5.13 Summary

Summary

1. Aristotle’s view that a force is necessary to keep a body in uniform motion is wrong. A force is necessary in practice to counter the opposing force of friction.
2. Galileo extrapolated simple observations on motion of bodies on inclined planes, and arrived at the law of inertia. Newton’s first law of motion is the same law rephrased thus: “Everybody continues to be in its state of rest or of uniform motion in a straight line, unless compelled by some external force to act otherwise”. In simple terms, the First Law is “If external force on a body is zero, its acceleration is zero”.
3. Momentum (p) of a body is the product of its mass \((m)\) and velocity \((\mathbf{v})\) :
\(
\mathbf{p}=m \mathbf{v}
\)
4. Newton’s second law of motion :
The rate of change of momentum of a body is proportional to the applied force and takes place in the direction in which the force acts. Thus
\(
\mathbf{F}=k \frac{\mathrm{d} \mathbf{p}}{\mathrm{d} t}=k m \mathbf{a}
\)
where \(\mathbf{F}\) is the net external force on the body and a its acceleration. We set the constant of proportionality \(k=1\) in SI units. Then
\(
\mathbf{F}=\frac{\mathrm{d} \mathbf{p}}{\mathrm{d} t}=m \mathbf{a}
\)
The SI unit of force is newton : \(1 \mathrm{~N}=1 \mathrm{~kg} \mathrm{~m} \mathrm{~s} \mathrm{~s}^{-2}\).
(a) The second law is consistent with the First Law \((\mathbf{F}=0\) implies \(\mathbf{a}=0)\)
(b) It is a vector equation
(c) It is applicable to a particle, and also to a body or a system of particles, provided \(\mathbf{F}\) is the total external force on the system and \(\mathbf{a}\) is the acceleration of the system as a whole.
(d) \(\mathbf{F}\) at a point at a certain instant determines \(\mathbf{a}\) at the same point at that instant. That is the Second Law is a local law; a at an instant does not depend on the history of motion.
5. Impulse is the product of force and time which equals change in momentum.
The notion of impulse is useful when a large force acts for a short time to produce a measurable change in momentum. Since the time of action of the force is very short, one can assume that there is no appreciable change in the position of the body during the action of the impulsive force.
6. Newton’s third law of motion:
To every action, there is always an equal and opposite reaction In simple terms, the law can be stated thus :
Forces in nature always occur between pairs of bodies. Force on a body A by body \(B\) is equal and opposite to the force on the body \(B\) by \(A\).

Action and reaction forces are simultaneous forces. There is no cause-effect relation between action and reaction. Any of the two mutual forces can be called action and the other reaction. Action and reaction act on different bodies and so they cannot be cancelled out. The internal action and reaction forces between different parts of a body do, however, sum to zero.
7. Law of Conservation of Momentum
The total momentum of an isolated system of particles is conserved. The law follows from the second and third law of motion.
8. Friction
Frictional force opposes (impending or actual) relative motion between two surfaces in contact. It is the component of the contact force along the common tangent to the surface in contact. Static friction \(f_{s}\) opposes impending relative motion; kinetic friction \(f_{k}\) opposes actual relative motion. They are independent of the area of contact and satisfy the following approximate laws:
\(
\begin{aligned}
&f_{\mathrm{S}} \leq\left(f_{\mathrm{S}}\right)_{\max }=\mu_{\mathrm{S}} R \\
&f_{\mathrm{k}}=\mu_{k} R
\end{aligned}
\)
\(\mu_{s}\) (co-efficient of static friction) and \(\mu_{k}\) (co-efficient of kinetic friction) are constants characteristic of the pair of surfaces in contact. It is found experimentally that \(\mu_{k}\) is less than \(\mu_{s}\).