horizontal
■ So we can write:
Magnitude of the displacement = Length AB
Direction of the displacement = A Direction which makes 30o with the horizontal
(ii) Consider another possibility:
• The displacement can be along AB' also
• Length AB = Length AB'
• But AB' makes an angle 60o with the horizontal
• If the object moves along AB, it's final position will be at B
• If the object moves along AB', it's final position will be at B'
(iii) Now, a stationary point P is also present in the problem
• Distance PB' > Distance PB
• So, if the displacement is along AB', the object will end up being at a greater distance from P
• If the displacement is along AB, the object will end up being at a lesser distance from P
• Note that, the distance travelled is the same in both cases
■ So this is a situation where direction also has to be considered
• Like distance, speed, area, mass, volume, temperature etc., are scalars
• Like displacement, velocity, force, acceleration, weight, momentum etc., are vectors
Let us learn more about vectors:
1. In print media, vectors are represented by boldface letters. For example:
• v represents the velocity vector
• a represents the acceleration vector
2. But when wriiten by hand, boldface is impractical. So in hand writting, we place an arrow above the letter. For example:
• $\vec{v}$ represents the velocity vector
• $\vec{a}$ represents the acceleration vector
3. Now we need a method to represent the magnitude of a vector. It can be explained with the help of an example:
• Given a force vector $\vec{F}$.
• Let it's magnitude be 15 N.
■Then we write:
|$\vec{F}$|= 15 N
Position vectors
1. Consider fig.4.1(b) above• An object is moving on the xy plane
• The green curve is the path along which the object moves
2. At time t1, the object is at P1
• Join the origin O and P1 using a straight line. This is shown in fig.4.1(c)
• Mark an arrow on the line OP1
• Then $\mathbf\small{\vec{OP_1}}$ is the position vector of the object at time t1
[The direction of the arrow on the line OP1 in the fig.4.1(c) is important
Based on the direction of that arrow, we say these:
♦ Tail of the position vector $\mathbf\small{\vec{OP_1}}$ is at O
♦ Tip of the position vector $\mathbf\small{\vec{OP_1}}$ is at P1]
■ We can write the following 3 points:
(i) Position vector of the object at time t1 is $\mathbf\small{\vec{OP_1}}$
(ii) Magnitude of $\vec{OP_1}$ :
$\mathbf\small{|\vec{OP_1}|}$ = Length OP1
(iii) Direction of $\vec{OP_1}$ :
The vector $\mathbf\small{\vec{OP_1}}$ makes an angle $\theta_1$ with the x axis
3. At time t2, the object is at P2
• Join the origin O and P2.
• Then $\mathbf\small{\vec{OP_2}}$ is the position vector of the object at time t2
■ We can write the following 3 points:
(i) Position vector of the object at time t2 is $\mathbf\small{\vec{OP_2}}$
(ii) Magnitude of $\mathbf\small{\vec{OP_2}}$:
$\mathbf\small{|\vec{OP_2}|}$ = Length OP2
(iii) Direction of $\mathbf\small{\vec{OP_2}}$:
The vector $\mathbf\small{\vec{OP_2}}$ makes an angle $\theta_2$ with the x axis
What is the practical application of position vectors?
Consider a situation:
1. A person gives us the following information:
(i) $\vec{OP_1}$ is the position vector of an object at time t1
(ii) Magnitude of $\vec{OP_1}$ is 250 m. That is., |$\vec{OP_1}$|= 250 m
(iii) Direction of $\vec{OP_1}$ is 70o with the x axis
2. We can use the above information to represent the 'position of the object at time t1' on a sheet of paper
Let us see how it is done:
(i) On a fresh sheet of paper (preferably a graph paper), draw x and y axis and mark distances to a suitable scale. (A scale of 1 cm = 40 m is appropriate for this problem)
(ii) From the origin O, draw a line OP1 such that:
• Length of OP1 = 250 m
• Angle which OP1 makes with the x axis = 70o
This is shown in fig.4.2 below:
 |
| Fig.4.2 |
• In this way, if we have information about $\vec{OP_2}$, we can show the position P2 also
In the next section, we will see displacement vector.
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