Class 9th Science: Chapter 9 Force and Laws of Motion
If an object does not change its position with respect to time and the surroundings, it is said to be at rest, else it is said to be in motion. Force is that which changes or tries to change the state of rest or of motion of an object by a push or pull.
The magnitude of force on an object is given by the product of the mass of the object (m) and its acceleration (a). Mathematically it is expressed by the equation, F = ma.
CGS unit of force is dyne and the SI unit is newton (N). The line along which a force acts on an object is called the line of action of the force. The point where the force acts on an object is called the point of application of the force.
When a number of forces act simultaneously on an object then their equivalent is the net force on an object. If the net force is zero the forces are said to be balanced which results in zero acceleration, else the forces are said to be unbalanced which results in acceleration of the object.
The force that opposes the relative motion between the surfaces of two objects in contact and acts along the surfaces in contact is called the force of friction or simply friction. According to the concepts developed by Galileo Galilei and Isaac Newton, if a body is either at rest or in uniform motion along a straight line path, then it is said to be in its natural state.
When the forces acting on an object are balanced, the net force or the resultant force acting on the body is zero. In such cases, the body continues to be in its natural state. If all the forces acting on a body result in an unbalanced force, then the unbalanced force can accelerate the body. It means that a net force or resulting force acting on a body can either change the magnitude of its velocity or change the direction of its velocity.
For example, when many forces are known to be acting on a body, and the body is found to be at rest, then we can conclude that the net force acting on the body is zero. Sometimes, balanced forces can cause a change in the shape of a body. The SI unit for force is the “newton,” and its CGS unit is the “dyne”.
Gram–centimetre per Second Square is known as “dyne”. One newton is equal to 105 dyne.
First Law of Motion
An object does not change its state of rest or uniform motion by its own. The inability of any object to change its state is called inertia. Newton’s first law of motion gives the concept of inertia and force. According to the law, any object in the universe remains in its state of rest or uniform motion. Inertia is of three types, namely Inertia of rest, Inertia of motion and Inertia of direction. Mass of an object is an intrinsic property of matter.
Mass is a measure of inertia of an object. If an object does not change its position with respect to time and the surroundings, it is said to be at rest and else it is said to be in motion. The rate of change in displacement of an object is called velocity. If an object covers equal displacements in equal intervals of time, the motion is called uniform motion, else it is non-uniform motion.
Mass and Inertia
According to Newton’s first law of motion, inertia is the natural tendency of an object to resist any change in its natural state of motion.Inertia of an object is not a physical quantity, and hence, we can’t measure it directly. Therefore, it does not have any unit.
In other words, all objects resist a change in their state of rest or motion.
In a qualitative way, the tendency of undisturbed objects to stay at rest or to keep moving with the same velocity is called inertia. This is why, the first law of motion is also known as the law of inertia. Inertia is of the following three types.
Inertia of Motion
The natural tendency of an object to resist a change in its state of motion is called inertia of motion.
We tend to remain at rest with respect to the seat until the driver applies a braking force to stop the motorcar due to the inertia of motion.
Inertia of Rest
The natural tendency of an object to resist a change in its state of rest is called inertia of rest.
The dust particles fly away from the blanket when we jerk it due to the inertia of rest.
Inertia of direction
The natural tendency of an object to resist a change in its direction of motion is called inertia of direction.
We tend to move out ward with respect to the seat at the cured roads as the vehicle takes a sudden due to the inertia of direction.
The mass of a body is the measure of its inertia. It means that heavier and more massive bodies offer more inertia than lighter and less massive bodies.
If a large body is at rest, then a large force is required to put it in motion. For example, even a small child can push a toy car. However, An adult also can’t push a loaded vehicle forward.
This is the reason why kicking a football is a pleasure, while kicking a large stone is very painful.
Galileo experimentally proved that objects in motion move with constant speed when there is no force acting on it. He performed many experiments with inclined planes. He noted that when a sphere is rolling down an inclined plane, its speed increases because of the gravitational pull acting on it. The speed of the sphere decreases as it climbs an inclined plane. When two inclined planes are placed side by side, and the sphere that rolls down the first inclined plane is made to climb the second inclined plane, it comes to rest after reaching a certain height. According to Galileo, if the force acting on the sphere is only gravitational force, then the height reached by the sphere must be equal to the height from which it was rolled. When the inclinations of the two planes are the same, the distance travelled by the sphere while rolling down is equal to the distance travelled by it while climbing up.
Now, if the inclination of the second plane is decreased slowly, then the sphere needs to travel over longer distances to reach the same height. If the second plane is made horizontal, then the sphere must travel forever trying to reach the required height.
This is the case when there is no unbalanced force acting on it. However, in reality, frictional forces bring the sphere to rest after it travels over a finite distance.
After further study, Newton, in his first law of motion, stated that all objects resist a change in their natural state of motion.
This tendency of resisting any change in the natural state of motion is called “inertia”.
Momentum and Second Law of Motion
When a cricketer catches a ball he moves his hand back while catching the ball. He does this to reduce the impact, due to the force of the ball on his hand. An object in motion has momentum. Momentum is defined as the product of mass and velocity of an object. The momentum of the object at the starting of the time interval is called the initial momentum and the momentum of the object at the end of the time interval is called the final momentum. The rate of change of momentum of an object is directly proportional to the applied force.
Newton's second law quantifies the force on an object. The magnitude of force is given by the equation,
F = ma, where 'm' is the mass of the object and 'a' is its acceleration. The CGS unit of force is dyne and the SI unit is newton (N).
A large amount of force acting on an object for a short interval of time is called impulse or impulsive force. Numerically impulse is the product of force and time. Impulse of an object is equal to the change in momentum of the object.
Impulse and Impulsive Force
The momentum of an object is the product of its mass and velocity. The force acting on a body causes a change in its momentum. In fact, according to Newton’s second law of motion, the rate of change in the momentum of a body is equal to the net external force acting on it.
Another useful quantity that we come across is “impulse”. “Impulse” is the product of the net external force acting on a body and the time for which the force is acted.
If a force “F” acts on a body for “t” seconds, then Impulse I = Ft.
In fact, this is also equal to the change in the momentum of the body. It means that due to the application of force, if the momentum of a body changes from “P” to “P ' ”, then impulse,
I = P ' - P.
For the same change in momentum, a small force can be made to act for a long period of time, or a large force can be made to act for a short period of time. A fielder in a cricket match uses the first method while catching the ball. He pulls his hand down along with the ball to decrease the impact of the ball on his hands.
In a cricket match, when a batsman hits a ball for a six, he applies a large force on the ball for a very short duration. Such large forces acting for a short time and producing a definite change in momentum are called “impulsive forces”.
Derivation of Newton’s Second Law of Motion
Newton’s second law of motion states that the rate of change of momentum of an object is Proportional to the applied unbalanced force in the direction of force.
Suppose an object of mass, m is moving along a straight line with an initial velocity, u. It is uniformly accelerated to velocity, v in time, t by the application of a constant force, F throughout the time t. The initial and final momentum of the object will be, p1 = mu and p2 = mv respectively.
The change in momentum = p2 – p1
The change in momentum = mv – mu
The change in momentum = m × (v – u).
The rate of change of momentum = m × (v - u)t
Or, the applied force,
F ∝ m × (v - u)t
F = km × (v - u)t = kma ---------------------------- (i)
Here a = [(v - u)t] is the acceleration, which is the rate of change of velocity. The quantity, k is a constant of proportionality. The SI units of mass and acceleration are kg and m s-2 respectively. The unit of force is so chosen that the value of the constant, k becomes one. For this, one unit of force is defined as the amount that produces an acceleration of 1 m s-2 in an object of 1 kg mass. That is,
1 unit of force = k × (1 kg) × (1 m s-2).
Thus, the value of k becomes 1. From Eq. (i)
F = ma
The unit of force is kg m s-2 or newton, represented as N.
Derivation of Newton’s first law of motion from Newton’s second law of motion
Newton's first law states that a body stays at rest if it is at rest and moves with a constant velocity if already moving, until a net force is applied to it. In other words, the state of motion of a body changes only on application of a net non-zero force.
Newton's second law states that the net force applied on a body is equal to the rate of change in its momentum. Mathematically,
F⃗ = dp⃗ dt
Where, F is the net force, and p is the momentum. Now, we can write the same as:
F⃗ = dp⃗ dt= d(mv−→)dt
⇒ F⃗ = d(mv⃗ )dt
⇒ F⃗ = m dv⃗ dt
So, if the net force, F is zero, change in the value of v is be zero i.e., a body at rest will be at rest and a body moving with constant velocity will continue with the same velocity, until a net force is applied. This conclusion is similar to the Newton’s first law of motion.
Thus, we can derive Newton’s first law of motion using Newton’s second law of motion.
Derivation of Newton’s third law of motion from Newton’s second law of motion
Consider an isolated system of two bodies A & B mutually interacting with each other, provided there is no external force acting on the system.
Let FAB, be the force exerted on body B by body A and FBA be the force exerted by body B on A.
Suppose that due to these forces FAB and FBA, dp1/dt and dp2/dt be the rate of the change of momentum of these bodies respectively.
Then, FBA = dp1dt ---------- (i)
=> FAB = dp2dt ---------- (ii)
Adding equations (i) and (ii), we get,
FBA + FAB = dp1dt + dp2dt
⇒ FBA + FAB = d(p1+p2)dt
If no external force acts on the system, then
d(p1+p2)dt = 0
⇒ FBA + FAB = 0
⇒ FBA = - FAB---------- (iii)
the above equation (iii) represents the Newton's third law of motion (i.e., for every action there is equal and opposite reaction)...
Third Law of Motion
For every action, there is an equal and opposite reaction is Newton’s third law of motion. This tells us that all forces in nature acts in pairs. These actions and reactions help in understanding the motion of bodies on which forces act. The law also helps in resolving issues in several applications of forces, namely when two bodies collide. The momentum of the bodies before collision and after collision can be worked out using the third law.
If we consider bodies moving along a straight path, the momentum they possess is called linear momentum. If two spheres in a linear motion collide, their momentum before and after the collision can be related using Newton’s third law of motion.
Law of Conservation of Momentum is derived from Newton's third law of motion using the mathematical expression of force, which is derived from Newton’s second law of motion, enunciates that in the absence of external forces, if two bodies collide, the total momentum of the bodies before the collision and after the collision remains the same. This is the law of conservation of linear momentum. After collision if the two bodies stick together, their common speed or velocity can be calculated by using the law of conservation of linear momentum.
Conservation of Momentum
There are very few laws in physics that are known to be valid in all situations. The “law of conservation of momentum” is one such very important law for which no exception has been found so far. According to the law of conservation of momentum, when no external unbalanced force is acting, the sum of the momenta of a system of particles is constant. This law is applied for a collision between two bodies, According to the law of conservation of momentum the total momentum of the colliding bodies before collision is equal to the total momentum after collision.
We can apply this law for different situations. For example, consider a boy standing on a boat at rest. When he jumps from the boat on to the bank, the boat will no longer be at rest. The subsequent motion of the boat and its velocity can be explained using the law of conservation of momentum.
When a bullet is fired from a gun, the gun gets some velocity in the opposite direction. The velocity of the gun, commonly called recoil velocity, can be calculated by applying the law of conservation of momentum.
• When one object exerts a force on another object, the second object instantaneously exerts a force back on the first.
• These two forces are always equal in magnitude but opposite in direction.
• These forces act on different objects but never on the same object.
• Every action has an equal and opposite reaction.
There are several applications of Newton’s third law of motion; the launching of satellites is one among them.
Derivation of law of conservation of momentum from Newton's third law:
Law of conservation of momentum statesthat if no external force acts on a system of particles, the algebraic sum of the linear momenta of the particles remains conserved."
Consider two particles A and B, Which collide head on as shown in figure The particles move in a straight line before and after collision.
Let particle A have initial velocity u1 and particle B has initial velocity u2 . The two particles will collide, if u1 > u2 . Let after collision the final velocities of A and B becomes v1 and v2 respectively . The two particles will seperate after collision, if v2 > v1 . Let the two particles A and B have m1 and m2 respectively.
Initial momentum of particle A is m1u1
Initial momentum of particle B is m2u2
Final momentum of particle A is m1v1
Final momentum of particle B is m2v2
Rate of change of momentum of particle A = m1v1-m1u1t = m1(v1-u1)t ........(1)
Rate of change of momentum of particle B = m2v2-m2u2t = m2(v2-u2)t ........(2)
Let FAB−→− and FBA−→− , be the forces exerted by particle B on A and particle A on B respectively.
According to Newton's second law of motion
FAB−→− = m1v1-m1u1t = m1(v1-u1)t and FBA−→− = m2v2-m2u2t = m2(v2-u2)t
Then, by Newton's third law of motion, we have
FAB + FBA = 0
FAB = - FBA
ie, m1v1-m1u1 = -(m2v2-m2u2)
⇒ m1u1+m2u2 = m1v1+m2v2
or initial momentum of the system = final momentum of the system
which is the law of conversation of momentum.
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