Introduction
Mechanics is the branch of physics that deals with the action of forces on bodies and motion, comprised of kinetics, statics, and kinematics. The branch of science concerned with the nature and properties of matter and energy. The subject matter of physics includes mechanics, heat, light and other radiation, sound, electricity, magnetism, and the structure of atoms.
Mechanics
- Those quantities which can describe the laws of physics and possible to measure are called physical quantities.
- The physical quantities which do not depend upon other physical quantities are called fundamental quantities.
- In Standard International (S.I.) system the fundamental quantities are mass, length, time, temperature, luminous intensity, electric current and amount of substance.
- The physical quantities which depend on fundamental quantities are called derived quantities e.g. speed, acceleration, force, etc.
Units
The unit of a physical quantity is the reference standard used to measure it.
Types of Units
1. Fundamental Units
The units defined for the fundamental quantities are called fundamental or base units
Derived Units
The units defined for the derived quantities are called derived units. e.g. unit of speed or velocity (metre per second), acceleration (metre per second2) etc.
| Fundamental Physical quantity | Fundamental unit |
|---|---|
| Mass (M) | kilogram (kg) |
| Length (L) | metre (m) |
| Time (T) | second (s) |
| Temperature(q or K) | kelvin (K) |
| Electric current (I) | ampere (A) |
| Luminous intensity | Candela (Cd) |
| Amount of substance | mole (mol) |
The limit of a derived quantity in terms of necessary basic units is called dimensional formula and the raised powers on the basic
units are dimensions.
| S.NO | Physical Quantity | Formula | Dimensional Formula | SI Unit |
|---|---|---|---|---|
| 1. | Area | Length × breadth | L × L = [latex]L^2 [/latex] = [latex]M^0L^2T^0 [/latex] | [latex]m^2[/latex] |
| 2. | Volume | Length × breadth × height | L × L × L = [latex]L^3 [/latex] = [latex]M^0L^3T^0 [/latex] | [latex]m^3[/latex] |
| 3. | Density | Mass/volume | [latex]M/L^3 [/latex] = [latex]ML^{-3}T^0 [/latex] | [latex]Kg/m^3[/latex] |
| 4. | Speed or velocity | Distance/time | [latex]L/T [/latex] = [latex]M^0LT^{-1} [/latex] | m/s |
| 5. | Linear momentum | Mass × velocity | [latex]MLT^{-1}[/latex] | kg m/s |
| 6. | Acceleration | [latex]\frac{Change in velocity}{time}[/latex] | [latex]\frac{LT^{-1}}{T}[/latex] = [latex]M^0LT^{-2} [/latex] | [latex]m/s^2[/latex] |
| 7. | Force | Mass × acceleration | [latex]MLT^{-2}[/latex] | newton(N) |
| 8. | Impulse | Force × time | [latex]MLT^{-2} [/latex] × T =[latex]MLT^{-1} [/latex] | Ns |
| 9. | Pressure | Force/area | [latex]\frac{ MLT^{-2} }{L^2}[/latex] = [latex]ML^{-1}T^{-2} [/latex] | [latex]Nm^2[/latex] |
| 10. | Work | Force × displacement | [latex]MLT^{-2} [/latex] × L = [latex]ML^2T^{-2} [/latex] | joule(J) |
| 11. | Energy | Mgh or [latex]\frac{1}{2}[/latex] [latex]mv^2 [/latex] | [latex]ML^2T^{-2} [/latex] | J |
| 12. | Power | Work/time | [latex]ML^2T^{-2}/T [/latex] = [latex]ML^2T^{-3} [/latex] | watt(W) |
| 13. | Moment of force | Force × distance | [latex]MLT^{-2} × L [/latex] =[latex]ML^2T^{-2} [/latex] | N-m |
| 14. | Universal gravitational | G=[latex]\frac{ Fr^2}{m_1m_2}[/latex] | G=[latex]\frac{MLT^{-2}×L^2 }{M^2}[/latex] =[latex]M^{-1}L^3T^{-2}[/latex] | [latex]Nm^2/kg^2[/latex] |
| 15. | Surface tension | Force/length | [latex]MLT^{-2} / L [/latex] = [latex]ML^0T^{-2} [/latex] | N/m |
| 16. | Surface energy | Energy/area | [latex]ML^2T^{-2} / L^2 [/latex] = [latex]ML^0T^{-2} [/latex] | [latex]J/m^2[/latex] |
| 17. | Thrust, Tension | Force | [latex]MLT^{-2}[/latex] | newton(N) |
| 18. | Stress | Force/area | [latex]MLT^{-2}/L^2 [/latex] =[latex]ML^{-1}T^{-2}[/latex] | [latex]N/m^2[/latex] |
| 19. | Strain | [latex]\frac{change in configuration}{initial configuration}[/latex] | [latex]M^0L^0T^0[/latex] (dimensionless) | No unit |
| 20. | Modulus of elasticity | Stress/strain | [latex]ML^{-1}T^{-2}[/latex] | [latex]N/m^2[/latex] |
| 21. | Radius of gyration | Length | [latex]M^0LT^0[/latex] | m |
| 22. | Moment of inertia | [latex]Mass × (distance)^2[/latex] | [latex]ML^2T^0[/latex] | [latex]kg m^2[/latex] |
| 23. | Angle | Length/radius | [latex]M^0L^0T^0[/latex] (dimensionless) | radian |
| 24. | Angular velocity | [latex]\frac{Angular displacement}{time}[/latex] | [latex]\frac{1}{T}[/latex]=[latex]M^0L^0T^{-1}[/latex] | rad/s |
| 25. | Angular acceleration | [latex]\frac{Angular velocity}{time}[/latex] | [latex]\frac{M^0L^0T^{-1}}{T}[/latex]= [latex]M^0L^0T^{-2}[/latex] | [latex]rad/s^2[/latex] |
| 26. | Angular momentum | Moment of inertia × angular velocity | [latex]ML^2T^{-1}[/latex] | [latex]kgm^2/s[/latex] |
| 27. | Torque | Moment of inertia × angular acceleration | [latex]ML^2T^{-2}[/latex] | N-m |
| 28. | Wavelength | Length | [latex]M^0LT^0[/latex] | m |
| 29. | Frequency | No. of vibrations/s | [latex]\frac{1}{T}[/latex]=[latex]M^0L^0T^{-1}[/latex] | [latex]s^{-1}[/latex] |
| 30. | Velocity gradient | Velocity/distance | [latex]\frac{LT^{-1}}{L}[/latex] =[latex]M^0L^0T^{-1}[/latex] | [latex]s^{-1}[/latex] |
Rules of Rounding off uncertain Digits
(a) The preceding digit is raised by 1 if the uncertain digit to
be dropped is more than 5 and is left unchanged if the better is less than 5.
Example : x = 5.68[latex]\underline{6}[/latex] is rounded off to 5.69 (as 6 > 5)
x = 3.46[latex]\underline{2}[/latex] is rounded off to 3.46 (as 2 < 5)
(b) If the uncertain digit to be dropped is 5, the preceding digit raised by 1 if it is odd and is left unchanged if it is even digit.
Example : 7.735 is rounded off to three significant figures becomes 7.74 as preceding digit is odd. 7.745 is rounded off to 7.74 as preceding digit is even.
Path Length or Distance
The length of the actual path between initial and final positions of a particle in a given interval of time is called distance covered by the particle.
The shortest distance from the initial position to the final position of the particle is called displacement.
Comparative Study of Displacement and Distance.
| S.No. | Displacement | Distance |
|---|---|---|
| 1. | It has single value between two points. | It may have more than one value between two points. |
| 2. | May be[latex] +i_ve, –i_ve [/latex]or zero. | It is always > 0 ([latex]+i_ve[/latex]) |
| 3. | It can decrease with time. | It can never decrease with time. |
| 4. | It is a vector quantity | It is a scalar quantity. |
Speed = [latex]\frac{Distance traveled}{Time taken}[/latex]
Average speed [latex]\overline{V}[/latex]= [latex]\frac{Total distance traveled}{Total time taken}[/latex]
Velocity = [latex]\frac{Displacement}{Time interval}[/latex]
Average velocity = [latex]\frac{Displacement}{Total time taken}[/latex]
Acceleration
Acceleration ([latex]\vec{a}[latex])= [latex]\frac{Change in velocity}{Time interval}[/latex]
= [latex]\frac{\vec{V'} - \vec{V}}{\vec{t'} - t}[/latex]
Average acceleration = [latex]\frac{Total change in velocity}{Total time taken}[/latex]
Kinematic Equations for Uniformly Accelerated Motion
Motion under uniform acceleration is described by the following
equations.
v = u + at ;
s = ut + [latex]\frac{1}{2}[/latex][latex]at^2[/latex] and
[latex]v^2[/latex]= [latex]u^2[/latex]+ 2as
Distance Travelled in nth Second of Uniformly
Accelerated Motion
[latex]S_{n^th}[/latex] = [latex]S_n - s_{n-1}[/latex] = (un + [latex]\frac{1}{2}[/latex]{an}^{2} - [u(n-1) + [latex]\frac{1}{2}[/latex][latex]a(n-1)^2[/latex]
[latex]S_{n^th}[/latex] = u + [latex]\frac{a}{2}[/latex](2n - 1)
Relative Velocity
If [latex]\vec{V_A}[/latex] and [latex]\vec{V_B}[/latex] be the respective velocities of object A and B then relative velocity of A w.r.t. B is
[latex]\vec{V_AB}[/latex] = [latex]\vec{V_A}[/latex] - [latex]\vec{V_B}[/latex]
Similarly, relative velocity of B w.r.t. A is
[latex]\vec{V_BA}[/latex] = [latex]\vec{V_A}[/latex] - [latex]\vec{V_B}[/latex]
Velocity = [latex]\frac{Displacement}{Time interval}[/latex]
Average velocity = [latex]\frac{Displacement}{Total time taken}[/latex]
Acceleration
Acceleration ([latex]\vec{a}[latex])= [latex]\frac{Change in velocity}{Time interval}[/latex]
= [latex]\frac{\vec{V'} - \vec{V}}{\vec{t'} - t}[/latex]
Average acceleration = [latex]\frac{Total change in velocity}{Total time taken}[/latex]
Kinematic Equations for Uniformly Accelerated Motion
Motion under uniform acceleration is described by the following
equations.
v = u + at ;
s = ut + [latex]\frac{1}{2}[/latex][latex]at^2[/latex] and
[latex]v^2[/latex]= [latex]u^2[/latex]+ 2as
Distance Travelled in nth Second of Uniformly
Accelerated Motion
[latex]S_{n^th}[/latex] = [latex]S_n - s_{n-1}[/latex] = (un + [latex]\frac{1}{2}[/latex]{an}^{2} - [u(n-1) + [latex]\frac{1}{2}[/latex][latex]a(n-1)^2[/latex]
[latex]S_{n^th}[/latex] = u + [latex]\frac{a}{2}[/latex](2n - 1)
Relative Velocity
If [latex]\vec{V_A}[/latex] and [latex]\vec{V_B}[/latex] be the respective velocities of object A and B then relative velocity of A w.r.t. B is
[latex]\vec{V_AB}[/latex] = [latex]\vec{V_A}[/latex] - [latex]\vec{V_B}[/latex]
Similarly, relative velocity of B w.r.t. A is
[latex]\vec{V_BA}[/latex] = [latex]\vec{V_A}[/latex] - [latex]\vec{V_B}[/latex]
Scalar and Vector quantities
- The physical quantities which require only magnitude to express, are called scalar quantities.
- Ex. Mass, distance, time, speed, volume, density, pressure, work, energy, power, charge, electric current, temperature, potential, specific heat, frequency, etc.
- Certain physical quantities have both magnitude and direction, they are called vector quantities.
- Ex. Displacement, velocity, acceleration, force, momentum, impulse, electric field, magnetic field, current density, etc.
1st law :
Every body continues to be in its state of rest or of uniform motion in a straight line unless compelled by an external force to change its state. This fundamental property of the body is called inertia. This law is known as Newton’s first law of motion or law of inertia.
Inertia
Inertia is the property of a body due to which it opposes the
change in its state. Inertia of a body is measured by mass of the
body. It is directly proportional to the mass of the body.
i.e, [latex]∝[/latex] mass
Momentum
The linear momentum of a body ([latex]\vec{P}[/latex]) is defined as the product of the mass of the body (m) and its velocity ([latex]\vec{V}[/latex])
i.e. [latex]\vec{P}[/latex] = m[latex]\vec{V}[/latex].
Relation between momentum and kinetic energy :
Consider a body of mass m moving with velocity v. Linear
momentum of the body, p = mv.
KE of a particle can be expressed as
E = [latex]\frac{P^2}{2m}[/latex]
P = [latex]\sqrt{2mE}[/latex]
2nd law :
The rate of change of momentum of a body is directly proportional to the unbalanced external force applied on it.
i.e. [latex]\vec{F}[/latex] [latex]∝[/latex] [latex]\frac{d\vec{p}}{dt}[/latex]
[latex]\vec{F}[/latex] [latex] = k [/latex] [latex]\frac{d\vec{p}}{dt}[/latex]
(or)
[latex]\vec{F}[/latex] = m[latex]\vec{a}[/latex].
Impulse :If a large force acts on a body or particle for a smaller time, then
impulse = product of force and time.
Impulse = [latex]\vec{F} ∇t[/latex]
3rd law :
According to this law, every action has equal and opposite reaction. Action and reaction act on two different bodies and they are simultaneous. There can be no reaction without action.
If the total external force acting on a system is equal to zero, then the final value of the total momentum of the system is equal to the initial value of the total momentum of the system.
[latex]\vec{P}[/latex] = constant
[latex]\vec{P}[/latex]f = [latex]\vec{P}[/latex]i
Motion in a Lift
Let a man of weight W = Mg be standing in a lift.
Case (a) : If the lift is moving with constant velocity v upwards or downwards.
In this case there is no accelerated motion hence no pseudo force
experienced by observer 'O' inside the lift. So apparent weight,
W' = actual weight W.
Case (b) : If the lift is accelerated i.e., a = constant and in upward
direction.
Then net forces acting on the man are
(i) weight W = Mg downward
(ii) fictitious force [latex]F_0[/latex] = Ma downward.
So apparent weight,
W' = W + [latex]F_0[/latex] = Mg + Ma = M (g + a)
Case (c) : If the lift is accelerated downward with acceleration a < g :
The fictitious force [latex]F_0[/latex] = Ma acts upward while weight of a man
W = Mg always acts downward,
therefore apparent weight,
W' = W + [latex]F_0[/latex] = Mg - Ma = M (g - a)
In this case there is no accelerated motion hence no pseudo force
experienced by observer 'O' inside the lift. So apparent weight,
W' = actual weight W.
Case (b) : If the lift is accelerated i.e., a = constant and in upward
direction.
Then net forces acting on the man are
(i) weight W = Mg downward
(ii) fictitious force [latex]F_0[/latex] = Ma downward.
So apparent weight,
W' = W + [latex]F_0[/latex] = Mg + Ma = M (g + a)
Case (c) : If the lift is accelerated downward with acceleration a < g :
The fictitious force [latex]F_0[/latex] = Ma acts upward while weight of a man
W = Mg always acts downward,
therefore apparent weight,
W' = W + [latex]F_0[/latex] = Mg - Ma = M (g - a)
Whenever a body moves or tends to move over the surface of
another body, a force comes into play which acts parallel to the surface of contact and opposes the relative motion. This opposing force is called friction.
Laws of Limiting Friction
(i) It depends on the nature of the surfaces in contact and
their state of polish.
(ii) It acts tangential to the two surfaces in contact and in a
direction opposite to the direction of motion of the body.
(iii) The value of limiting friction is independent of the area of the surface in contact so long as the normal reaction remains the same.
(iv) The limiting friction ([latex]f_s max[/latex])is directly proportional to the normal reaction R between the two surfaces.
i.e,
[latex]f_s max[/latex][latex]∝ μ_{s}[/latex] R
[latex]μ_{s}[/latex] = [latex]\frac{f_s max}{R}[/latex] = [latex]\frac{
Limiting friction}{Normal reaction}[/latex]
The force directed towards the centre required for traversing a circular path is called centripetal force.
Centripetal force = F = [latex]\frac{mv^2}{r}[/latex]= [latex]mω^2r[/latex]
Motion of car on banked road.
In a banked path with curvature ([latex] θ[/latex]) with friction, the safe velocity is given by
V = [latex]\sqrt{[rg (tanθ + μ)/(1 - μtanθ)]}
[/latex]
The angle of banking, if small, is h/d where h- height to
which outer edge of the road,h/d = [latex]\frac{V^2}{rg} [/latex] is raised and d- width of the road, since for small angle tan[latex]θ[/latex] = [latex]\frac{V^2}{rg} [/latex]
Bending of cyclist :
In order to take a circular turn of radius r with speed v, the cyclist should bend himself through an angle [latex] θ[/latex] from the vertical such that
tan[latex]θ[/latex] = [latex]\frac{V^2}{rg} [/latex]
Work done by a force on a body is defined as the product of force and the displacement of the body in the direction of force. SI unit of work is joule.
Work done (W) by a constant force
([latex]\vec{F}[/latex]) producing a displacement ([latex]\vec{F}[/latex]) is
W = [latex]\vec{F}[/latex].[latex]\vec{S}[/latex] = F COS[latex]θ[/latex]
where [latex]θ[/latex] is the angle between [latex]\vec{F}[/latex] & [latex]\vec{S}[/latex].
Work done by a variable force [latex]\int\limits_{x_1}^{x_2}[/latex]Fdx
where F = f (x) and [latex] x_1 and x_2 [/latex] are initial and final positions.
Work done by a stretched spring
W =[latex]\int[/latex]kx dx = [latex]\frac{1}{2}[/latex]k[latex]x^2[/latex]
Where k is spring constant.
Power
Power is the rate of doing work.
P = [latex]\frac{dW}{dt}[/latex] = [latex]\frac{\vec{F}.d.\vec{S}}{dt}[/latex] = [latex]\vec{F}\vec{V}[/latex]
=FVCOS[latex]θ[/latex]
therefore [latex]\vec{V}[/latex] = [latex]\frac{d\vec{S}}{dt} [/latex]
where q is the angle between [latex]\vec{F}[/latex] and [latex]\vec{v}[/latex] .
Its SI unit is watt.
1 Horse power [1HP] = 746 W, 1calorie = 4.2J and 1 kW h
= [latex]3.6 × 10^6[/latex] J
Energy
Energy is the capacity of doing work. It is also a scalar quantity.
The SI unit is joule.
Work-energy theorem states that the work done on a body is equal to the change in its kinetic energy.
Kinetic energy : K.E. is the energy possessed by the body due to its motion.
K.E. = [latex]\frac{1}{2}[/latex][latex]mv^2[/latex]
where v is the velocity of the body
K.E. of rotation motion = [latex]\frac{1}{2}[/latex][latex]Iω^2[/latex]
Potential energy : P.E. is the energy possessed by the body due to its position or shape.
Gravitational P.E. = mgh (due to change in position)
(or)
P.E. = [latex]\frac{1}{2}[/latex][latex]K x^2[/latex] (due to change in shape).
k is spring constant
Law of conservation of energy:
Law of conservation of energy states that energy can neither be created nor be destroyed but it can be transformed from one form to another.
Mass-energy equivalence:
According to this theorem mass and energy are inter-convertible.
E = [latex] mc^2 [/latex]
where c= [latex]3 × 10^8 ms^{–1} [/latex] is velocity of light in vaccum or air.
If the path of a body is affected by another body when two bodies physically come in contact, then collision is said to have taken place.
Elastic collision:
Both momentum and K.E. are conserved.
For elastic collision in one dimension,
Inelastic collision:
Only momentum is conserved.
Coefficient of restitution
It is defined as the ratio of velocity of separation to the velocity of approach.
Coefficient of restitution = e = [latex]\frac{v_2 - v_1}{u_1 - u_2}[/latex]
e = 1 for perfectly elastic collision
e = 0 for perfectly inelastic collision
Center of mass is an imaginary point at which the whole mass of a body is supposed to be concentrated.
Characteristics of centre of mass:
(i) It need not hold mass physically, e.g. for a hollow sphere, centre of mass is at the geometrical centre of the sphere although there is no mass present physically at the centre.
(ii) Location of centre of mass depends on the distribution of masses and their individual location. For regular geometrical shaped bodies, having uniform distribution of mass, the centre of mass is located at their centres.
(iii) When no external force acts on a body, centre of mass has
constant velocity and constant angular momentum. Acceleration is zero.
If C.M. is the origin, then [latex]\sum_{i=1}^{n} M_i \vec{r_i}[/latex] = 0
Torque
Torque is the moment of force. It is the cross product of the force with the perpendicular distance between the axis of rotation and the point of application of force with the force.
Torque = [latex]\vec{τ}[/latex] = [latex]\vec{r}[/latex] × [latex]\vec{F}[/latex] S.I. unit is N – m
Angular momentum
Angular Momentum is the moment of linear momentum. It is also the product of the linear momentum and the perpendicular distance of the mass from the axis of rotation.
[latex]\vec{τ}[/latex] = [latex]\vec{r}[/latex] × [latex]\vec{P}[/latex]
where [latex]\vec{r}[/latex] = position relative to origin
[latex]\vec{P}[/latex] = linear momentum at position.
Angular momentum = L = [latex]\vec{r}[/latex] × [latex]\vec{P}[/latex]
S.I unit [latex]kgm^2/s[/latex]
Relation between torque and angular momentum
[latex]\vec{τ}[/latex] = [latex]\frac{d \vec{L}}{dt}[/latex]
It is equivalent to mass in rotational motion. It is defined as the sum of the product of the constituent masses and the square of their perpendicular distances from the axis of rotation.
For an n-particle system having mass points [latex]m_1,m_2,m_3.......m_n.[/latex] at perpendicular distances [latex]r_1,r_2,r_3.......r_n.[/latex], moment of inertia.
I = [latex]m_1 {r_1}^2 + m_2 {r_2}^2 + m_3 {r_3}^2 +.......+m_n {r_n}^2[/latex] = [latex]\sum_{i=1}^{n} M_i {r_i}^2[/latex]
S.I. unit is [latex]kgm^2[/latex] and it is a scalar quantity.
Radius of Gyration
It is the root mean square of the perpendicular distances of the constituent masses. It is the perpendicular distance of the point where the whole mass is concentrated form the axis of rotation.
Radius of gyration = k = [latex]\sqrt{\frac{{r_1}^2 + {r_2}^2 +{r_3}^2 +....+{r_n}^2}{n}}[/latex]
Moment of inertia I = [latex]Mk^2[/latex]
Moment of inertia of various objects:
| Name of the object | Axis | Moment of inertia |
|---|---|---|
| (a) Rod | (i) about an axis passing through its C.M. and [latex]\bot[/latex] to its length (ii) about an axis passing through one edge of the rod | [latex]Ml^2/12[/latex] [latex]Ml^2/3[/latex] |
| (b) Ring | (i) about an axis passing through C.M & [latex]\bot[/latex] to its plane (ii) about any diameter (iii) about a tangent in the plane of the ring (iv) about a tangent of the ring [latex]\bot[/latex] to the plane | [latex]MR^2[/latex] [latex]MR^2/2[/latex] [latex]\frac {3}{2}[/latex][latex]MR^2[/latex] [latex]2MR^2[/latex] |
| (c) Disc | (i) about an axis passing through C.M. and [latex]\bot[/latex] to its plane (ii) about its diameter | [latex]MR^2/2[/latex] [latex]MR^2/4[/latex] |
| (d) Solid cylinder | about its axis | [latex]MR^2/2[/latex] |
| (e) Hollow cylinder | about its axis | [latex]MR^2[/latex] |
| (f) Hollow sphere | about its diameter | [latex]\frac {2}{3}[/latex][latex]MR^2[/latex] |
| (g) Solid sphere | about its diameter | [latex]\frac {2}{5}[/latex][latex]MR^2[/latex] |
Gravitaion:
It is the force of attraction between any two bodies.
Newton's Universal Law of Gravitation:
Every body in this universe attracts every other body with a force which is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
F[latex]∝m_1 m_2[/latex] and
F[latex]∝[/latex][latex]\frac{1}{r^2}[/latex]
F[latex]∝[/latex][latex]\frac{m_1m_2}{r^2}[/latex]
therefore F= G[latex]\frac{m_1m_2}{r^2}[/latex]
G = Universal gravitational constant = [latex]6.67 × 10^{-11}Nm^2/kg^2[/latex]
Acceleration due to gravity:
The acceleration produced in a body due to gravitational force of the earth is called acceleration due to gravity (g).
g = [latex]\frac{GM_e}{{R_e}^2}[/latex] (on the surface of the earth);
[latex]M_e[/latex]= mass on the earth.
[latex]R_e[/latex] = radius of the earth.
Mass and density of the earth:
[latex]M_e[/latex] = [latex]\frac{g {R_e}^2}{G}[/latex] = [latex]6 × 10_24 kg [/latex]
[latex]R_e[/latex] = 6400 km;
Density = [latex]ρ[/latex] = [latex]\frac{3g}{4πR_eG}[/latex] = 5.5 × 10^3 kg/m^3
Variation of 'g':
(a) With height:
g' = g[1-[latex]\frac{2h}{R}[/latex]]
g' = g[latex]{[\frac{R}{R+h}]}^2[/latex]
when [latex]h ≈ R[/latex]
therefore g decreases with height
(b) With depth:
g' = g[1-[latex]\frac{d}{R}[/latex]]
therefore g decreases with depth.
At the centre of the earth d = R, g' = 0
(c) Effect of latitude:
g' = g - R[latex]ω^2cos^2 θ[/latex]
where [latex]θ[/latex] is latitude of the point.
[latex]ω[/latex] is angular velocity of the earth.
(d) Effect of the shape of the earth:
The equatorial radius is 21 km (approx) greater than the
polar radius.
therefore [latex]g∝[/latex][latex]\frac {1}{R^2}[/latex]
therefore g increases from equator to poles.
i.e., g is maximum at poles and least (zero) at equator.
Gravitational Potential : Gravitational potential at a point in a gravitational field is defined as the work done in taking a unit mass from infinity to the point.
Gravitational potential = V = -[latex]\frac {GM}{r}[/latex]
Gravitational P.E. = Gravitational potential × mass of the object
= -[latex]\frac{GMm}{r}[/latex]
Escape Speed
Minimum speed required to escape the earth’s gravitational pull.
[latex]V_e[/latex] = [latex]\sqrt{2gR}[/latex] = [latex]\sqrt{2} × V_0[/latex]
(For earth [latex]V_e[/latex] = 11.2 km/s)
where [latex]v_0[/latex] = oribital speed
It is a heavenly body or an artificial object which revolves round a planet in a particular orbit. The required centripetal force is provided by the gravitational force. Kepler’s laws of planetary motion are applicable to them.
(a) Orbital velocity of a satellite:
Velocity with which the satellite orbits around the planet
[latex]V_0[/latex] = [latex]\sqrt{\frac{GM}{R+h}}[/latex]
h is height of the orbit from the surface of the planet;
R is Radius of the planet.
If [latex]h<<R[/latex] [latex]V_0[/latex] = [latex]\sqrt{\frac{GM}{R}}[/latex] = [latex]\sqrt{gR}[/latex]
(b) Time period of a satellite: Time taken by it to complete one revolution around the planet.
T = [latex]\sqrt{\frac{3π(R + h)^3}{GρR^3}}[/latex] = [latex]\frac {2π}{R}[/latex][latex]\sqrt{\frac{(R + h)^3}{g}}[/latex]
where [latex]ρ[/latex] mean density of the planet:
For h << R; T = [latex]\sqrt{\frac{3π}{Gp}}[/latex]
(c) Height of a satellite above the surface of the planet:
H = [latex][\frac{T^2R^2g}{4{π^2}}]^{1/3}[/latex] - R
(d) “Total energy of a satellite
Orbiting on a circular path is negative” with potential energy being negative but twice as the magnitude of positive kinetic energy.
(e) Binding energy
Binding Energy of a satellite is the energy required to remove it from its orbit to infinity.
B.E. = [latex]\frac{GMm}{2r}[/latex] No energy is required to keep the satellite in its orbit. Geostationary satellites : The satellites in a circular orbit around the earth in the equatorial plane with a time period of 24 hours, appears to be fixed from any point on earth are called geostationary satellite. For geostationary satellite, height above the earth’s surface = 35800 km and orbital velocity = 3.1 km/s. Polar Satellites : A satellite that revolves in a polar orbit along north-south direction while the earth rotates around its axis in east west direction. Weightlessness A situation where the effective weight of the object becomes zero. An astronaut experiences weightlessness in space satellite because the astronaut as well as the satellite are in a free fall state towards the earth.
B.E. = [latex]\frac{GMm}{2r}[/latex] No energy is required to keep the satellite in its orbit. Geostationary satellites : The satellites in a circular orbit around the earth in the equatorial plane with a time period of 24 hours, appears to be fixed from any point on earth are called geostationary satellite. For geostationary satellite, height above the earth’s surface = 35800 km and orbital velocity = 3.1 km/s. Polar Satellites : A satellite that revolves in a polar orbit along north-south direction while the earth rotates around its axis in east west direction. Weightlessness A situation where the effective weight of the object becomes zero. An astronaut experiences weightlessness in space satellite because the astronaut as well as the satellite are in a free fall state towards the earth.
Basic Terms
Some of the frequently used terms in Physics related to matter and energy are described below.
Mechanics:
Matter:
Anything that occupies space and Possesses weight is called Matter. Matter or Material Substances consists of atoms and molecules. Atoms combine to form molecules of a substance.
Phases of Matter: One of the most obvious ways that matter varies in its phase. Matter exists in three phases are solid, liquid, gas each of which can change into one of the other according to changes in temperature and Pressure.
Solid -> Liquid -> Gas -> Plasma
Mass: The SI unit of mass is kilogram. It is the measure of inertia possessed by a piece of matter that is quantity of matter in a body.
Weight: Weight is the product of acceleration due to gravity and mass of the body.
The force of gravity on an object is its weight. Because all objects at a given position experience the same gravitational acceleration, weight is proportional to mass.
W = mg (weight on the earth)
Density: Density is an important property of a material, whether liquid, solid or gaseous and is the measure of its compactness. Density is mass per unit volume of a substance and is expressed is the SI unit as kg/[latex]m^{3}[/latex]
Density = [latex]\frac{Mass}{Volume}[/latex]
Force:
Force is that which makes a body change its state of rest or uniform motion in a straight line - it causes objects to remain stationary, to continue moving steadily or to move faster.
Centrifugal Force:
Which appears to act on a body moving in a circular path and is directed away from the centre around which the body is moving.
Centripetal Force:
A force which acts on a body moving in a circular path and is directed towards the centre around which the body is moving.
Springs are best suited force-measuring devices, stretching for compressing in reply to the application of force. The stretch or compression is directly proportional to the force for an ideal spring.
[latex]F_{spring}=-kx[/latex] (Hooke's law)
Friction:
Friction is a force that resists the movement of one surface over another. It is the opponent of motion. If an object moves left, friction acts, on it to the right. If an object moves upward, friction pushes it downward. Whenever, two objects are in contact, friction acts in such a way as to prevent or to slow their relative motion.
Energy:
The capacity of doing work is called energy. It can exist in a number of forms, for example mechanical, electrical, potential, chemical, kinetic, nuclear. The energy possessed by a body owing to its position is called potential energy.
Surface Tension:
Surface Tension is known to be due to inter-molecular attraction in the liquid surface and these forces produce a skin effect on the surface. It is a surface tension which causes water to climb up a narrow capillary.
The surface tension of a liquid decreases with increase in temperature and vanishes at the critical temperature.
Motion:
Motion almost every event that takes place in the universe involves movement or motion of one kind or the other. Motion is the change of position of a body with respect to its surroundings.
It is important that force causes change in motion, not motion itself.
Newton's first law of motion:
A body in uniform motion remains in uniform motion, and a body at rest remains at rest, unless acted on a non-zero net force.
Newton's Second law of motion:
The rate at which a body's momentum changes is equal to the net force acting on a body.
F = ma
Newton's third law of motion:
Newton's third law of motion says that forces come in pairs. that is an action has an equal and opposite reaction.
Work:
When a force creates motion in a body it implies work has been done, that is work is done by moving force. It is equal to the product of the force and the distance it moves along its line of action.
Kinetic energy is a scalar quantity that depends on an objects mass and speed.
K = [latex]\frac{1}{2}mv^{2}[/latex]
Power:
Power is the rate at which work is done or energy is used.
The unit power is Watt(W), is equal to the 1 joule/second.
P = [latex]\frac{dW}{dt} = F.v [/latex]
Newton's law of Universal Gravitation:
This law describes the attractiveness force (F) between two masses ([latex]m_{1}[/latex]) and ([latex]m_{2}[/latex]) located a distance (r).
F = [latex]\frac{Gm_{1}m_{2}}{r^{2}}[/latex]
Escape Velocity:
A total energy, kinetic + potential, of zero marks the dividing line between closed and open orbits. An object located at a distance (r) from a gravitating mass (M) must have at least the escape velocity to achieve an open orbit and escape M's vicinity forever.
[latex]v_{esc} = \sqrt{\frac{2GM}{r}}[/latex]
Quiz
1. Which law describes the attractiveness force(F) between two masses, located a distance(r) apart?
- A. Newton's law of Universal gravitation
B. Newton's first law of motion
C. Newton's second law of motion
D. Gravitational field
- A. Coulomb
B. Ohm
C. Watt
D. Kilogram
- A. Friction
B. Energy
C. Surface tension
D. Motion
- A. F=a
B. F=[latex]ma^2[/latex]
C. F=m
D. F=ma
- A. Coulomb's law
B. Watt
C. Gram
D. Kilogram
- A. Particle
B. Matter
C. Pressure
D. Climate
- A. Matter
B. Nucleus
C. Atom
D. Molecules
- A. Elements
B. Mater
C. Plasma
D. Solid
- A. Plastic
B. Steel
C. Synthetic
D. Wood
- A. Speed
B. Velocity
C. Acceleration
D. Power