Syllabus:
- Linear momentum, Impulse
- Conservation of linear momentum
- Application of Newton’s laws
- Moment, torque and equilibrium
- Solid friction: Laws of solid friction and their verifications.
Laws of motion
Inertia of rest and motion:
Inertia of rest:
It is the inability of body to change by itself, its state of rest. This means that body at rest will remain at rest and cannot start moving on its own.
Inertia of motion:
It is the inability of the body to change its state of uniform motion, which cannot be accelerated or retarded on its own and comes to rest.
Momentum:
Moving body tries to exert force on anything that tries to stop it. The faster the object is travelling the harder it is to stop. So, this quality of a moving body is called momentum. It is a vector quantity, so it has direction. The S.I unit of momentum is kilograms per second (kg·m/s). Momentum can also be expressed as Newton second (Ns).
Angle of friction:
The angle of a plane to the horizontal, when the body placed on the plane will just start to slide. This phenomenon is called the angle of friction. For example, consider a block placed on a rough surface. The reaction force acting on the block will be \(\vec{R}\), because it is equal and opposite to the weight \(\vec{W}\). Now by applying the force in horizontal direction \(\vec{P}\), the block will just begin to slide. Under such situation, friction force will be equal to the limiting force. When this condition is fulfilled, the resultant angle will be the angle of friction.
Formula of angle of friction:
tanθ = μ
Angle of repose:
The maximum slope, measured in degrees from the horizontal, at which loose materials will remain in place without sliding. This is known as angle of repose. For example, when the platform of a truck containing stones is raised to a certain angle so that the stones begin to slide down, this is because the gravitational force along the platform just overcomes the frictional force. This angle, to which the platform is inclined with the horizontal, will be the angle of repose.
Relationship between angle of friction and angle of repose:
Angle of repose is defined as a minimum angle of inclination of a plane structure with the horizontal such that a body kept on it just begins to slide down the plane.
Let us consider a mass m is kept on the plane surface AB and it is inclined to angle θ such that the object begins to slide.
The component of the weight \(mg\) normal to the surface is \(mg \cosθ\) which is balanced by the reaction force \(R\) and the component \(mg \sinθ\) acts along the inclined surface which is balanced by frictional force.
\(mg \sinθ = F_s\) ................................i
\(mg \cosθ = R\) ..................ii
From above:
\(\frac{mg \sinθ}{mg \cosθ} = \frac{F_s}{R}\)
tanθ = \(\frac{F_s}{R}\)
As μ = \(\frac{F_s}{R}\)
tanθ = μ
This shows that tangent of angle of repose is equal to the coefficient of friction between the surfaces.
Momentum and impulse:
Momentum
The momentum of an object is defined as the product of its mass and velocity. It is a vector quantity, so it has direction. The S.I unit of momentum is kilograms per second (kg·m/s). Momentum can also be expressed as Newton second (Ns).
Impulse:
Change in the momentum of the body is called Impulse. For example, when a bat hits a cricket ball, the force certainly varies during this collision. In this case, \(F \times t\) will be called as force of the impulse.
Impulse = \(F \times t = mv_f - mv_i\)
Newton’s third law of motion:
Newton's third law states for every action, there is an equal and opposite reaction. The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. The size of the forces on the first object equals the size of the force on the second object.
1. When we sit in a chair, our body exerts a downward force on the chair and the chair exerts an upward force on our body. These two forces are called action and reaction forces.
2. Suppose a person slams the wall with his fist. The force applied by the person through his fist on the wall is equal to the force applied by the wall on the fist. The harder the person slams the wall, the more he gets hurt. A similar procedure is followed when a person kicks a football.
3. The force exerted by the hammer on the nail is action and the force exerted by the nail on the hammer is reaction. Here, the force exerted by the hammer on the nail is equal in magnitude and opposite in direction to the force applied by the nail on the hammer.
4. When a rocket moves in space, it pushes the gas outside from it, i.e., the rocket applies force on the gases in the backward direction. As a reaction, the gases put an equal amount of force on the rocket in the opposite direction and the rocket moves in the forward direction.
5. A ball follows the projectile motion and accelerates towards the earth due to the force of gravity applied by the earth on the ball. Similarly, the ball also applies the same force to the earth and tries to attract the earth towards it. But the earth's mass is very large, so the acceleration produced in the ball is very small or negligible.
Newton’s laws of motion:
First law of motion:
Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.
Second law of motion:
It states that the rate of change of momentum of the body with respect to time is directly proportional to the external force applied and the changes take place in the direction of force.
\(F ∝ \frac{dp}{dt}\)
\(F = K \frac{dp}{dt}\)
Where k is a proportionality constant whose value is chosen to be 1
\(F = \frac{d(mv)}{dt} = F = \frac{mdv}{dt}\)
∴ \(F = ma\)
Newton’s third law of motion:
Newton's third law states for every action, there is an equal and opposite reaction. The statement means that in every interaction, there is a pair of forces acting on the two interacting objects. The size of the forces on the first object equals the size of the force on the second object.
1. When we sit in a chair, our body exerts a downward force on the chair and the chair exerts an upward force on our body. These two forces are called action and reaction forces.
2. Suppose a person slams the wall with his fist. The force applied by the person through his fist on the wall is equal to the force applied by the wall on the fist. The harder the person slams the wall, the more he gets hurt. A similar procedure is followed when a person kicks a football.
3. The force exerted by the hammer on the nail is action and the force exerted by the nail on the hammer is reaction. Here, the force exerted by the hammer on the nail is equal in magnitude and opposite in direction to the force applied by the nail on the hammer.
4. When a rocket moves in space, it pushes the gas outside from it, i.e., the rocket applies force on the gases in the backward direction. As a reaction, the gases put an equal amount of force on the rocket in the opposite direction and the rocket moves in the forward direction.
5. A ball follows the projectile motion and accelerates towards the earth due to the force of gravity applied by the earth on the ball. Similarly, the ball also applies the same force to the earth and tries to attract the earth towards it. But the earth's mass is very large, so the acceleration produced in the ball is very small or negligible.
Verification of first law of motion:
The first law of motion can be verified using an air track. An air track is a device that eliminates friction. When an air track is kept horizontal, a glider placed on it remains at rest. When the glider is given a small push, it slides along the air track with uniform speed.
Verification of second law of motion:
Second law of motion is verified by means of Atwood's machine. It consists of two masses m1 and m2 attached to a string, which passes over a light frictionless pulley. The two masses are initially held at the same level. When m1 > m2, the larger mass m1 will accelerate downwards, pulling the smaller mass m2 upwards. The difference in their masses is proportional to the acceleration of the system.
Verification of third law of motion:
Third law of motion is verified using a balloon. When air is released from a balloon, it moves in the opposite direction to the direction of escaping air. The force exerted by the escaping air on the balloon is equal and opposite to the force exerted by the balloon on the air.
Applications of Newton’s laws:
Applications of first law:
1. A passenger standing in a moving bus is thrown forward when the bus suddenly stops. This is because, when the bus stops, the lower part of the body in contact with the bus comes to rest. The upper part continues to move forward due to inertia of motion.
2. When a coin is placed on a card and the card is suddenly pulled, the coin falls into the glass placed below it. This is because the coin remains at rest due to inertia of rest and the card moves away due to sudden pull.
3. When a branch of tree is shaken, the fruits fall down. This is because the branch comes into motion and the fruits remain at rest due to inertia of rest.
4. A passenger standing in a moving bus leans backward when the bus suddenly moves. This is because, when the bus starts moving, the lower part of the body in contact with the bus comes into motion. The upper part remains at rest due to inertia of rest.
Applications of second law:
1. A cricketer lowers his hands while catching a ball. By doing so, he increases the time of impact and decreases the force of impact.
2. A karate player breaks a pile of tiles with a single blow. This is because the blow is so fast that the time of impact is very small and hence the force of impact is very large.
3. While falling from a certain height, if we land on our feet, we bend our knees. This increases the time of impact and decreases the force of impact.
Applications of third law:
1. When we walk, we push the ground backwards with our feet. As a reaction, the ground pushes us forward.
2. When a boatman pushes the bank with a pole, the boat moves forward. The force exerted by the boatman on the bank is equal and opposite to the force exerted by the bank on the boat.
3. When a gun is fired, it recoils. The force exerted by the bullet on the gun is equal and opposite to the force exerted by the gun on the bullet.
4. When we swim, we push the water backwards with our hands. As a reaction, the water pushes us forward.
Applications of law of conservation of momentum:
Recoil of a gun:
When a bullet is fired from a gun, the gun exerts a force on the bullet in forward direction. As a reaction, the bullet exerts an equal and opposite force on the gun in the backward direction. This causes the gun to recoil.
Rocket propulsion:
When a rocket is fired, the fuel burns and produces hot gases. These gases are expelled at high speed in the downward direction. As a reaction, the rocket moves upward.
Collision of two bodies:
When two bodies collide, the total momentum of the system before collision is equal to the total momentum of the system after collision.
Proof of conservation of linear momentum:
Let two bodies A and B of masses m1 and m2 move with initial velocities u1 and u2 respectively. After collision, let their final velocities be v1 and v2 respectively. The total initial momentum of the system is \(m1u1 + m2u2\) and the total final momentum of the system is \(m1v1 + m2v2\). According to the law of conservation of momentum, total initial momentum = total final momentum.
\(m1u1 + m2u2 = m1v1 + m2v2\)