Newton’s Laws of Motion | Principle of Conservation of Momentum

Motion is a change of position. It is described in terms of distance moved or displacement. Distance-time graphs represent the motion of objects in pictorial form. If the distance-time graph is a straight line, it shows that the object is moving at a constant speed. An Odometer is a meter that measures the distance moved by the vehicle.

1. Basic terms

1.1. Scalar, Vector and Tensor quantities

Scalar quantities have magnitude only. They have no direction. Vector quantities have magnitude and direction both. Tensor quantities have magnitude, direction and a plane in which they act.

1.2. Displacement

Distance moved by an object is called displacement. Displacement and distance have the same unit (metre). Example- A person moving towards the right or left relative to the platform.

1.3. Speed

The speed of an object is the distance covered per unit time. Its SI unit is metre/second (m/s). Speedometer records the speed of the vehicle. Example- Speedometer of a car reading 120 km/hr shows the speed of the car.

1.4. Velocity

The velocity of an object is the displacement per unit time. Its SI unit is metre/second (m/s).  

\(Velocity ={ Displacement \over Time}\)

1.5 Acceleration

The acceleration of an object is the change in velocity per unit time. The SI unit of acceleration is ms^–2.

\( Acceleration = {Change in velocity \over Time}\)

1.6 Inertia

Inertia is the property of the body through which it resists change in its state of rest or uniform motion. Examples of Inertia are listed below.

• We get thrown backward when we are standing in a stationary bus and the driver starts the bus suddenly.
• When a running car stops suddenly, we bend forward.
• The leaves fall off a branch when it is shaken.
• When the elevator starts suddenly, there is a jerk (shaking).

1.7. Momentum

Momentum of a body is the product of its mass and velocity. It is denoted by p. Its SI unit is kilogram metre per second (kg-m/s or kg-ms^-1). Due to its higher velocity and momentum, a bullet from a gun pierces through the target more easily than a stone thrown by hand.

Momentum (p) = mass (m) × velocity (v)  

1.8. Impulse

Impulse is the product of force and time which equals the change in momentum. Its SI unit is newton second (Ns) or kg-m/s. Airbags in cars, wrapping of Chinawares in paper or straw pieces and the presence of buffers within bogies of train are common applications of impulse.    

2. Equations of Motion

1. In case of motion with uniform velocity, displacement (s) = final velocity (v) × time (t)
2. In case of motion with uniform acceleration
   1. \(v = u + at\)
   2. \(s = ut+{1\over2}{at^2}={({{u+v}\over2})t}\)
   3. \(v^2= u^2 + 2as\)
3. In the case of motion with uniform acceleration, average velocity = \({(u+v)\over2}\)

In the above equation, ‘u’ is initial velocity; ‘v’ is final velocity; ‘a’ is acceleration; ‘s’ is displacement and ‘t’ is time.

3. Newton’s Laws of Motion

The three laws of motion given by Sir Isaac Newton show the relationship between object moving and forces applied to it. The three laws are as follows:

3.1. Newton’s First law of motion

Acceleration of the object is zero if the net external force on the object is zero. This is also called the law of inertia. This implies that an object will continue in its rest position or uniform motion in a straight line until and unless some force is applied on it.

Let F₁, F₂, F₃… are the external forces acting on a body and F_net = F₁ + F₂ + F₃ +….., i.e. vector sum of all forces, then if F_net = 0, it implies that the object’s velocity must not be changing. The reason could be either the object is not moving or it is moving with constant velocity.

F_net = 0 implies, \({{\partial v}\over{\partial t}}=0\)

The law is valid only in an intertial reference frame.

Fig.1: Three laws of motion

3.2. Newton’s Second law of motion

• According to Newton’s second law of motion, the rate of change of momentum of a body is directly proportional to the applied force and occurs in the direction in which the force acts.
• Newton’s second law of motion is also expressed as

\(p=mv, a={{\partial v}\over{\partial t}}\)

\(F=ma\)

\(f={{\partial p}\over{\partial t}}\)

• SI unit of the force is newton (N). 1 N is the force that causes an acceleration of 1 ms^-2 to a mass of 1 kg.
• This law explains why a cricketer pulls their hands back at the time of catching a fast cricket ball. This decreases the velocity of the ball, rate of change of momentum and hence the impact of catching the fast ball.
• This law also explains why athletes jump on cushioned or sand beds during a high jump event.

3.3. Newton’s Third law of motion

• According to Newton’s third law of motion, there is always an equal and opposite reaction to every action. It is also called action-reaction law.
• In other words, forces always occur in pairs. Example: F_A is the force applied by an object A on another object B is equal in magnitude and opposite in direction to F_B (force by object B on A).

F_A = -F_B

• Action and reaction forces are simultaneous forces. There is no cause-effect relationship between them.
• The application of this law is seen while swimming and while walking on ground and sand.
• Swimmer pushes the water backward (action) and water pushes the swimmer forward (reaction).
• Walking on sand is difficult due to the displacement of sand by feet and due to small reactions from sandy ground.
• The motion of a jet engine produces thrust and hot exhaust gases flow out the back of the engine, and a thrusting force is produced in the opposite direction.

4. Principle of Conservation of Momentum

• This law states that the total momentum of an isolated system (i.e. a system with no external force) of interacting particles remains unchanged.
• The working of the rocket is based on the principle of conservation of linear momentum.

Let there be two objects of masses m_A and m_B travelling in a straight line in the same direction and at velocities u_A and u_B respectively. Let u_A  > u_B and the two objects collide with each other. The collision lasts for time t and object A applies a force F_AB on object B and object B applies a force F_BA on object A. After the collision, the velocities of the two objects are v_A and v_B . The momentum of object A before and after the collision will be m_Au_A and m_Av_A respectively.

Similarly, the momentum of object B before and after collision will be m_Bu_B and m_Bv_B respectively.

As per the third law of motion, the force exerted by object A on object B should be equal and opposite to the force exerted by B on A.

F_AB=-F_BA

\(m_A{{(v_A-u_A)} \over t}=-m_B{{(v_B-u_B)} \over t}\)

\(m_Au_A+m_Bu_B=m_Av_A+m_Bv_B\)

Thus, we can say that the total momentum of the objects remains conserved provided there is no external force acting on them.

5. Friction

Friction is the force that opposes relative motion between two surfaces in contact. There are two main types of friction- Static and Kinetic.

Static friction opposes impending relative motion.  Kinetic friction opposes actual relative motion. Rolling friction is the friction that resists the motion of a body rolling over a surface.

• Static friction and kinetic friction are independent of the area of contact.
• Friction increases with the increase in roughness of surface.
• Lubricants are used to reduce kinetic friction in machines.
• Ball bearings between two moving parts of a Machine are also used to reduce friction. 
• Kinetic friction is important for quickly stopping relative motion. It is used by brakes in machines and automobiles.
• Friction causes wastage of energy. It is necessary for walking, brakes of vehicles, writing on blackboard etc.  

6. Centripetal Force and Centrifugal Force

• Centripetal force is the force directed towards centre of the circle in which the body is moving.
• Centrifugal force is a pseudo force that is equal and opposite of centripetal force.
• The working of the cream separator and centrifugal drier is based on centrifugal force.
• Tension in the string provides centripetal force if a stone joined to a string is rotated in a circular path.
• Sun’s gravitational force provides centripetal force for the movement of planets around the sun.

Summary

• Straight line on distance time graph shows that the object is moving with a constant speed.
• Odometer measures the distance moved by the vehicle. Speedometer records the speed of vehicle.
• Scalar quantities have magnitude only and Vector quantities have magnitude and direction both.
• Displacement, Speed and Velocity: Displacement (distance moved by an object), Speed (distance covered per unit time) and Velocity (displacement per unit time).
• Acceleration and Inertia: Acceleration (change in velocity per unit time) and Inertia (property of resistance to change).
• Momentum and Impulse: Momentum (product of its mass and velocity) and Impulse (product of force and time).
• Newton’s laws of motion
  • First law: Acceleration of the object is zero if net external force on the object is zero.
  • Second law: The rate of change of momentum of a body is directly proportional to the applied force and occurs in the direction in which the force acts.
  • Third law: There is always an equal and opposite reaction to every action.
• Principle of conservation of momentum:  Total momentum of an isolated system of interacting particles is conserved. The working of the rocket is based on this principle.
• Friction: The force that opposes relative motion between two surfaces in contact.
• Centripetal and centrifugal force: Centripetal force is the force directed towards centre. Centrifugal force is equal and opposite of centripetal force. The working of the cream separator and centrifugal drier is based on centrifugal force.

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