Scalars and Vectors
- A quantity that has only magnitude and no direction is called a scalar quantity, e.g. length, area, volume, mass, density, speed, work, energy etc.
- A quantity that has both magnitude and direction is called a vector quantity, e.g. displacement, velocity, acceleration, force, momentum etc.
- A vector quantity is usually represented by a line with arrowhead. The magnitude of the vector is shown by the length of the line and the direction of the arrow represents the direction. To find the resultant of scalar quantities, they are simply added algebraically. To find the resultant of two or more vectors, the triangle law, parallelogram law or polygon law is applied. A single vector may also be resolved into two or more components.
1. Speed: The rate at which a body is displaced in position is called its speed. Speed is a scalar quantity.
2. Velocity: The rate of displacement of a body in a given direction is called its velocity. It is a vector quantity. Unit of both speed and velocity is m/s.
3. Acceleration: It is the rate of change of velocity. The change may be in speed or direction or both. Unit: m/s2.
4. Mass: The mass of a body is the quantity of matter contained in it. Unit: kilogram (kg).
5. Inertia: There is a natural tendency for things to continue doing what they are already doing. A body that is at rest tends to remain at rest. A body that is moving tends to continue moving at the same speed in the same direction. This tendency of a body in equilibrium, to continue in the same state is due to its inertia. It is a property possessed by all bodies, which shows the reluctance to change their state of equilibrium. It is a quality and has no units. Inertia can be measured only in terms of mass, which is a scalar quantity.
6. Force: Application of force changes or tends to change the state of rest or uniform motion of a body in a straight line. In other words, force is that which changes or tends to change the momentum of a body. F = ma. Unit: Newton.
7. Momentum: The momentum of a body is the quantity of motion contained in a body and is the product of the mass and velocity of the body. Momentum is a vector quantity having the same direction as the velocity vector, and may be resolved into components, or combined with another momentum to give a resultant, like other vector quantities. Momentum should not be confused with inertia which is merely a quality related to the mass of the body, not its velocity. Momentum = mv. Unit: kg.m/s.
Laws of Motion. Sir Issac Newton propounded three laws of motion:
- First Law. A body will continue in its state of rest or uniform motion in a straight line unless compelled to change that state by an external force.
- Second Law. The rate of change of momentum is proportional to the applied force, and the change of momentum takes place in the direction of the applied force.
- Third Law. Action and reaction are equal and opposite. (Note that action and reaction refer in this case to different bodies).
8. Newton’s second law is equivalent to defining force as that which causes acceleration. From the second law we can say that F ∝ rate of change of momentum. But momentum = mv, and if m is constant for a given body, F ∝ m X rate of change of v, or F ∝ ma. If the unit of force is chosen at that force which gives unit acceleration to unit mass, the equation can be written as F = ma
9. Impulse of a Force. Suppose a force F acts on a body of mass m for a short time t (e.g. when a bat strikes a ball), such that the velocity of the body changes from u to v.
As F = ma,
The acceleration ‘a’ produced by the force will be F/m.
Substituting a in equation v = u + at
v = u + F Δ t/ m
F Δ t = mv – mu
10. The expression on the left (F.δt) is known as the impulse of the force, and is equal to the change of momentum caused by the force.
11. Conservation of Momentum. Consider two bodies travelling in the same direction which collide with each other, the duration of the collision being a short time say δt. Throughout the collision each will experience a force equal and opposite of that experienced by the other (Newton’s third law). The impulse of the force is F δ t and is the same for each body. Thus the change of momentum will be the same for each body. If at the time of collision, body A was overtaking body B, it is apparent that the effect of the impact will be to decrease the momentum of A and increase that of B, and the total momentum of the system of two bodies will remain unchanged.
12. The Law of Conservation of Momentum. The effects of the interaction of parts of a closed system are summarized in the law of conservation of momentum, which states that the total momentum in any given direction before impact is equal to the total momentum in that direction after impact.
13. Work: A force is said to do work on a body when it moves the body along its line of action. The amount of work done is equal to the product of the force and the distance moved (s) in the direction of the force. W = Fs. Unit: Nm or Joules.
14. Energy: The energy of a body is its capacity to do work. E = Fs. Unit: Joules.
15. Power: The power of an engine is its rate of doing work.
16. Newton’s Universal Law Of Gravitation: It states that any two bodies in the universe attract each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
∴ F ∝ M1 M2/ S²
17. Acceleration Due to Gravity ‘g’. It can be seen from the universal law of gravitation that any body of some mass will be attracted towards any other body. If ‘M1’ be the mass of the earth and ‘M2’ be the mass of a body then it will be attracted towards the earth (assuming this is the only body attracting it) with a force which is given by equation above. Here ‘s’ is the distance between the given body and the centre of the earth. This implies that different bodies at the same distance from the centre of the earth, having different masses will have correspondingly different forces of gravity acting upon them. However all of them will experience the same acceleration due to gravity, provided that only the force of gravity acts upon them. This might sound a little complex however the following proof shall make the understanding easier.
As per Newton’s Law Of Gravitation we know
F ∝ M1 M2/ S²
Also if a body of mass ‘M2’ is being accelerated then the force acting on that body is:
F = M2 x a
Where in this case ‘a’ is acceleration due to earth’s gravity. Hence ‘a’ may be replaced by ‘g’ which denotes acceleration due to gravity. Therefore, we have:
F = M2 x g
Now replacing the value of F in Newton’s formula we get
M2 x g = M1 M2/ S²
Since M2 gets eliminated on both the sides of the equation, in all the cases where ‘S’ remains same the acceleration due to gravity ‘g’ shall remain same as the mass of the earth is constant.
18. Weight. The force with which a body is attracted towards the centre of the earth is called its weight. It is equal to the product of mass and acceleration due to gravity at that point. W = mg, where g is the acceleration due to gravity at that point. Since g is not constant over the earth’s surface (more at poles and lesser at equator), therefore the weight of a body is variable, unlike its mass which remains constant.
19. Density: Density is the mass per unit volume of a substance. It is denoted by the symbol ρ. Unit: kg/m³.
20. Pressure: Pressure is the force exerted per unit area. Unit: N/m2 or Pascals.
21. Stress: Stress is the force exerted between two contacting bodies or parts of a body. It is measured as the force per unit area. Unit: N/m2.
22. Strain: Strain is the deformation caused by stress. It is recorded as the change of size over the original size. Since it is a ratio, it does not have any unit.
23. Motion on Curved Path.
- Refer Fig above. Consider a body travelling along a circle of radius ‘r’ with velocity ‘v’.
- In a small time say ‘t’ the body travels from point A to point B describing an arc S which subtends an angle ‘θ’ at the centre O.
- Considering the radian theory, the relationship between the arc, angle subtended and the radius is
S = rθ
or θ = S/r
By definition we know that the rate of change of angle per unit time is angular velocity (ω), therefore:-
ω = θ/t
ω = S/rt replacing the value of θ
ω = v/r because S/t = v (Rate of change of distance is velocity)
- Now ‘v’ is the linear velocity of the body. If θ is very small, it can be considered, that the two vectors (AD and BC) representing motion at these two points (Pt A and Pt B) are of the same length. Though the magnitude has remained the same, the direction of the velocity has changed. Therefore, by definition, the body is subjected to acceleration. Vector DC represents the change in velocity between Pt A and Pt B in time ‘t’ and is represented by ‘∆v’. Therefore, the acceleration is given by DC/t. i.e. ∆ v / t, and this is the centripetal acceleration (CPA) ‘a’.
- From the figure, using radian theory, it can be said that:
- Δv = v x θ
- CPA = Δv/t
- a = v x θ/t
- a = v x ω because ω = θ/t
- a = v x v/r because ω = v/r
- a = v²/r
24. Centripetal Force. It was shown that a body travelling with uniform speed in a circle has an acceleration of v²/r towards the centre of the circle. The force producing this acceleration is termed as the centripetal force, and for a body of mass ‘m’ the centripetal force is mv²/r towards the centre of the circle.
as F = M a
F = Mv²/r because in circular motion a = v²/r
or f = W v²/ rg As Mass = weight/ acceleration due to gravity
25. Centrifugal Force. It should be noted that in the case of a body travelling in a circular path at the end of a string, while the mass is experiencing centripetal force towards the hand, there is an equal and opposite reaction on the hand holding the string, known as centrifugal force. It is the outward force on the body travelling along a curved path and represents the resistance of the mass to centripetal acceleration. Centrifugal Force is an inertia force and it exists only as an equal and opposite reaction to centripetal force.
26. Let us take the simple example of a stone on the end of a piece of string. If the stone is whirled round so as to make one revolution per second, and the length of the string is 1 meter, the distance travelled by the stone per second will be 2πr, i.e., 2π X 1 or 6.28 m.
Therefore v = 6.28 m/s, r = 1 m.
Therefore, acceleration towards centre = v²/r
- (6.28 X 6.28) / 1
- 5 m/s² (approx)
27. Notice that this is nearly four times the acceleration due to gravity, or nearly 4g. Since we are only using this example as an illustration of principles, let us simplify matters by assuming that the answer is 4g i.e. 39.24 m/s².
28. This means that the velocity of the stone towards the centre is changing at a rate 4 times as great as that of a falling body. Yet it never gets any nearer to the centre! No, but what would have happened to the stone if it had not been attached to the string? It would have obeyed the tendency to go straight on, and in doing so would have departed farther and farther from the centre. The acceleration of 4g may, in a sense be taken as the rate at which it is being prevented from doing this.
Now lets see what centripetal force will be required to produce this acceleration of 4g?
CPF = The mass of the stone X 4g.
So, if the mass is ½ kg, the centripetal force will be ½ X 4g = ½ X 4 X 9.81 = 19.62, say 20 Newtons.
Therefore the pull in the string is 20 N in order to give the mass of ½ kg, an acceleration of 4g.
Notice that the force is 20 Newtons, very roughly 2X9.81 m/s² or 2kgf, the acceleration is 4g. There is a tendency to talk about “g” as if it were a force, it is not, it is acceleration.
28. Now this is all very easy provided the centripetal force is the only force acting upon the mass of the stone – but is it? Unfortunately, no. There must, in the first place, be a force of gravity acting upon it.