Kerala Plus One Physics Notes Chapter 4 Motion in a Plane
If a particle is free to move in a plane, its position can be located with two coordinates. We shall choose the plane of motion as the XY plane. To describe motion in two dimensions, that is, motion in a plane, we need vectors. We learn to add, substract and multiply vectors.
A quantity which has only magnitude and no direction is called a scalar quantity, eg., length, volume, mass, time, work etc.
Vectors are quantities which have both magnitude and direction, eg., velocity, acceleration, momentum, force, etc.
Vectors are represented by a straight line with arrow on it. The length of the line represents its magnitude and arrow represents its direction.
A vector can be represented by a single letter with head on its top.eg., F (force vector).
Few definitions in Vector Algebra
1. Modulus of a vector:
The magnitude of a vector is called modulus of that vector. The modulus of a vector is represented by or A.
2. Unit vector:
A unit vector is a vector of unit magnitude drawn in the direction of the given vector.
A unit vector in the direction of the given vector is found by dividing the given vector by its modulus (magnitude). Thus a unit vector in the direction of A is given by
where the unit vector A is read as ‘A cap’ or ‘A hat’.
(i) Here unit magnitude means 1 times the SI unit of the given quantity, eg., 1 kg, 1 m, 1 N, etc.
(ii) In cartesian coordinates are the unit vectors along x-axis, y-axis and z-axis respectively.
Two vectors are said to be equal if they have the same magnitude and same direction.
Here are equal vectors.
4. Negative of a vector:
The negative of a vector is defined as another vector having the same magnitude but opposite direction.
The above two are opposite vectors.
5. Co-initial vectors:
The vectors are said to co-initial if their initial point is common (same).
6. Collinear vectors:
These are vectors having equal or unequal magnitudes and are acting along parallel straight lines.
Here are collinear in all three cases.
Position and displacement vectors:
Consider the motion of a object in the X-Y plane with origin at O. Let at time t1 the
object be at point A and at t2 it be at B.
The position vector of the object at A is given by . The position vector of the object at B is given by . As the object moves from A to B. is called the displacement vector of the object in the time interval (t2-t1). If the coordinates of points A and B are (x1, y1) and (x2,y2) respectively, then position vector, and displacement vector.
Multiplication of vectors by a real number:
If a vector is multiplied by a real number n it becomes another vector nA. Its magnitude becomes n times the original magnitude but the direction remains the same or opposite depending on the positive or negative sign of n.Thus
if n=2 .we see
Addition and subtraction of vectors:
Composition of vector.
We have understand that the resultant of two or more vectors is a single vector which produces the same effect as the individual vectors together would produce. This way of adding vectors is called composition of vectors.
Consider 3 forces acting on a body which makes the body move in a particular direction with an acceleration a. We can apply a single force with a particular magnitude that will make the body move in the same direction with the same acceleration a. Addition of vectors will help us in doing that.
Triangle law of vector addition:
“If two vectors are represented both in magnitude and direction by the two sides of a triangle take in same order, then their resultant is represented both in magnitude and direction, by the third side of the triangle taken in the opposite order. ”
Example. If we want to add two vectors A and B given in the figure below,
then we draw a vector equal and parallel to . Now, from the head P of draw a vector pg equal and parallel to Then the resultant vector is given by .
Parallelogram law of vector addition:
“If two vectors are represented both in magnitude and direction by the two adjacent (next- to-next) sides of a parallelogram drawn from a common point, then their resultant is completely represented, both in magnitude and direction, by the diagonal of the parallelogram passing through that point.”
Example. If we want to add two vectors and B given in the figure below given in the figure below.
then from a common point O, draw a vector OP equal and parallel to and equal and parallel to . Complete the parallelogram OPSQ. According to the parallelogram law of vector addition, the diagonal will give the resultant vector .
A river 800 m wide flows at the rate of 5 kmh-1. A swimmer who can swim at 10 kmh-1 in still water, wishes to cross the river straight
(i) Along what direction must he strike?
(ii) What should be his resultant velocity?
(iii) How much time he would take?
the resultant velocity v is perpendicular to the bank of the river. This will be possible if the swimmer moves making an angle 9 with the upstream of the river.
In right ΔOCB
Let us add two vectors . Using triangle law of vector law of vector addition we write, , where is the resultant vector.
When two vectors are perpendicular to each other then,
Properties of vector addition:
- Vector addition is commutative
- Vector addition is associative,
Subtraction of vectors:
Let there be two vectors . We can write
So to subtract from we reverse the direction of and then add it to .
Note: Vector subtraction is not commutative nor associative.
We get this when we add two equal and opposite vectors.
eg., for a motion along a complete circle, j the displacement vector is a null vector.
Resolution of vectors:
We can add two vectors to get a third vector. Similarly we can resolve a single vector into two components. The vector and its components should lie in the same plane.
Rectangular components of a vector:
If we split a vector into two components which are at right angles to each other, then they are called the rectangular components of the vector.
can be resolved into two compounds from the above fig
Find the vector AB and its magnitude if it has initial point A (1, 2, -1) and final point B (3,2, 2).
Two forces are acting simultaneously at a point. What is the magnitude of the resultant force?
Scalar Product or Dot product:
Scalar product of two vectors is defined as the product of the magnitude of two vectors and the cosine of the smaller angle between them.
In the above figure, scalar product is,
The scalar product of two vectors is always a scalar.
Find the angle between the vectors
Vector product or cross product:
It is defined as the product of magnitude of the two vectors and the sine of the small angle between them.
The direction of vector product is perpendicular to the plane containing A and a . Note :
Relative velocity in two dimensions:
The concept here is same as that in one ‘dimension motion.
A boat is moving with a velocity with respect to ground. The water in the river is moving with a velocity with respect to ground. What is the relative velocity of boat with respect to river?
A body thrown into space (air) with some initial velocity and is allowed to move under the influence of gravity only is called a projectile.
Consider a body which is projected into air with a velocity u at an angle with the horizontal. The initial velocity ‘if can be resolved into two components u cos θalong horizontal direction and u sin θ along vertical direction.
Time of flight
The time taken by the projectile to cover the path OAB is called time of flight Here time of flight is the total time taken to reach B.
OB = x = ut cos θ
The y- coordinate at the point B is zero Thus, writing the equation for vertical motion,
y = ut + 1/2at2
Taking vertical displacement y = 0, a.= -g and initial vertical velocity = u sin θ, we get
0 = u sint θ – 1/2 gt2
1/2gt2=u sin θ t
In the figure the maximum height is shown at A. The vertical component of velocity at A is 0. The maximum height is the y- coordinate at A,
v2 = u2 + 2as
When we substitute v – 0, a = -g, s = H and u = u sin θ, we get
0 = (u sin θ)2+ 2 (- g) H
2gH = u2sin20
The distance OB (R) is the horizontal range. It is the distance travelled by the particle in time, of flight).
Horizontal velocity = u cos θ
Hence the horizontal distance (R) can be Time of found as, R = horizontal velocity xtime of flight.
Note: Fora given velocity, a projectile has the I same horizontal range for the angles of
projection 0 and (90° – θ). (3) shows that, R is maximum when sin 20 is maximum,
ie., when 0 = 45°
The maximum horizontal range,
Equation for path of projectile:
The path of projectile is parabola. The vertical displacement of projectile at any time to be found using the formula.
s = ut + 1/2 at2
y – u sinθ t-1/2 gt2………….(1)
In this equation g, 0 and u are constants. Hence eq. (4) can be written in the form
y = ax + bx2 . where a and b are constants. This is the equation of parabola, ie., the path of the projectile is a parabola.
A body is projected with a velocity of 30 ms-1 at an angle of 300 with the vertical. Find the maximum height, time of flight and the horizontal range.
Uniform circular motion
If a particle moves along a circular path with a constant speed (it cover equal.distances on the circumference of the circle in equal intervals of time), then the motion of the object is said to be uniform circular motion.
1. Angular displacement’s (θ):
The angle swept over by the radius vector in a give interval of time is called angular displacement θ.
2. Angular velocity :
It is the rate of change of angular displacement. If A0 is the angle turned in time At then the angular velocity,
Instantaneous angular velocity is given by
3.Time period (T) :
It is the time taken by the particle to complete one revolution.
It is the number of revolutions made by the object in one second.
(for one while revolution)
Relation between linear velocity and angular velocity:
Relation between linear acceleration and angular acceleration.:
When a body moves in a circular path, its velocity changes continuously at every point. Hence it experiences acceleration. The direction of acceleration is towards the center of the circle. So it is called centripetal acceleration. For uniform circular motion, centripetal
acceleration is given by,
Hence, centripetal force,
A body of mass 10 kg revolves in a circle of diameter 0.40 m, making 1000 revolutions per minute. Calculate its liner velocity and centripetal acceleration.