forces in common for mechanics

8/7/2019 Forces in common for mechanics

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Forces in common for mechanics

Gravitational Force : Every object on the earth experiences the force of gravity due to the

earth. Gravity also governs the motion of celestial bodies. The gravitational force can act at a

distance without the need of any intervening medium.

All the other forces common in mechanics are contact forces.A contact force on an object

arises due to contact with some other object: solid or fluid. When bodies are in contact (e.g. abook resting on a table, a system of rigid bodies connected by rods, hinges and other types of

supports), there are mutual contact forces (for each pair of bodies) satisfying the third law.

The component of contact force normal to the surfaces in contact is called normal reaction.

The component parallel to the surfaces in contact is called friction. Contact forces arise also

when solids are in contact with fluids. For example, for a solid immersed in a fluid, there is an

upward bouyant force equal to the weight of the fluid displaced. The viscous force, air

resistance, etc are also examples of contact forces.

Two other common forces are tension in a string and the force due to spring. When aspring is compressed or extended by an external force, a restoring force is generated. This

force is usually proportional to the compression or elongation (for small displacements). The

spring force F is written as F = k x where x is the displacement and k is the force constant.

The negative sign denotes that the force is opposite to the displacement from the unstretched

state. For an inextensible string, the force constant is very high. The restoring force in a string

is called tension. It is customary to use a constant tension T throughout the string. This

assumption is true for a string of negligible mass.

The different contact forces of mechanics mentioned above fundamentally arise fromelectrical forces. At the microscopic level, all bodies are made of charged constituents (nuclei

and electrons) and the various contact forces arising due to elasticity of bodies, molecular

collisions and impacts, etc. can ultimately be traced to the electrical forces between the

charged constituents of different bodies.

Newton First Law of motion-Mechanics

Inertia is the property of any body because of which it always continue its state and always oppose its

change.

It is incorrect to assume that a net force is needed to keep a body in uniform motion. To maintain a body in

uniform motion, we need to apply an external force to encounter the frictional force, so that the two forces

sum up to zero net external force.

If the net external force is zero, a body at rest continues to remain at rest and a body in motion continues to

move with a uniform velocity. This property of the body is called inertia. Inertia means resistance to

change.

A body does not change its state of rest or uniform motion, unless an external force compels it to change that

state.

Newton's First Law of motion: Every body continues to be in its state of rest or of uniform motion in a

straight line unless compelled by some external force to act otherwise.
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The state of rest or uniform linear motion both imply zero acceleration. The first law of motion can,

therefore, be simply expressed as: If the net external force on a body is zero, its acceleration is zero.

Acceleration can be non zero only if there is a net external force on the body.

For example, a spaceship out in interstellar space, far from all other objects and with all its rockets turned

off, has no net external force acting on it. Its acceleration, according to the First Law, must be zero. If it is in

motion, it must continue to move with a uniform velocity.

For terrestrial phenomena, in particular, every object experiences gravitational force due to the earth. Alsoobjects in motion generally experience friction, viscous drag, etc. If then, on earth, an object is at rest or in

uniform linear motion, it is not because there are no forces acting on it, but because the various external

forces cancel out i.e. add up to zero net external force.

Consider a book at rest on a horizontal surface Fig. a. It is subject to two external forces :

The force due to gravity (i.e. its weight W) acting downward and the upward force on the book by the table,

the normal force R . R is a self-adjusting force. We observe the book to be at rest. Therefore, we conclude

from the first law that the magnitude of R equals that of W. A statement often encountered is :

Since W = R, forces cancel and, therefore, the book is at rest. This is incorrect reasoning. The correct

statement is : Since the book is observed to be at rest, the net external force on it must be zero, according to

the first law. This implies that the normal force R must be equal and opposite to the weight W.

Consider the motion of a car starting from rest, picking up speed and then moving on a smooth straight road

with uniform speed Fig. b. When the car is stationary, there is no net force acting on it. During pick-up, it

accelerates. This must happen due to a net external force. Note, it has to be an external force.

The acceleration of the car cannot be accounted for by any internal force. The only conceivable external

force along the road is the force of friction. It is the frictional force that accelerates the car as a whole. When

the car moves with constant velocity, there is no net external force.

The property of inertia contained in the First law is evident in many situations. Suppose we are standing in a

stationary bus and the driver starts the bus suddenly. We get thrown backward with a jerk.It is because our

feet are in touch with the floor. If there were no friction, we would

remain where we were, while the floor of the bus would simply slip forward under our feet and the back of

the bus would hit us.

However, fortunately, there is some friction between the feet and the floor. If the start is not too sudden, i.e.

if the acceleration is moderate, the frictional force would be enough to accelerate our feet along with the bus.

But our body is not strictly a rigid body. It is deformable, i.e. it allows some relative displacement between

different parts.

What this means is that while our feet go with the bus, the rest of the body remains where it is due to inertia.

Relative to the bus, therefore, we are thrown backward. As soon as that happens, however, the muscular

forces on the rest of the body (by the feet) come into play to move the body along with the bus.

A similar thing happens when the bus suddenly stops. Our feet stop due to the friction which does not allow

relative motion between the feet and the floor of the bus. But the rest of the body continues to move forward

due to inertia. We are thrown forward.The restoring muscular forces again come into play and bring the

body to


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Uniform Circular Motion

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

circular motion. The word uniform refers to the speed, which is uniform (constant) throughout the motion.

Suppose an object is moving with uniform speed v in a circle of radius R as shown in Fig. Since the velocity

of the object is changing continuously in direction, the object undergoes acceleration.

Let r and r be the position vectors and v and v the velocities of the object when it is at point P and P asshown in Fig. (a). By definition, velocity at a point is along the tangent at that point in the direction of

motion. The velocity vectors v and v are as shown in Fig. (a1). v is obtained in Fig. (a2) using the triangle

law of vector addition. Since the path is circular, v is perpendicular to r and so is v to r .

Therefore, v is perpendicular to r. Since average acceleration is along v, the average acceleration a is

perpendicular to r. If we place v on the line that bisects the angle between r and r , we see that it is

directed towards the centre of the circle. Figure (b) shows the same quantities for smaller time interval. v

and hence a is again directed towards the centre.

In Fig.(c), t tends to zero and the average acceleration becomes the instantaneous acceleration. It is

directed towards the centre.

Thus, we find that the acceleration of an object in uniform circular motion is always directed towards the

centre of the circle.

a = change in velocity /time.

Let the angle between position vectors r and r

be . Since the velocity vectors v and v are always perpendicular to the position vectors, the angle

between them is also . Therefore, the

triangle CPP formed by the position vectors and the triangle GHI formed by the velocity vectors v, v and

v are similar (Fig. a). Therefore, the ratio of the base-length to side-length for

one of the triangles is equal to that of the other triangle. That is :
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Therefore, the centripetal acceleration ac=v2/R

Thus, the acceleration of an object moving with speed v in a circle of radius R has a magnitude v2/R and is

always directed towards the centre.

This is why this acceleration is called centripetal acceleration . Centripetal comes from a Greek term

which means centre-seeking. Since v and R are constant, the magnitude of the centripetal acceleration is

also constant. However, the direction changes, pointing always towards the centre. Therefore, a centripetal

acceleration is not a constant vector.

Kinematics Linear Motion

Previously we have discussed regarding speed,velocity, acceleration ofone dimensional motion.Here we are

going to extend that more in detail and going through the above posts will definitely help in understanding

the present concepts.

Basic definitions of Kinematics :

1. The study of motion of objects without any reference to the cause of motion is called kinematics.

2. The actual path traversed by a body is called the distance traveled.

3. The shortest distance between the initial and final positions of a body is called displacement.

4. Displacement of a body may be zero, or positive or negative but distance traveled is always positive.

5. The speed of a body is the rate at which it describes its path.

6. The rate of change of displacement is called velocity.

7. Average speed = total distance / total time

8. Average velocity = net displacement / total time

Linear Motion :

It is nothing but one dimensional motion where body always moves along a line.We can derive thefollowing formula in the case of linear motion.

1 . Average velocity V = Total displacement / Total time
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Avg.velocity = s1+s2+s3/t1+t2+t3 where t1 is time in the first case where as s1 is the displacement in the first

case and so on.

2 . If a body travels with a velocity v1 for the first half of the journey time and with a velocity v2 for the

second half of the journey time, then the average velocity is equal to v1+v2/2.

3 . If a body covers first half of its journey with uniform velocity v1 and the second half of the journey with

uniform velocity v2, then the average velocity is equal to 2v1v2/v1+v2 .

4 . If a body travels first one third of the distance with a speed v 1, and second one third of the distance with a

speed v2 and the last one third of the distance with a speed v3 then the average velocity is

3v1v2v3/v1v2+v2v3+v3v1.

5. The rate of change of velocity is called acceleration.

Equations of motion for a body moving with uniform acceleration

The following equations represent the motion of a body under constant acceleration.

If a body starts from rest and having uniform

acceleration then the above equations can be modified as shown below.

Please click on the screen for a better view.
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Notes :

1 . If the velocity of a body becomes 1/n of original velocity after a displacement x then it will come to rest

after covering a further displacement of X/n2 - 1 .

2 . A body is describing uniform circular motion with a speed v. When it describes an angle q at the

center then the change in velocity is dv = 2vsin (q/2)

3 . If the displacement of a body is proportional to the square of time, then its initial velocity is zero.

4 . Starting from rest a body travels with an acceleration a for some time and then with deceleration b and

finally comes to rest. If the total time of journey is t, then the maximum velocity and displacement and

average velocity are respectively then

i) Maximum velocity = ab t/a + b

ii) Displacement s = abt2/2(a+b).

iii)Average Velocity = maximum velocity / 2.

5 . If a particle starts from rest and moves with uniform acceleration a such that it travels distances X and

Y in the m and n particular seconds then

sn/s = X-Y/m-n where n is the particular second of journey.

6 . A particle starts from rest and moves along a straight line with uniform acceleration. If s is the distance

traveled by it n seconds and S is the distance travel led in the particular n th second then s n/s = 2n-1 /n2

Kinematics Acceleration
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The previous post about kinematics deals with SPEED AND VELOCITYand it can be browsed

here.

Acceleration as the rate of change of velocity with time.

The average acceleration a over a time interval is defined as the change of velocity dividedby the time interval :

a = v / t .

where v2 and v1 are the instantaneous velocities or simply velocities at time t2 and t1. It is

the average change of velocity per unit time. The SI unit of acceleration is m s2 .

On a plot of velocity versus time, the average acceleration is the slope of the straight line

connecting the points corresponding to (v2, t2) and (v1, t1). The average acceleration for

velocity-time graph shown in figure for different time intervals 0 s - 10 s, 10 s 18 s, and 18 s

and 20 s are :

0 s - 10 s a = 24 - 0 /10 - 0 = 2.4 m s2 .

10 s - 18 s a = 24 - 24 / 18 - 10 = 0 m s2 .

18 s - 20 s a = 0 - 24 / 20 - 18 = -12 m s2.

Instantaneous acceleration is defined asrate of change of velocity at any particular instant.It is shown as a = dv/dt .

The acceleration at an instant is the slope of the tangent to the vt curve at that instant.

Since velocity is a quantity having both magnitude and direction, a change in velocity may

involve either or both of these factors. Acceleration, therefore, may result from a change in

speed (magnitude), a change in direction or changes in both. Like velocity, acceleration can

also be positive, negative or zero.

Position-time graphs for motion with positive, negative and zero acceleration are shown in

figures (a), (b) and (c), respectively.
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The graph curves upward for positive

acceleration; downward for negative acceleration and it is a straight line for zero acceleration.

The area under the curve represents the displacement over a given time interval.of velocity-

time graph for any moving object .

Let us check for a simple case as follows.

Let an object moving with constant velocity u. Its velocity-time graph is as shown in figure.

The v-t curve is a straight line parallel to the time axis and

the area under it between t = 0 and t = T is the area of the rectangle of height u and base T.

Therefore, area = u T = uT which is the displacement in this time interval.

Kinematics Velocity and Speed

The previous post in kinematics deals with one dimensional motion concept and you can

browse it here.

AVERAGE VELOCITY AND AVERAGE SPEED :

Average velocity is defined as the change in position or displacement (x) divided by the time

intervals (t),

v = X2 - X1/t2 - t1

where x2 and x1 are the positions of the object at time t2and t1, respectively. Here bar over

the symbol for velocity is a standard notation used to indicate an average quantity. The SI

unit for velocity is m/s or , although km/h is used in many everyday applications.

The average velocity can be positive or negative depending upon the sign of the

displacement. It is zero if the displacement is zero. The following fugures shows the x-t graphs

for an object, moving with positive velocity (Fig. a), moving with negative velocity (Fig. b) and

at rest (Fig. c).
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Average speed is defined as the total path length

travelled divided by the total time interval during which the motion has taken place :

Average speed = Total path length/Total time interval

Average speed has obviously the same unit(m/s) as that of velocity. But it does not tell us in

what direction an object is moving. Thus, it is always positive (in contrast to the average

velocity which can be positive or negative). If the motion of an object is along a straight line

and in the same direction, the magnitude of displacement is equal to the total path length.In

that case, the magnitude of average velocity is equal to the average speed.

INSTANTANEOUS VELOCITY AND SPEED :

The velocity at an instant is defined as the limit of the average velocity as the time interval t

becomes infinitesimally small.

v = dx/dt

For uniformmotion, velocity is the same as the average velocity at all instants.

Instantaneous speed or simply speed is the magnitude of velocity.

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Projectile Motion

It is nothing but two dimensional motion.It could be either from ground or from a certain height.If a body is

changing its position along X and Y axis with respect to time then it is called projectile motion.

An object that is in flight after being thrown or projected is called a projectile. Such a projectile might be a

football, a cricket ball, a baseball or any other object. The motion of a projectile may be thought of as the

result of two separate, simultaneously occurring components of motions. One component is along a

horizontal direction without any acceleration and the other along the vertical direction with constantacceleration due to the force of gravity.

we shall assume that the air resistance has negligible effect on the motion of the projectile for a simple

projectile equation.
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Let us assume that the projectile is launched with velocity vo that makes an angle o with the x-axis as

shown in Fig. below. After the object has beenprojected, the acceleration acting on it is that due to gravity which is directed vertically downward:

a = g j which means that

ax = 0, ay = g

The components of initial velocity vo are :

v along x axis is = vo cos o

v along y axis is = vo sin o

If we take the initial position to be the origin of the reference frame as shown in Fig. we have :

One of the components of velocity, i.e. x-

component remains constant throughout the motion and only the y- component changes, like an object in

free fall in vertical direction. This is shown graphically at few instants in Fig. below Note that at the point of

maximum height, vy= 0 and therefore, tan = y component of velocity /x component of velocity = 0.
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The equation for the path of a projectile canbe derived as shown below.

Vector Resolution

A vector will be generally having components in different directions like x,y and z. Dividing the vector into

components along this directions is called resolution of vectors.

We are able to identify value of vectors along the corresponding directions with the help of components ofvectors.

Let a and b be any two non-zero vectors in a plane with different directions and let A be another vector in

the same plane. A can be expressed as a sum of two vectors one obtained by multiplying a by a real

number and the other obtained by multiplying b by another real number.

To see this, let O and P be the tail and head of the vector A. Then, through O, draw a straight line parallel to

a, and through P, a straight line parallel to b. Let them intersect at Q. Then, we have

A = OP = OQ + QP

But since OQ is parallel to a, and QP is parallel to b, we can write :

OQ = a, and QP = b where and are real numbers.
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Therefore, A = a + b.

We say that A has been resolved into two component vectors a and

b along a and b respectively. Using this method one can resolve a given vector into two component vectors

along

a set of two vectors all the three lie in the same plane. It is convenient to resolve a general vector along the

axes of a rectangular coordinate system using vectors of unit magnitude.

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 and z-axes of a

rectangular coordinate system are denoted by i , j and k , respectively, as shown in Figure below.

These unit vectors are perpendicular to each other.

If we multiply a unit vector, say n by a scalar, the result is a vector = n . In general, a vector A can be

written as A = |A|n.

Vector resolution in two dimensions basing on Unit vectors :

Consider a vector A that lies in x-y plane as shown in Figure below. We draw lines from the head of A

perpendicular to the coordinate axes and get vectors A1 and A2 such that A1 + A2 = A. Since A1 is parallelto I and A2 is parallel to J , we have :

A1 = Ax i, A2 = Ay j where Ax and Ay are real numbers.

So we can represent the vector as shown.

A = Ax i+ Ay j
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Using simple trigonometry, we can express

Ax and Ay in terms of the magnitude of A and the angle it makes with the x-axis :

Ax = A cos

Ay = A sin

As is clear from Eq. a component of a vector can be positive, negative or zero depending on the value of .

Now, we have two ways to specify a vector A in a plane. It can be specified by :

(i) its magnitude A and the direction it makes with the x-axis; or

(ii) its components Ax and Ay If A and are given, Ax and Ay can be obtained using Eq. If Ax and Ay are

given, A and . Then we can deduce the following relations .

The previous topics of vectors can be browsed

here below.

Vectors Cross Product

The previous post of the blog deals with dot product of vectors.Cross product is another way of

multiplying two vectors . Here the the result of product is a vector which will have both magnitude and

direction.

1 . When the perpendicular component of one vector with respect to the another vector is effective then the

cross product is taken.

2 . The cross product of two vectors is a vector and its direction is given by right hand cork screw rule.

3 . If a and b are two vectors and the angle between them is then the cross product of and is given by ab = |
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a| |b| sin ( ) where n s a unit vector perpendicular to the plane containing a and b .

4 . If two vectors are parallel i.e. = 0 or 180 then a b = 0 .

5 . If two vectors are perpendicular to each other a b = ab and it is maximum .

6 . If i , j and k are unit vectors then

APPLICATIONS OF CROSS PRODUCT OFVECTORS :
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VECTORS CONCEPTS

1.Physical quantities are mainly classified into two types a) Scalars b) Vectors .

2. Scalar quantities re those which have only magnitude.

3. Physical quantities which have both magnitude and direction are called vectors and they should satisfy the parallelogram law of vector addition.

4. Mathematically any directed line segment is called a vector. It has three characteristics.

a) Support (base)

b) Sense

c) Length (mangnitude or modulus)

5. The magnitude of a vector is a scalar.

6. Electric current, velocity of light have both magnitude and direction but they do not obey the laws ofvector addition. Hence they are scalars.

DIFFERENT TYPES OF VECTORS

7. EQUAL VECTORS: Two vectors are said to be equal when their magnitude and direction are equal.

8. NEGATIVE VECTOR: Negative vectors are those which are equal in magnitude but opposite in

direction.

9. NULL VECTOR (ZERO VECTOR): It is a vector whose magnitude zero and direction is unspecified.

Examples :

a) Displacement after one complete revolution.

b) Velocity of vertically projected body at the highest point.

10. UNIT VECTOR: It is a vector whose magnitude is unity. A unit vector parallel to a given vector R is

given by R r = R

11. REAL VECTOR OR POLAR VECTOR: If the direction of a vector is independent of the coordinatesystem, then it is called a polar vector.

Example : linear velocity, linear momentum, force, etc.

12. PSEUDO VECTOR: Vectors associated with rotation about an axis and whose direction is changed

when the co-ordinate system is changed from left to right, are called axial vectors (or) pseudo vectors.

Example : Torque, Angular momentum, Angular velocity, etc.

13. POSITION VECTOR: It is a vector that represents the position of a particle with respect to the origin

of a co-ordinate system. The Position Vection of a point (x, y, z) is r = x i+yj+zk .

ADDITION OF VECTORS

14. There are three laws of addition of vectors.
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a) Commutative law : A + B = B + A

b) Associative law : A + (B+C) = (A + B) + C

c) Distributive law : m(A + B) = mA + mB where m is a scalar

RESULTANT OF NUMBER OF VECTORS

15. Resultant is a single vector that gives the total effect of number of vectors.

16. Resultant can be found by using a) Triangle law of vectors b) Parallelogram law of vectors c) Polygon

law of vectors .

17. TRIANGLE LAW OF VECTORS: If two given vectors are represented both in magnitude and

direction by the two adjacent sides of a triangle, then closing side (third side) taken in the reverse order will

give the resultant both in magnitude and direction.

APPLICATIONS OF TRIANGLE LAW :

a) MOTION OF A BOAT CROSSING THE RIVER IN SHORTEST TIME :

If velocities of boat and river are represented with B and R subscripts with V then to cross the river in

shortest time, the boat is to be rowed across the river i.e., along normal to the banks of the river.
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MOTION OF A BOAT CROSSING THE RIVER IN SHORTEST DISTANCE :

Vector Concepts part two

This lesson is in continuation with Vectors concepts part oneand going through that first will give more

convenience to understand the present topic.

c)If three forces (vectors) are to be in equilibrium, then the sum of magnitudes of any two forces must begreater than the magnitude of third force.

d)Lami's theorem:
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If a body is in equilibrium under the action of three coplanar forces P,Q,R at angles as shown in the figure,

18. A body of mass 'm' is suspended by a

string of length 'l' from a rigid support. It is pulled aside by distance 'x' so that it makes an angle with the

vertical by applying a horizontal force F. When the body is in equilibrium.

19. PARALLELOGRAM LAW OF VECTORS

(OR FORCES):

"If two vectors acting at a point making an angle with each other are represented both in magnitude and

direction by the adjacent sides of a parallelogram, then the diagonal drawn from the same point will give the

resultant both in magnitude and direction" .
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22. POLYGON LAW OF VECTORS :

" If number of vectors acting at a point in the same plane in different directions are represented both in

magnitude and direction by the adjacent sides of a polygon taken in order, then the closing side taken in thereverse order will give the resultant both in magnitude and direction".

APPLICATIONS OF POLYGON LAW

1. If 'n' equal forces act on a body such that each force makes an angle 2 / n with the previous one and the

polygon is closed, then the resultant is zero.

If each force of magnitude 'F' makes an angle with previous one, then

a) the resultant is zero, if the number of forces is 2/

b) the resultant is 'F', if the number of forces are 2/ - 1

34

Dot Product of Vectors

The previous post of the vector topic is regarding parallelogram lawand definition ofdifferent kinds of

vectors.Here we are going to discuss product of vectors.Here there are three possibilities.

1. Vector multiplied with scalar gives a resultant of vector.2. Vector multiplied with vector gives a resultant of scalar(Dot Product)

3. Vector multiplied with vector gives a resultant of vector(Cross Product)

Here is the explanation in detail for each time of multiplication.

CASE ONE :

1. When a vector is multiplied by a scalar its products is a vector whose magnitude is equal to the scalar

times the magnitude of the given vector.

2.The direction of a vector is same as the given vector, if the scalar is positive and opposite if the scalar isnegative.

Example :
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P = m v where P is momentum, m is mass and v is velocity.

F = m a where F is force , m is mass and a is acceleration.

NOTE :

A vector multipllied by another vector may give a scalar (or) a vector. Hence there are two types of products

for multiplication of two types of products for multiplication of two vectors.

a) dot product (or) Scalar product

b) cross product (or) vector product

CASE TWO SCALAR PRODUCT (OR) DOT PRODUCT PROPERTIES :

1 . When the magnitude of one vector along another vector is effective then the dot product of two vectors is

taken.

2. The dot product of two vectors is a scalar.

3. The scalar product of two vectors and is a.b = ab cos

4. Scalar product is commutative i.e. a.b = b.a

5 . Scalar product is distributive i.e a.(b+c) = a.b + a.c

6 . The scalar product of two parallel vectors is maximum I.e when = 0

7 . The scalar product of two opposite vectors is negative i.e when = 180.

8 . The scalar product of two perpendicular vectors is zero when = 90.

9 . In case of unit vectors i.i = j.j = k.k = 1 i.e i.i = 1*1*cos 0 = 1

10 . Similarly i.j = j.k = k.i = 0 since i.j = 1 *1* cos 90 = 0.

11 . In terms of Components A.B = AxBx + AyxBy + AzBz .

APPLICATIONS OF DOT PRODUCT :

1 . W = F.S Dot product of force and displacement is work .

2 . P = F.V Dot product of force and velocity is power.

3 . E = mgh Dot product of gravitational force and vertical displacement is P.E.

4 . = B.A Dot product of area vector and magnetic flux density vector.

5. Angle between the two vectors a and b is a.b/|a| |b| .

forces in common for mechanics

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