NATURE AND TYPES OF FORCES
Force is an external activity which pushes or pulls a body or tends to change the state of rest or of uniform motion of a body.
SI unit if force is Newton (N). It can be expressed as kg-m/s2
CHARACTERISTICS OF FORCE
- It produces or tries to produce motion in a body at rest.
- It stops or tries to stop a moving body.
- It changes or tries to change the direction of motion of body.
- It can produce a change in the shape of a body.
These are the forces in which contact between two bodies is not necessary. Gravitational force between two bodies, electrostatic forces between two bodies are two examples of field forces.
Earth applies gravitational force on a mass which is known as weight (W). Magnitude of weight is W = mg where m is the mass of the body and g is the magnitude of free fall acceleration on the surface of earth.
Direction of weight is always taken towards the vertically downward direction. But actually it acts towards the centre of the earth.
The force exerted by one surface over the surface of another body when they are physically in contact with each other is known as contact force.
Two bodies in contact exert equal and opposite forces on each other. If contact force is frictionless contact force is perpendicular to the common surface and known as normal reaction.
If the objects are in rough contact and move relative to each other without losing contact then frictional force arises which opposes such motion.
Each object exerts a frictional force on each other and the two forces are equal and opposite.
This force is perpendicular to normal reaction. Thus the contact force (F) between two objects is made up of two forces.
- Normal Force
- Friction Force
The forces F1 and F2 shown in the diagram acting, respectively, on bodies A and B act away from the surfaces of contact and prevent the two bodies from “occupying the same space”.
If F1 is the action then F2 is the reaction : they are equal in magnitude and opposite in direction. F1 and F2 are both perpendicular to the surfaces in contact and act on two different bodies .
In the diagram given below few examples are given demonstrating normal forces and resolution of normal forces.
ATTACHMENT TO ANOTHER BODY
When a string , thread or wire is held tight , the ends of the string or thread pull on whatever bodies are attached to them in the direction of string. This force is known as Tension.
- If a string is inextensible the magnitude of acceleration of ay number of masses connected through string is always same.
- If a string is massless , the tension in it is same everywhere . However , if a string has a mass , tension at different points will be different.
- If there is friction between pulley and string . tension will be same on both sides of pulley .
SPRING FORCE : Whenever a spring is compressed or elongated , the elastic force developed in the spring which helps the spring to restore its original length is known as spring force .
Force is an extended (or compressed) spring is proportional to the magnitude of extension (or compression)
F = kx , where k is a positive constant (spring constant) and x is the elongation or compression from the natural length .
Aristotelian law of motion may be phrased thus: An external force is required to keep a body in motion.
NEWTONS LAWS OF MOTION:
Galileo concluded that an object moving on a frictionless horizontal plane must neither have acceleration nor retardation, i.e. it should move with constant velocity.
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.
NEWTONS 1’ST LAW:
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.
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.
Inertia of Rest
The tendency of a body to remain at rest is known as inertia of rest.
Example: Consider a stack of coins and you hit the bottom coin it slides bu the rest of the stack lies in its own position.
Inertia of Motion
The tendency of a body to remain in the state of motion is known as inertia of motion.
Example: While standing/sitting on a running bus, when breaks are applied, we bend forward.
Inertia of Direction
The tendency of a body to retain its direction is known as inertia of direction
Example: A stone tied to a string gets thrown in tangential direction.
NEWTON’S SECOND LAW OF MOTION:
Statement: The rate of change of momentum is proportional to the magnitude of force applied and takes place in the direction of the straight line in which that force is applied.
Momentum of a body is defined to be the product of its mass m and velocity v, and is denoted by p
p = m v
Momentum is clearly a vector quantity.
A seasoned cricketer catches a cricket ball coming in with great speed far more easily than a novice, who can hurt his hands in the act.
One reason is that the cricketer allows a longer time for his hands to stop the ball. As you may have noticed, he draws in the hands backward in the act of catching the ball.
Mathematical Expression of Newton’s Second Law :
Thus, if under the action of a force F for time interval ∆t, the velocity of a body of mass m changes from v to v + ∆v i.e. its initial momentum.
p = m v changes by ∆ p = m ∆v
According to the Second Law,
F is proportional to rate of change of momentum
F = k(dp/dt)
Where k is a constant of proportionality.
Taking the limit ᅀt tends to 0 , then
Becomes derivative or differential coefficient of p with respect to t , denoted by , Thus
For a fixed body of mass m ,
Then Second Law can be written as F= kma
which shows that force is proportional to the product of mass m and acceleration a.
We, therefore, have the liberty to choose any constant value for k. For simplicity, we choose k = 1. The second law then is
F= dp/dt = ma
In SI unit force is one that causes an acceleration of 1 ms-2 to a mass of 1 kg. This unit is known as Newton: 1 N = 1 kg ms-2
- In the second law, F = 0 implies a = 0. The second law is obviously consistent with the first law.
- The second law of motion is a vector law. It is equivalent to three equations, one for each component of the vectors:
This means that if a force is not parallel to the velocity of the body, but makes some angle with it, it changes only the component of velocity along the direction of force. The component of velocity normal to the force remains unchanged.
For example, in the motion of a projectile under the vertical gravitational force, the horizontal component of velocity remains unchanged.
- The second law of motion is applicable to a single point particle. The force F in the law stands for the net external force on the particle and a stands for acceleration of the particle. a is the acceleration of the centre of mass of the system.
- Any internal forces in the system are not to be included in F.
- The second law of motion is a local relation which means that force F at a point in space (location of the particle) at a certain instant of time is related to a at that point at that instant.
Acceleration here and now is determined by the force here and now, not by any history of the motion of the particle.
The product of force and time, which is the change in momentum of the body remains a measurable quantity. This product is called impulse:
Impulse = Force x Time
= Change In Momentum
A large force acting for a short time to produce a finite change in momentum is called an impulsive force.
Impulsive force is like any other force – except that it is large and acts for a short time.
NEWTON’S THIRD LAW OF MOTION
To every action, there is always an equal and opposite reaction.
- Forces always occur in pairs. Force on a body A by B is equal and opposite to the force on the body B by A.
- There is no cause effect relation implied in the third law. The force on A by B and the force on B by A act at the same instant.
- Action and reaction forces act on different bodies, not on the same body. Consider a pair of bodies A and B. According to the third law.
FAB = – FBA
(force on A by B) = – (force on B by A)
CONSERVATION OF MOMENTUM
Law of conservation of momentum : The total momentum of an isolated system of interacting particles is conserved.
An important example of the application of the law of conservation of momentum is the collision of two bodies. Consider two bodies A and B, with initial momenta PA and PB. The bodies collide, get apart, with final momenta P′A and P′B respectively. By the Second Law .
FABᅀt = P’A – PA
FBAᅀt= P’B – PB
Since , FAB=-FBA ( by third law)
Now , P’A – PA = P’B – PB
P’A + P’B = PA + PB
Torque is the measure of force that causes an object to rotate. It’s also known as moment of force and is calculated by multiplying together the magnitude of the force and perpendicular distance r from the axis of rotation. It is denotes by C or tau
C = Fxr
Direction of Torque
- The angular direction of torque is the sense of the rotation it would cause.
This is a technique which consists of diagrammatic representation of all the forces acting on a body (isolated from the surroundings).
We can apply vector techniques with Newton’s Laws of motion to study the motion of the body.
Step-1 : The first step is to decide the system . A system must have identical motion for all its particles.
Step-2: Once the system is decided, make a list of all the forces acting on the system due to all the objects other than the system itself. Any forces applied by the system on the surrounding bodies should not be included in the list of forces.
Step-3 : Now represent the system by a point in a separate diagram and draw vectors representing the forces acting on the system with this point a common origin.
Consider a triangular wedge of mass M and a block of mass m placed on it as shown in the figure. Draw free body diagram of m and triangular block M.
APPLICATION OF NEWTONS LAWS OF MOTION
Newton’s laws can be applied to study the motion of a body under the influence of a force or a set of forces.
- Isolate the system and draw the forces acting on the system . Here replace the actual system by a point mass . This is a free body diagram. Show the direction of acceleration with an arrow.
- With reference to the direction of motion of a system , select a suitable coordinate system.
- Consider the origin of forces acting on each object . To do this find out the field forces acting on the object.
- Draw the forces in the given problem , decide whether the system can accelerate or not.
- If it accelerates , find its direction.
- In case of composite problem of a number of bodies connected by strings and pulleys , find the relationship of accelerations of different bodies by constraint relationship.
- Resolve all the external forces along the chosen axes. Set up equation of motion along individual axes and calculate the net acceleration.
EQUILIBRIUM OF A PARTICLE
Equilibrium of a particle in mechanics refers to the situation when the net external force on the particle is zero.
According to the first law, this means that, the particle is either at rest or in uniform motion.
If two forces F1 and F2, act on a particle, equilibrium requires F1 = − F2
Equilibrium under three concurrent forces F1, F2 and F3 requires that the vector sum of the three forces is zero.
F1 + F2 + F3 = 0
F1x + F2x + F3x = 0
F1y + F2y + F3y = 0
F1z + F2z + F3z = 0
Suppose two unequal masses m and 2m are attached to the inextensible strings which passes over a smooth massless pulley.
We have to find the acceleration of the system. We can assume the mass 2m is pulled downwards by a force equal to its weight, i.e. 2mg.
Similarly, the mass m is being pulled by a force of mg downwards .
Using free body diagrams
T – mg = ma
2mg – T = 2ma
Solving these two equations , we get
a = g/3
Before studying pseudo force , lets understand frame of reference first.
Frame Of Reference
It is a conveniently chosen coordinate system , which describes the position and motion of a body in space.
Inertial Frame of Reference
A reference frame in which Newton’s laws are strictly valid is called as inertial frame of reference.
Any reference frame which is either or rest or in uniform motion with respect to inertial frame is itself inertial.
Non-Inertial Frame of Reference
A reference frame which accelerates or rotates with respect to an inertial reference frame. Newton’s laws are not valid in non-inertial frame. It can be applied only after applying an additional force (called pseudo force) on the body.
Motion of a particle (P) is studied from two frames of reference S and S’ . S is an inertial frame of reference and S’ is a non-inertial frame of reference . From the vector triangle OO’P , we get
r’ = r – R
Differentiating this equation twice with respect to time , we get
a’= a – A
a’ = acceleration of the particle P relative to S’
a= acceleration of the particle relative to S
A= acceleration of S’ relative to S
Multiplying the above equation by m (mass of the particle) , we get
ma’= ma- mA
F’= F(real) – mA
In a non-inertial frame of reference , an extra force is taken into account in order to apply Newton’s Laws of motion.
The magnitude of this force is equal to the product of mass of the body and acceleration of the frame and it is always directed opposite to the acceleration of the frame. This force is known as Pseudo Force, because this force doesn’t exist in the inertial frame of reference.
In case of lift going upwards the weight appears to be increased.
R= m(g+a) the weight of the man in a lift going upwards with some acceleration.
In our daily lives, we can see a football rolling over the floor stops after sometime. Similarly, it’s seen that in a slippery surface cyclist falls.
Cause: Whenever one body starts sliding on the surface of another body, some force gets developed between them which tries to oppose this relative motion.
This is called friction or frictional force, it acts whenever there is relative motion or relative motion is tried to be started. Friction can be static, sliding or rolling.
Types of Friction
- Static Friction: As the applied force F increases, fs (frictional force) also increases, remaining equal and opposite of the applied force (up to a certain limit), keeping the body at rest. Hence, it is called static friction. Static friction opposes impending motion.
- Kinetic Friction: When the body is at the verge of starting its motion, the frictional force acting at this stage is called limiting friction or kinetic friction.
It is found experimentally that the limiting value of static friction (fs) max is independent of the area of contact and varies with the normal force (N) approximately as:
(fs )max = µs N
Where µs is a constant of proportionality depending only on the nature of the surfaces in contact. The constant µs is called the coefficient of static friction.
Laws of Static Friction:
The static frictional force can take its value from anything in between zero and µs N depending on the requirement to cease relative sliding. Whenever needed it is calculated by Newton’s Laws.
Laws of Kinetic Friction
- Limiting frictional force is independent of the apparent area of contact till the value of the normal reaction remains the same.
- The direction of limiting frictional force is opposite to the direction in which one body is on the verge of starting its motion.
- The limiting frictional force depends upon the nature of surfaces in contact.
- Quantitatively, the magnitude of the force of limiting force of limiting friction (f) on one of the bodies in contact, is directly proportional to the normal reaction (N) on this body due to the other.
- f N or f = μN
Where μ is the constant of proportionality called coefficient of friction.
Angle of Friction:
The angle of friction between any two surfaces in contact is defined as the angle which the resultant of the force of limiting friction F and normal reaction N makes with the direction of normal reaction N. It is represented as θ.
Angle of Repose or Angle of sliding:
Angle of repose is defined as the minimum angle of inclination of a plane with the horizontal such that a body placed on the plane just begins to slide down.
It is represented as α. Its value depends on the material and nature of surfaces in contact.
It is the force of resistance to the motion of body when it rolls over a surface.
BANKING OF ROADS
Mostly it is seen that when a road is straight it is horizontal too. However when a sharp turn comes, the surface doesn’t remain horizontal. This is called banking of roads.
Purpose of Banking:
- To contribute in providing necessary centripetal force.
- To reduce frictional wear in tyres
- To avoid skidding
- To avoid overturning of vehicles
Case: 1 When μ= 0 (on a level road)
It is the case when there is no friction between tyres and road , yet we can take a turn.
In the given figure vertical Ncosθ component of the normal reaction N will be equal to mg and the horizontal Nsinθ component will provide necessary centripetal force.
Dividing each equation we get ,
Case-II When μ not equal to 0
In the figure shows a section of the banked road and the view of a vehicle from the rear end .
The total forces acting are
N1 and N2 = Normal Reactions
F1 and F2 = Frictional Forces
mg = weight
r= radius of the turn
θ= angle of the banking
Let N = Resultant of N1 and N2
Let F = Resultant of F1 and F2
Vector resolution in vertical direction
Ncosθ= Fsinθ + mg ….. (1)
The resultant of horizontal components , however , becomes the net external force acting on the vehicle in the radially inward direction of the round turn.
Fcosθ + Nsinθ = mv2 / r …….(2)
Further , F = μN ….. (3)
Putting (3) in equation (1)
Ncosθ = μNsinθ + mg
N(cosθ-μsinθ) = mg
N = mg /(cosθ– μsinθ) …. (4)
Putting(3) and (4) in equation (2) gives
(μmgcosθ + mgsinθ)/ (cosθ-μsinθ) = mv2 /r
Solving we get
For detailed understanding of concepts of laws of motion , pseudo forces , banking of roads , freebody diagram download the pdf given above.