Laws of Motion Physics Notes
we have learnt to describe the motion of an object in terms of its displacement, velocity and acceleration. Now, the important question arises: what makes an object move? Or what makes a ball roll along the ground to come to halt? We know from our everyday experience that we need to push or pull an object if we want to change its position.
The Greek thinker Aristotle (384 B.C. – 322 B.C.) held the view that if a body is moving, something external is needed to keep it moving.
However, there are some situations where the cause behind an action is not visible. For example: Why raindrops fall to the ground? Why does the Earth move around the Sun?
In this chapter, we will learn the basic laws of motion and discover the forces that cause motion. The concept of force discussed in this chapter will be useful in different branches of Physics.
Newton demonstrated that force and motion are closely connected. The laws of motion are fundamental and make us understand everyday phenomenon.
→ Force: It is a push or a pull, which either changes or tends to change the state of rest or of uniform motion of a body. It is a vector quantity and is devoted by \(\vec{F}\).
→ Inertia: It is the property of matter in which an object that is at rest wants to remain at rest, and object that is moving wants to remain moving in a straight line unless another force acts upon it.
In other words inertia is the resistance of any physical object to any change in its state of motion. This includes changes to the object’s speed, direction or state of rest.
→ Momentum: It is the quantity of motion contained in a body and is measured as the product of the mass of the body and its velocity. It is a vector quantity and is denoted by \(\vec{P}\) mathematically, \(\vec{p}\) = m \(\vec{v}\)
→ Newton’s first law of motion: It states that an object at rest, stays at rest and an object in motion stays in motion with the same speed in the same direction unless acted upon by an unbalanced force. It is sometimes referred to as the law of inertia.
→ Newton’s second law of motion: It states that the time rate of change of momentum of a body is directly proportional to the external force applied on it and the change in momentum takes place in the direction of force.
Mathematically,
Absolute unit of force: In SI, the absolute unit of force is Newton (N):
1 Newton (1N) = 1 kg 1m/ s2 =1 kgm/s2
→ Newton’s third law of motion: It states that to every action, there is an equal and opposite
reaction. Mathematically, \(\overrightarrow{F_{A B}}=-\vec{F}_{B A}\)
→ Impulse: It is defined as the product of the average force acting during the impact and the time for which the force lasts. In Classical Mechanics, the impulse is the integral of a force F, over the time interval, t, for which it acts. Since force is a vector quantity, the impulse is also a vector in the same direction.
It is also equal to the total change in momentum produced during the impact.
If \(\vec{F}_{a v}\) is the average force acting during the impact then \(\vec{I}=\vec{F}_{a v} t=\overrightarrow{p_{2}}-\overrightarrow{p_{1}}\)
Unit: SI, unit of impulse is Ns or kg m/s.
→ Law of conservation of linear momentum: It no external force acts on a system, then its total linear momentum remains conserved.
Mathematically, \(m_{1} \overrightarrow{u_{1}}+m_{2} \overrightarrow{u_{2}}=m_{1} \overrightarrow{v_{1}}+m_{2} \overrightarrow{v_{2}}\) provided Fext =0.
Linear momentum depends on frame of reference but law of conservation of linear momentum is independent of frame of reference.
Newton’s law of motion are valid only in inertial frame of reference.
→ Rocket Propulsion: Rocket is an example of variable mass, following law of conservation of momentum.
(a) Thrust on rocket at any instant: F = -v\(\frac{d m}{d t}\)
Where v = exhaust speed of the burnt gases and \(\frac{d m}{d t}\) = rate of gases combustion of fuel
(b) Velocity of rocket at any instant is given by
v = v0 + vg loge\(\left(\frac{m_{0}}{m}\right)\)
where v0 = initial velocity of rocket
m0 = initial mass of the rocket
m = present mass of the rocket
If the effect of gravity is taken in to account, then speed of rocket ,
v = v0 + vg loge\(\left(\frac{m_{0}}{m}\right)\) – gt
→ Friction: A force acting on the point of contact of the objects, which opposes the relative motion is called friction. It acts parallel to the contact surfaces.
Frictional forces are produced due to inter-molecular interaction acting between the molecules of the bodies in contact. Friction is of three types:
(a) Static friction: It is an opposing force that comes into play when one body tends to move over the surface of the other body but the actual motion is not taking place.
Static friction is a self-adjusting force that increases as the applied force is increased.
(b) Limiting friction: It is the maximum value of friction when the body is at the verge of starting motion.
Limiting friction (fs)max = μsN = μsmg
Where, μs = coefficient of limiting friction and N = normal reaction
Limiting friction does not depend on the area of contact surfaces but depends on their nature i. e., smoothness or roughness.
(c) Kinetic friction: If the body begins to slide on the surface, the magnitude of the frictional force rapidly decreases to a constant value f which is called kinetic friction.
Kinetic friction fk = μkV
= μkmg
where μk = coefficient of kinetic friction and
N = normal force
Kinetic friction is of two types:
- Sliding friction
- Rolling friction
As rolling friction < sliding friction,
Therefore it is easier to roll a body than to slide.
→ Centripetal force: An external force required to make a body move along the circular path with uniform speed is called centripetal force.
It acts along a radius and towards the centre of the circular path.
Mathematically: Centripetal force
= \(\frac{m v^{2}}{r}\) = mω2r
= 4mπv2r
→ A vehicle taking circular turns on a level road: If the coefficient of friction between the tyre and the road is p, the maximum velocity which vehicle can take a circular turn of radius r is given by
vmax = \(\sqrt{\mu \mathrm{rg}}\)
→ Banking of tracks (roads): So that vehicles can move on a curved track of radius r with a maximum speed v, the track is banked through an angle θ given by
v2 = rgtanθ
or tan θ = \(\frac{v^{2}}{r g}\)
The maximum permissible speed of a vehicle on a banked road at angle θ is given by
vmax = \(\left[\frac{r g(\mu+\tan \theta)}{1-\mu \tan \theta}\right]^{1 / 2}\)
where µ = coefficient of the friction between the road and tyres of the vehicle.
→ Motion in a vertical circle: For a body of mass ‘m’ to just loop a vertical circle of radius r,
(a) Minimum velocity of the body at the lowest point;
v1 = \(\sqrt{5 g r}\)
(b) Velocity at the highest point, u2 = \(\sqrt{g r}\)
(c) Tension in the string at the lowest point, T = 6 mg
→ Concurrent Forces: The forces acting at the same point are called concurrent forces.
→ Equilibrium of concurrent forces: A number of concurrent forces acting on a body are said to be in equilibrium, if the resultant of these forces in zero or if the concurrent forces can be represented completely by the sides of a closed polygon taken in the same order
Mathematically: \(\vec{F}+\vec{F}_{2}+\vec{F}_{3}+\ldots+\vec{F}_{n}\) = 0
→ Free body diagram: A diagram for each body’ of the system depicting all the forces on the body by the remaining part of the system is called the free body diagram.
→ Force: Is an external agency which changes or tends to change the state of a body.
→ Inertia: Is the property of an object which resist to the change.
→ Momentum: The product of mass and velocity of a body which is moving.
→ Newton: The force which produces the acceleration of 1m/s 2in a mass of 1 kg, is equal to one Newton force.
→ Impulse: The total effect of force on the motion of a body is called impulse.
→ Friction: The force that resists the relative motion of solid surfaces, fluid layers and material elements that slide against each other is known as friction.
→ Concurrent forces: Are two or more forces whose lines of action intersect at a common point.