Force and Laws of Motion
Category : 9th Class
Force and Laws of Motion
Chapter Overview
"Gives us the knowledge of Laws of nature and both future and past will reveal their secrets".
-Issac Newton
In this chapter, we shall investigate the cause of motion and it will be found that a body at rest may be set into motion, when some force is applied on it.
The motion of the bodies can be understood by studying the relation between the force applied on a body and the effect produced by the applied force.
Do You Know
Force cannot be seen but the effect of force on an object can be seen or felt.
Force: It is a push or pull which either changes or tends to change the state of rest or of uniform motion of a body.
In our everyday life, we use force, quite frequently to perform various activities, some examples are kicking a football, squeezing a tube of toothpaste, lifting a book, catching a ball, opening a door etc.
Force can:
(i) Move a body lying at rest:
(a) Kicking a stationary football.
(b) Lifting a book kept on a table top.
(c) Hitting a stationary ball with a bat.
(ii) Stop a moving body:
(a) A moving ball can be stopped by the force of our hand.
(b) A moving ball can stop on its own due to the force of friction.
(c) A rotating top can be stopped by the force exerted by our hands.
(iii) Change the speed of a moving body:
(a) If a person pushes a moving swing, if will move faster
(b) When more force is applied on the pedals by a cyclist, speed of cycle increases,
(iv) Change the direction of a moving body
(a) When a batsman hits the ball with his bat, the direction of the moving ball changes.
(b) A carom counter changes its direction after a collision.
(c) A football player hitting a ball coming towards him, towards the goal post.
(v) Change the shape and size of an object:
(a) When we squeeze a toothpaste tube, its gets flattened.
(b) When we stretch a rubber band, its shape and size changes.
(c) On stretching a spring, its length changes.
1. Contact force: A force involving direct contact of two bodies is called a contact force. It is also divided into two categories:
(a) Frictional force: The force which acts to reduce relative motion between the surfaces of contact is called the frictional force.
OR
Friction is the opposing force that is set up between the surfaces of contact, when one body slides or rolls or tends to do so on the surface of another body.
Fig. 3.1.
Do You Know
The friction between two surfaces increases (rather than to decrease when the surfaces are made highly smooth.)
(b) Normal force: The force acting on a body perpendicular to the surface of contact is called a normal force.
Now, we discuss the application of normal force using some more examples:
(i) Spring force: A spring is a coiled metallic wire having a definite size and shape.
When the spring is neither stretched nor compressed, is said to be in its natural length,
Spring applies force on both ends. These forces are equal in magnitude but opposite in direction.
Fig. 3.2.
(ii) Spring balance: A spring balance consists of a spring inserted in a metal tube. The metal tube has a vertical slot through which the pointer comes out. The object to be weighed is suspended on the hook at the bottom. When the object comes to equilibrium it stretches the spring so the pointer come down and shows the reading against a scale.
Fig. 3.3.
Spring balance measures the weight of a body. The weight in this case is 12 kg.
(iii) Tension: Tension of a string has the effect of pulling the bodies tied to its ends.
Internal forces (of molecular nature) are developed in the string to oppose the extension.
These forces are called tension. We will discuss about tension in higher classes.
Fig.3.4.
Balanced forces: When two forces of equal magnitude but acting in opposite direction on an object simultaneously then the object continues in its state of rest or of uniform motion in a straight line. Such forces acting on the object are known as balanced forces.
Fig. 4.1.: A car at rest on which balanced forces are acting.
If we apply equal forces \[{{F}_{1}}\] and \[{{F}_{2}}\]simultaneously on the toy car, then the car will not move. It will remain in the state of rest. Such forces are called balanced force.
Example: (1) A student is holding a school bag in his hand. The downward force due to the weight (gravity) of the school bag is balanced by the force of pull applied by the hand in the upward direction by the boy.
(2) When we try to push a heavy box on a rough surface it does not move.
The pushing force applied by the boy is balanced by the large force of friction opposing the motion of the box.
Fig. 4.2.
(3) A rubber ball is pressed by a person in his hands by applying two equal and opposite forces. The shape of the ball changes from spherical to oblong.
Fig. 4.3.: Rubber ball becomes oblong under the action of balanced forces.
Unbalanced forces: When two forces of unequal magnitudes act in opposite directions on an object simultaneously then the object moves in the direction of the force with larger magnitude forces. These forces acting on the object are known as unbalanced forces.
Fig. 4.4.
From the above figure, it is observed that when two forces of unequal magnitudes are applied in opposite directions, on an object, then the object moves in the direction of larger force.
Remember: The net force acting on an object is not zero, whenever on it, while in case of balanced forces acting on it, while in case of balanced force, the net force acting on it is zero.
Remember these points
Aristotle (384-322) believed that the "natural state of all bodies is the state of rest and an external force is always required to move a body with a constant speed along a straight line."
But in 17th century, Galileo (1564-1642) said that "A uniform motion in a straight line did not require any force to maintain it, so if a body is in motion, it will continue to move with the same speed and along the same straight line provided it is left to itself and no external force acts on it.
The work of Galileo was further explored by Sir Issac Newton. Newton was born in England, the same year (1642) in which Galileo died. The main credit of a better understanding of the concept of force and its relation with motion goes to Newton.
He formulated three laws of motion which are the basic building blocks of dynamics.
According to this law, everybody continues in state of rest or uniform motion in a straight line unless compelled by some external force (i.e., unbalanced force) to change that state.
Newton's first law of motion defines inertia as the inherent property of a body by virtue of which it resists any change in its state of rest or of uniform motion in a straight line on its own is called its inertia. Newton's first law of motion is also known as Law of Interia.
Types of Inertia: Inertia can be divided into three types:
Inertia of Rest: The tendency of a body to oppose any change on its state of rest is known as inertia of rest.
Examples of Inertia of rest:
(1) A person standing in a bus falls backward when the bus suddenly starts moving forward.
When the bus moves, the lower part of his body begins to move along the bus while the upper part of his body continues to remain at rest due to inertia.
That is why, a person falls backward when the bus starts.
(2) The carpet is beaten with a stick to remove the dust particles.
When the carpet is beaten with a stick, the fibres of the carpet come in motion and hence move forward. On the other hand, the dust particles remain at rest due to inertia of rest.
Therefore, they fall down.
(3) When a tree is vigorously shaken some of the leaves fall from the tree.
When a tree is vigorusly shaken, the branches of the tree come in motion but the leaves tends to continue in their state of rest due to inertia of rest. As a result of this, leaves get separated from the branches of the tree and hence fall down.
(4) On hitting a striker on a pile of carrom coins, it has been observed that only the lowest coin moves away, the rest of the pile remains in the original position.
Mass of a body is the measure of inertia. If a body has more mass it has more inertia i.e., it is more difficult to change its state of rest or of uniform motion. For example, if we kick a football, it flies a long way. If we kick a stone of the same size, it hardly moves. The stone opposes the change its motion better than the football because of its more mass. Thus, stone has more inertia than football.
Memories: Inertia is directly proportional to the mass of the body.
The tendency of a body to oppose any change in its state of uniform motion is known as inertia of motion.
Examples of Inertia of Motion:
(i) When a horse running fast suddenly stops, the rider is thrown forward, if he is not firmly seated.
This is because the lower part of the rider's body, which is in contact with the horse, comes to rest while the upper part of his body tends to keep moving due to inertia of motion.
(ii) A person getting out of a moving bus or train falls in the forward direction.
As the man jumps out of a moving bus, his feet suddenly come to rest on touching the ground while the upper part of his body continues to move forward. That is why he falls with his head forward. In order to save himself from falling down, he should run in the forward direction through some distance.
(iii) An athlete runs for a certain distance before taking a long jump.
The inertia of motion gained by him at the time of jumping adds to his muscular effort and helps him in taking a longer jump.
(iv) A ball thrown upward in a moving train comes back into the thrower's hands.
The ball acquires the horizontal velocity of the train and maintain it (inertia of motion) during its upward and downward motion.
In this period the ball covers the same horizontal distance as the train, so it comes back into the thrower's hands.
The tendency of a body to oppose any change in its direction of motion is known as inertia of direction.
Examples of Inertia of direction:
(1) When a fast moving bus negotiate a curve on the road, passengers fall towards the centre of the curved road.
This is due to the tendency of the passengers to continue to move in a straight line due to inertia of direction.
(2) When a vehicle moves, the mudguard sticking to its wheels flies off tangentially.
This is due to inertia of direction. The mud guards over the wheels are provided to stop this mud to protect the clothes of the driver or persons passing near the vehicles.
(3) During the sharpening of a knife, the sparks coming from the grind stone fly off tangentially to the rim of the rotating stone. This is due to the inertia of direction.
(4) When a dog chases a hare, the hare runs along a zig-zag path.
It becomes difficult for the dog to catch the hare. This is because the dog has more mass and hence more inertia of direction than that of hare.
Demonstration: Place a water-filled tumbler on a tray. Hold the tray and turn around as fast as you can. What do you see? It will be observed that the water spills out because initially the water filled in the tumbler placed on the tray was in the state of rest. On applying a force the tray is made to move (turn around). The tumbler which is in contact with the tray also starts moving along with the tray. But the water placed in the tumbler tends to remain in its state of rest due to inertia of rest. As a result the water spills out.
Fig. 9.1.
Do You Know
Inertia of a body depends on its mass and equal to the mass of the body.
Applications of Newton's first law: A body is said to be in translational equilibrium, when:
(a) It is at rest, or
(b) When it is moving with a constant velocity.
Misconception: A state of equilibrium means that a body must be at rest.
Conception: Equilibrium means that the acceleration is zero.
Concept of Momentum: When a small piece of stone is dropped from a small height on a glass pane placed on a table, it does not break the glass pane while, when a heavy stone is dropped from the same height, the glass pane breaks. Here the small and the heavy stones have the same velocity when they fall on the glass pane.
So, the above example shows that the effect of motion of a body depends both on its mass and velocity.
Definition: Momentum of a body is equal to the product of mass (m) of the body and the velocity (v) of the body.
It is denoted by\[\overset{\to }{\mathop{p}}\,\].
Momentum = mass x velocity
\[\overset{\to }{\mathop{p}}\,=\overset{\to }{\mathop{mv}}\,\]
Momentum (p) is a vector quantity.
In magnitude form p = mass \[\times \] speed
= m \[\times \] v
Direction of momentum of a body is same as that of the direction of the velocity of the body.
Units of momentum: S.I. units of momentum is\[kgm{{s}^{-1}}\].
In C.G.S., unit of momentum is\[gcm{{s}^{-1}}\].
Case 1. When \[{{v}_{1}}>{{v}_{2}}\]
\[{{p}_{1}}=m{{v}_{1}}\]and\[{{p}_{2}}=m{{v}_{2}}\]
\[\frac{{{p}_{1}}}{{{p}_{2}}}=\frac{m{{v}_{1}}}{m{{v}_{2}}}=\frac{{{v}_{1}}}{{{v}_{2}}}\]
As \[{{v}_{1}}>{{v}_{2}}\]so\[{{p}_{1}}>{{v}_{2}}\]
\[{{m}_{1}}>{{m}_{2}},v={{v}_{2}}=v\]
\[{{p}_{1}}={{m}_{1}},v\And {{p}_{2}}={{m}_{2}}v\]
\[\frac{{{p}_{1}}}{{{p}_{2}}}=\frac{{{m}_{1}}}{{{m}_{2}}}\]
\[\because \] \[{{m}_{1}}>{{m}_{2}}\]so\[{{p}_{1}}>{{p}_{2}}\]
So, when two objects having equal linear momentum moves the lighter object will move faster than the heavier one.
Physical significance of linear momentum: It measure the motion of an object. The momentum of the moving body is proportional to the:
(i) Mass of the body
(ii) Velocity of the body.
According to this law, the change in momentum of a body per unit time is directly proportional to the unbalanced force acting on the body and the change in momentum takes place in the direction of the unbalanced-force on the body.
\[F=\frac{{{p}_{f}}-{{p}_{i}}}{t}\]
Where
\[{{p}_{f}}-\] final momentum
\[{{p}_{f}}-\]initial momentum
t – time taken for this change \[({{p}_{f}}-{{p}_{i}})\]in momentum.
Fig.: 12.1.
13. Mathematical Formulation of Second Law of Motion
Fig, 13.1
Consider a body of mass m moving with initial velocity u. Let a force F applied on the body for time t so that object acquired velocity v after time t.
Initial momentum \[{{p}_{i}}=mu\]
Final momentum \[{{p}_{f}}=mv\]
Change in momentum\[={{p}_{f}}-{{p}_{i}}\]
= mu - mu
=m (v-u)
Time taken to change this momentum
\[=(t-0)=t\]
Rate of change of momentum\[\text{=}\frac{\text{Change in momentum}}{\text{Time}\,\text{taken}}\]
\[=\frac{m(v-u)}{t}\]
According to the definition of Newton's Second Law of Motion
\[F\propto =\frac{m(v-u)}{t}\] … (i)
\[\because \] \[a=\frac{v-u}{t}\]
Fig.:13.2.
Force can be measured using Second Law of Motion.
So, eq. (i) can be written as
\[F\propto m.a\] ...(ii)
\[F=Kma\]
where K is constant of proportionality
if F =1 unit
m = 1 unit
a = 1 unit
= K or h = 1
Put this value of K = 1 in equation (ii), we get
F = ma
So, force acting on the body is directly proportional to (i) its mass, (ii) its acceleration.
If there is no external force, then F = 0
\[\therefore \]From equation (i) mv = mv =0
then mv = mv
So v = v
If u = 0, then v is also zero. It means the body will remain at rest if no external force is applied on the body.
Unit of force: As F = ma
S.I. unit of force \[=1\text{ }kg\times 1\text{ }m{{s}^{-}}^{2}=1\text{ }kgm{{s}^{-}}^{2}\]
\[kg\text{ }m{{s}^{-}}^{2}\] is known as 1 Newton (N).
Definition of ‘N’: The force is said to be 1 Newton if it produces \[1m{{s}^{-}}^{2}\]acceleration in a today of 1 kg mass.
C.G.S. unit of forces: F = ma
C.G.S. unit of force \[=1g\times 1cm{{s}^{-2}}\]
\[=1g\,cm{{s}^{-2}}\]
\[1dyne=1g\,cm{{s}^{-2}}\]
Relation between Newton and Dyne,
1 Newton \[(N)=1kgm{{s}^{-2}}\] ...(i)
\[1\text{ }kg=1000g={{10}^{3}}g\]
\[m=100cm={{10}^{2}}cm\]
Put the value in equation (i).
\[1N={{10}^{3}}g\times {{10}^{2}}cm{{s}^{-2}}\]
\[={{10}^{5}}gcm{{s}^{-2}}\]
\[1N={{10}^{5}}dyne.\]
(1) A cricket player lowers his hands while catching the ball.
We know that \[\text{F=}\frac{\text{Change in momentum}}{\text{t}}\]
When a player lowers his hands, the time to stop the ball is increased and hence, less force has to apply to cause the some change in the momentum of the ball, so the hands of the player are not injured.
(2) A person falling on a cemented floor gets injured but a person falling on a heap of sand is not injured.
When a person falls on a cemented floor, the rate of change of momentum is very large, so the large force is exerted on the body of the person and he gets injured.
But on the other hand, when a person fallen a heap of sand, the rate of change of momentum is less. Hence, a small force is exerted on the body of the person when he fall on the heap of sand.
(3) The vehicles are fitted with shockers. The shockers increase the time of transmission of the force jerk to reach the floor of the vehicle. Hence, less force is experienced by the passengers in the vehicle.
Change in momentum of a body = – force applied on the body \[\times \] time for which the force is applied = Impulse
Newton's third law of motion describes the relationship between the forces that come into play when the two bodies interact with one another.
Statement: "To every action, there is an equal and opposite reaction."
Forces in nature always occur between pairs of bodies. Force exerted on body A by body B is equal and opposite to the force exerted on the body B by A.
Fig. 16.1.
Force exerted on A by B = ? Force exerted on B by A
Some Important Points about third Law of Motion:
(1) Newton's third law of motion is applicable irrespective of the nature of the forces.
(2) Action and reaction always act on different bodies.
If they acted on the same body, the resultant force would be zero and there could never be accelerated motion.
(3) The forces of action and reaction cannot cancel each other.
This is because action and reaction though are equal and opposite always act on different bodies and so cannot balance each other.
(4) No action can occur in the absence of a reaction: For example, in a tug-of-war, one am can pull the rope only if the other team is pulling the other end of the rope. No force can e exerted if the other end is free.
One team exerts the force of action and the other team provides the force of reaction.
(1) Boatman pushes the river bank with a bamboo pole to take his boat into the river.
When the boatman pushes the river bank with a bamboo pole, the river bank offer an equal and opposite reaction. This reaction helps the boat to move into the river.
Fig. 17.1.
(2) When gun recoils according to Newton's third law of motion, an equal and opposite force is exerted on the gun.
Fig. 17.2.
The force exerted on the bullet is known as action and force exerted on the gun is known is reaction since, the mass of gun is larger than the mass of a bullet, so the gun recoils with a velocity much lesser than the velocity with which the bullet moves forward.
(3) Motion of Rokets and Jet Aero planes are also based on Newton's third law. According to Newton's third law of motion an equal and opposite reaction acting on the rocket pushes the rocket in the upward direction.
Fig. 17.3.
Statement: The total momentum of a system remains constant, if no. net external unbalanced force acts on the system.
p = constant if \[{{F}_{ext}}=0\]
Proof of the law of conservation of momentum.
Fig. 17.4.
Let the two bodies A and B have masses \[{{m}_{1}}\]and \[{{m}_{2}}\]
Let their velocities be\[{{u}_{1}}\]and\[{{u}_{2}}\]\[({{u}_{1}}>{{u}_{2}})\]
According to the definition of momentum,
momentum of body \[A={{m}_{1}}{{u}_{2}}\]
momentum of body\[B={{m}_{2}}{{u}_{2}}\]
\[\therefore \]Initial momentum of the system\[={{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}\]
Now let the body A collide with body B and after collision their velocities be\[{{v}_{1}}\]and \[{{v}_{2}}\]respectively.
So, final momentum of the system\[={{m}_{1}}{{v}_{2}}+{{m}_{2}}{{v}_{2}}\]
Body A exerts a force of action fab on body B
Body B exerts a force of action \[{{F}_{AB}}\]on body A
\[{{F}_{AB}}=\]Rate of change of momentum of body A
\[=\frac{{{m}_{1}}({{v}_{1}}-{{u}_{1}})}{t}\]
\[{{F}_{BA}}=\]Kate of change of momentum of body \[B=\frac{{{m}_{2}}({{v}_{2}}-{{u}_{2}})}{t}\]
According to Newton's third law of motion their is equal and opposite reaction
\[\therefore \] \[{{F}_{AB}}={{F}_{BA}}\]
\[\frac{{{m}_{1}}({{v}_{1}}-{{u}_{1}})}{t}=\frac{-{{m}_{2}}({{v}_{2}}-{{u}_{2}})}{t}\]
\[{{m}_{1}}{{v}_{1}}-{{m}_{1}}{{u}_{1}}=-{{m}_{2}}{{v}_{2}}+{{m}_{2}}{{u}_{2}}\]
\[{{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}={{m}_{1}}{{v}_{1}}+{{m}_{2}}{{u}_{2}}\]
\[\therefore \]Momentum of the system after the collision
= Momentum of the system before collision
This is the law of conservation of momentum.
Momentum = Mass \[\times \]Velocity or\[p=mv\]
Force = Mass \[\times \]Acceleration
or \[F=ma=m\left( \frac{v-u}{t} \right)=\frac{{{p}_{f}}-{{p}_{i}}}{t}\]
This law defines unit of force. One unit force is that force which produces a unit acceleration in a body of unit mass.
1 Newton \[=1kg\times 1m{{s}^{-2}}\]or \[1N=m{{s}^{-2}}\].
\[{{m}_{1}}{{u}_{1}}+{{m}_{2}}{{u}_{2}}={{m}_{1}}{{v}_{1}}+{{m}_{2}}{{v}_{2}}\]
Friction: Whenever a body slides or rolls over the surface of another body, a force comes into action which acts in the opposite direction of the motion of a body. This opposing force is called ‘friction'.
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