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 2nd 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 3rd 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 $.
