Newton's Laws and Force

1. Newton’s three laws of motion form the basis of mechanics. According to Ist law, A body continues to be in its state of rest or of uniform motion along a straight line, unless it is acted upon by some external force to change the state. This law defines force and is also called law of inertia.

According to second law, the rate of change of linear momentum of a body is directly proportional to the external force applied on the body, and this change takes place in the direction of the applied force. This law gives us a measure of force. i.e. $ F \propto \frac{d\vec{p}}{dt} $. 

According to third law, To every action, there is always an equal and opposite reaction. This law gives us the nature of force.

2. Inertia is the inability of a body to change by itself, its state of rest, or its state of uniform motion along the straight line. Inertia is of three types: (i) Inertia of rest (ii) Inertia of motion, (iii)Inertia of direction.

3. From Newton’s 2^nd law, we obtain $ \vec{F_{ext}} = m \vec{a} $ i.e. an external force is the product of mass and acceleration of the body.

4. The absolute unit of force on SI in newton (N) and on cgs system, it is dyne. 

5. According to the principle of conservation of linear momentum, the vector sum of linear momentum of all the bodies in an isolated system is conserved and is not affected due to their mutual action and reaction. An isolated system is that on which no external force is acting. In other words, If external forces acting on the system is zero then it's linear momentum is constant.  Flight of rockets, jet planes, recoiling of a gun, etc. are explained on the basis of this principle. Newton’s 3^rd law of motion can also be derived from this principle and vice-versa.

6. Apparent weight of a man in an elevator is given by $ W' = m(g \pm a) $ where mg is real weight of the man. Acceleration is (+ a), when the lift is accelerating upward and (-a) when the lift is accelerating downwards. When lift is moving uniformly (upwards/downwards). a = 0. W’ = m g = real weight. In free fall, a = g,  W' = m (g – g) = 0 i.e. apparent weight becomes zero.

7. When two bodies of masses m₁ and m₂ are tied at the ends of an inextensible string passing over a light frictionless pulley, acceleration of the system is given by, \[a = \frac{|m_1 - m_2|}{m_1+m_2}g\], Tension is given by, \[T = \frac{2m_1m_2}{m_1+m_2}g\]

8. Impulse \[\vec{I} = \vec{F_{av}} \times t = \vec{P_2}-\vec{P_1}\] where t is the time for which average force acts $ (\vec{P_2 } – \vec{P_1})$ is change in linear momentum of the body.

9. The force which are acting at a point are called concurrent forces. They are said to be in equilibrium when their resultant is zero.

FRICTION

10. Friction is the opposing force that comes into play when one body is actually moving over the surface of another body or one body is trying to move over the surface of the other. Two causes of friction are: the roughness of surfaces in contact; Force of adhesion between the molecules of the surfaces in contact.

11. Limiting friction is the maximum value of static friction. Dynamic/Kinetic friction is somewhat less than the force of limiting friction.

12. Static friction is a self adjusting force.

13. Rolling friction is less than sliding friction.

14. Laws of limiting friction are: 

(i) $ F \propto R$, where R is normal reaction and F is the friction force.

(ii) Direction of F is opposite to the direction of motion.

(iii) F does not depend upon the actual area of contact.

(iv) F depends upon the nature of material and nature of polish of the surfaces in contact.

15. Coefficient of friction is given by, $ \mu  = \frac{F}{R} $.

16. Angle of Repose ($ \theta $) is the minimum angle of inclination of a plane with the horizontal, such that a body placed on the plane just begins to slide down.

17. Acceleration of the body down a rough inclined plane, \[a = g(sin\theta - \mu cos\theta)\]

18. Work done in moving a body over a rough horizontal surface, \[W = \mu mgd \]Work done in moving a body over a rough inclined plane, \[W = mg(sin\theta + \mu cos\theta)d\]

19. Friction is a necessary evil. Some of the methods of reducing friction are polishing, lubrication; streamlining the shape etc.

20. Centripetal force is the force required to move a body uniformly in a circle. The magnitude of this force is $ F = \frac{mv^2}{r}=mr\omega^2 $. It acts along the radius and towards the centre of the circle.

21. Centrifugal force is a force that arises when a body is moving actually along a circular path, by virtue of tendency of the body to regain its natural straight line path. Centrifugal force can be treated as the reaction of centripetal force. The magnitude of centrifugal force is same as that of centripetal force. The direction of centripetal force is along the radius and away from the centre of the circle.

22. While rounding a level curved road, the necessary centripetal force is provided by the force f friction between the tyres and the road. The maximum velocity with which a vehicle can go round a level curve without skidding is $ v = \sqrt{\mu rg}$. To avoid dependence on friction, curved roads are usually banked i.e. outer edge of the curved road is raised suitably above the inner edge. If θ is the angle of banking, then $ tan\theta = \frac{v^2}{rg}$.

23. While rounding a banked curved road, the maximum permissible speed is given by \[v_{max} = \sqrt{\frac{rg(\mu_s + tan\theta)}{(1-\mu_s tan\theta)}}\]When frictional force is ignored, the optimum speed is, \[v_{max} = \sqrt{rg tan\theta }\].

24. Motion along a vertical circle is a non-uniform circular motion. Tension in the string at any position is $ T = \frac{mv^2}{r} + mgcos\theta $ where θ is the angle with vertical line through the lowest point of the circle.

1.                   For looping the vertical loop, the velocity of projection at lowest point L is $ v_L \geq \sqrt{5rg}$.

2.                   The value of velocity at the highest point H is $ v_H \geq \sqrt{rg}$.

3.                   Difference in tension in the string at lowest point and highest point of vertical circle is, $ T_L - T_H = 6mg $.

4.                   For oscillation over the arc of vertical circle $ 0 < v_L \leq \sqrt{2rg} $.

5.                   For leaving the vertical circle somewhat between $ 90^{\circ} < \theta < 180^{\circ} $, $ \sqrt{2rg} < v_L < \sqrt{5rg} $.

6.                   The minimum height h through which a motor cyclist has to descend to loop a vertical loop of radius r is, $ h = \frac{5}{2}r $.