SPEED AND VELOCITY
Body at rest
When a body does not change its position, with respect to its surroundings, it is said to be at rest.
Body at motion
When a body changes its position, with respect to its surroundings, it is said to be in motion. The motion may be linear if the body moves in a straight line or it may be circular when it moves in a curved path.
Displacement
When a body is in motion from one place to another, the displacement is the distance from the starting position to the final position.
Speed
$\displaystyle \small \bullet$ It is the rate of change of displacement of a body in motion.
$\displaystyle \small \bullet$ It has got no direction and it is a scalar quantity.
$\displaystyle \small Speed=\frac{Distance\; travelled}{Time\; taken}=\frac{s}{t}$
$\displaystyle \small \bullet$ Units are generally km/hour or cm/sec
Velocity
$\displaystyle \small \bullet$ It is the rate of change of displacement of a body in motion in a given direction.
$\displaystyle \small \bullet$ It is a vector quantity and can be represented both in magnitude and direction by a straight line.
$\displaystyle \small \bullet$ Velocity may be linear or angular.
$\displaystyle \small \bullet$ Units are generally km/hr, m/sec or cm/sec
Acceleration
$\displaystyle \small \bullet$ Rate of change of velocity is known as acceleration or it is the change of velocity in unit time.
$\displaystyle \small Acceleration=\frac{Change\; in \; velocity}{Time}$
$\displaystyle \small a=\frac{v-u}{t}$
$\displaystyle \small \bullet$ Its unit is $\displaystyle \small m/sec^{2}$.
u = Initial velocity in metre per second(m/sec)
v = Final velocity in metre per second(m/sec)
s = Distance in metre (m)
t = Time in second (sec)
a = Acceleration $\displaystyle \small m/sec^{2}$ (positive value)
R = Retardation $\displaystyle \small m/sec^{2}$ (negative value of acceleration)
Retardation
When the final velocity is lesser than the initial velocity the body is said to be in retardation.
$\displaystyle \small R=\frac{u-v}{t}$
Equations of Motion
$\displaystyle \small v=u+at$
$\displaystyle \small Average\; velocity=\frac{u+v}{t}$
Distance moved = Average velocity X Time
$\displaystyle \small s=ut+\frac{1}{2}at^{2}$
$\displaystyle \small v^{2}-u^{2}=2as$
Stroke speed
$\displaystyle \small \bullet$ For one revolution of the point k of the crank pin the distance the power saw blade moves = 2 X s
$\displaystyle \small \bullet$ Therefore for ‘n’ revolutions in a minute, the distance = 2 X s X n.
Average Speed, $\displaystyle \small V_{m}=2\times s\times n$
Piston speed
$\displaystyle \small \bullet$ As the piston moves backward and forward, its speed constantly changes between the upper and lower dead centres.
$\displaystyle \small \bullet$ Hence in this case also the average speed Vm =2 X s X n.
$\displaystyle \small \bullet$ Since s is expressed in mm and n in number of revolutions/per minute,
Average Speed, $\displaystyle \small V_{m}=\frac{2\times s\times n}{1000\times 60}$ m/s
Momentum
$\displaystyle \small \bullet$ It is the quantity of motion possessed by a body and is equal to the product of its mass and the velocity with which it is moving.
Momentum = mass X velocity
$\displaystyle \small \bullet$ Unit of momentum is kg metre/sec.
Newton’s Laws of Motion
Newton’s First Law of Motion
Also known as Law of Inertia. A body continues to remain in the state of rest or uniform motion in a straight line unless some external force changes its state
Newton’s Second Law of Motion
The rate of change of momentum of a body is proportional to the external force acting upon it and takes place in the direction in which the force acts.
Newton’s Third Law of Motion
To every action there is an equal and opposite reaction
Motion under gravity
$\displaystyle \small \bullet$ A body falling from a height, from rest, has its velocity increasing and it will be maximum when it hits the ground. Therefore a body falling freely under gravity has a uniform acceleration.
$\displaystyle \small \bullet$ When the motion is upward, the body is subjected to a gravitational retardation.
$\displaystyle \small \bullet$ The acceleration due to gravity is denoted with ‘g’. Its value is taken as 9.8 $\displaystyle \small m/sec^{2}$ or 32 $\displaystyle \small ft/sec^{2}$.
Equations of motion under gravity
When a body does not change its position, with respect to its surroundings, it is said to be at rest.
Body at motion
When a body changes its position, with respect to its surroundings, it is said to be in motion. The motion may be linear if the body moves in a straight line or it may be circular when it moves in a curved path.
Displacement
When a body is in motion from one place to another, the displacement is the distance from the starting position to the final position.
Speed
$\displaystyle \small \bullet$ It is the rate of change of displacement of a body in motion.
$\displaystyle \small \bullet$ It has got no direction and it is a scalar quantity.
$\displaystyle \small Speed=\frac{Distance\; travelled}{Time\; taken}=\frac{s}{t}$
$\displaystyle \small \bullet$ Units are generally km/hour or cm/sec
Velocity
$\displaystyle \small \bullet$ It is the rate of change of displacement of a body in motion in a given direction.
$\displaystyle \small \bullet$ It is a vector quantity and can be represented both in magnitude and direction by a straight line.
$\displaystyle \small \bullet$ Velocity may be linear or angular.
Acceleration
$\displaystyle \small \bullet$ Rate of change of velocity is known as acceleration or it is the change of velocity in unit time.
$\displaystyle \small Acceleration=\frac{Change\; in \; velocity}{Time}$
$\displaystyle \small a=\frac{v-u}{t}$
$\displaystyle \small \bullet$ Its unit is $\displaystyle \small m/sec^{2}$.
u = Initial velocity in metre per second(m/sec)
v = Final velocity in metre per second(m/sec)
s = Distance in metre (m)
t = Time in second (sec)
a = Acceleration $\displaystyle \small m/sec^{2}$ (positive value)
R = Retardation $\displaystyle \small m/sec^{2}$ (negative value of acceleration)
Retardation
When the final velocity is lesser than the initial velocity the body is said to be in retardation.
$\displaystyle \small R=\frac{u-v}{t}$
Equations of Motion
$\displaystyle \small v=u+at$
$\displaystyle \small Average\; velocity=\frac{u+v}{t}$
Distance moved = Average velocity X Time
$\displaystyle \small s=ut+\frac{1}{2}at^{2}$
$\displaystyle \small v^{2}-u^{2}=2as$
Stroke speed
$\displaystyle \small \bullet$ For one revolution of the point k of the crank pin the distance the power saw blade moves = 2 X s
$\displaystyle \small \bullet$ Therefore for ‘n’ revolutions in a minute, the distance = 2 X s X n.
Average Speed, $\displaystyle \small V_{m}=2\times s\times n$
Piston speed
$\displaystyle \small \bullet$ As the piston moves backward and forward, its speed constantly changes between the upper and lower dead centres.
$\displaystyle \small \bullet$ Hence in this case also the average speed Vm =2 X s X n.
$\displaystyle \small \bullet$ Since s is expressed in mm and n in number of revolutions/per minute,
Average Speed, $\displaystyle \small V_{m}=\frac{2\times s\times n}{1000\times 60}$ m/s
Momentum
$\displaystyle \small \bullet$ It is the quantity of motion possessed by a body and is equal to the product of its mass and the velocity with which it is moving.
Momentum = mass X velocity
$\displaystyle \small \bullet$ Unit of momentum is kg metre/sec.
Newton’s Laws of Motion
Newton’s First Law of Motion
Also known as Law of Inertia. A body continues to remain in the state of rest or uniform motion in a straight line unless some external force changes its state
Newton’s Second Law of Motion
The rate of change of momentum of a body is proportional to the external force acting upon it and takes place in the direction in which the force acts.
Newton’s Third Law of Motion
To every action there is an equal and opposite reaction
Motion under gravity
$\displaystyle \small \bullet$ A body falling from a height, from rest, has its velocity increasing and it will be maximum when it hits the ground. Therefore a body falling freely under gravity has a uniform acceleration.
$\displaystyle \small \bullet$ When the motion is upward, the body is subjected to a gravitational retardation.
$\displaystyle \small \bullet$ The acceleration due to gravity is denoted with ‘g’. Its value is taken as 9.8 $\displaystyle \small m/sec^{2}$ or 32 $\displaystyle \small ft/sec^{2}$.
Equations of motion under gravity
| Downward | Upward |
|---|---|
| $\displaystyle \small v=u+gt$ | $\displaystyle \small v=u-gt$ |
| $\displaystyle \small s=ut+\frac{1}{2}gt^{2}$ | $\displaystyle \small s=ut-\frac{1}{2}gt^{2}$ |
| $\displaystyle \small v^{2}-u^{2}=2gs$ | $\displaystyle \small u^{2}-v^{2}=2gs$ |
WORK, POWER AND ENERGY
Work
$\displaystyle \small \bullet$ When the point of application of force moves in the direction of the applied force, we say that work is done.
$\displaystyle \small \bullet$ Applied force ‘F’ moves a body through a distance’s.
Work done=Applied force X Displacement
W=Fs cosθ or W=Fs
$\displaystyle \small \bullet$ Unit is Joule = Nm
$\displaystyle \small \bullet$ Also 1 joule = 1 Nm = 105 dynes X 100 cm = $\displaystyle \small 10^{7}$ dynes cm = $\displaystyle \small 10^{7}$ ergs.
Force
$\displaystyle \small \bullet$ A Force is that which changes or tends to change the state of rest or motion of a body.
Force = Mass x Acceleration = m X a
$\displaystyle \small \bullet$ Units
MKS: $\displaystyle \small kg-m/sec^{2}$ = Newton (SI unit)
CGS: $\displaystyle \small gm-cm/sec^{2}$ = Dyne
FPS: $\displaystyle \small lb-ft/sec^{2}$ = Poundal
Power
$\displaystyle \small \bullet$ It is the rate of doing work or rate at which a machine can perform work.
$\displaystyle \small Power=\frac{Work\; done}{Time}$
$\displaystyle \small \bullet$ SI unit: Watts = Nm/sec = Joule/sec
Horse Power of Engines
$\displaystyle \small \bullet$ It is the practical unit of power
$\displaystyle \small \bullet$ 1HP (metric) = 735.5 Watts
$\displaystyle \small \bullet$ 1HP (British) = 746 Watts = 0.746 KW
$\displaystyle \small \bullet$ 1 KW = 1.34 HP
Indicated Horse Power (I.H.P.)
I.H.P. is the actual power generated in the engine cylinder.
Brake Horse Power (B.H.P.)
All the power generated by engine cylinder is not available for useful work because, part of it is always utilized in overcoming the internal friction of moving parts of the engine. The net output of the engine is B.H.P.
B.H.P.= I.H.P.- Losses
Mechanical Efficiency
$\displaystyle \small \bullet$ Power input is the power given to a machine to do work. Power output is what we get out of the machine.
$\displaystyle \small \bullet$ Power output is always less than power input due to friction in the machine.
$\displaystyle \small \bullet$ The ratio between power output to power input is efficiency of the machine and it is expressed in percentage
$\displaystyle \small Efficiency=\frac{Power\; output}{Power\; input}\times 100%$
$\displaystyle \small \bullet$ Mechanical efficiency is the ratio of B.H.P. to I.H.P.
$\displaystyle \small Mechanical\; Efficiency=\frac{BHP}{IHP}\times 100%$
Energy
$\displaystyle \small \bullet$ The capacity of a body to do work is called energy.
$\displaystyle \small \bullet$ Energy can neither be created nor destroyed and can only be converted from one form to another. If one form of energy disappears, it reappears in another form. This principle is known as law of conservation of energy.
$\displaystyle \small \bullet$ Types of energy
Kinetic energy
Kinetic energy is the energy a body possesses by virtue of its motion
$\displaystyle \small KE=\frac{1}{2}mv^{2}$ Joules
$\displaystyle \small KE=\frac{mv^{2}}{2g}$ kg meters
where,
m = mass of body (kg)
v = velocity of body (m/s)
g = acceleration due to gravity
Ex: moving train, flowing water, blowing wind, rotating wheels etc.
Potential energy
Potential energy is the energy a body possesses by virtue of its position
$\displaystyle \small PE=mgh$ Joules
$\displaystyle \small PE=mh$ kg meters
where,
m = mass of body (kg)
g = acceleration due to gravity
h = height (m)
Ex: a body placed at height, water in overhead tank, gas stored in overhead tank, springs of clock etc.
$\displaystyle \small \bullet$ When the point of application of force moves in the direction of the applied force, we say that work is done.
$\displaystyle \small \bullet$ Applied force ‘F’ moves a body through a distance’s.
Work done=Applied force X Displacement
W=Fs cosθ or W=Fs
$\displaystyle \small \bullet$ Unit is Joule = Nm
$\displaystyle \small \bullet$ Also 1 joule = 1 Nm = 105 dynes X 100 cm = $\displaystyle \small 10^{7}$ dynes cm = $\displaystyle \small 10^{7}$ ergs.
Force
$\displaystyle \small \bullet$ A Force is that which changes or tends to change the state of rest or motion of a body.
Force = Mass x Acceleration = m X a
$\displaystyle \small \bullet$ Units
MKS: $\displaystyle \small kg-m/sec^{2}$ = Newton (SI unit)
CGS: $\displaystyle \small gm-cm/sec^{2}$ = Dyne
FPS: $\displaystyle \small lb-ft/sec^{2}$ = Poundal
Power
$\displaystyle \small \bullet$ It is the rate of doing work or rate at which a machine can perform work.
$\displaystyle \small Power=\frac{Work\; done}{Time}$
$\displaystyle \small \bullet$ SI unit: Watts = Nm/sec = Joule/sec
Horse Power of Engines
$\displaystyle \small \bullet$ It is the practical unit of power
$\displaystyle \small \bullet$ 1HP (metric) = 735.5 Watts
$\displaystyle \small \bullet$ 1HP (British) = 746 Watts = 0.746 KW
$\displaystyle \small \bullet$ 1 KW = 1.34 HP
Indicated Horse Power (I.H.P.)
I.H.P. is the actual power generated in the engine cylinder.
Brake Horse Power (B.H.P.)
All the power generated by engine cylinder is not available for useful work because, part of it is always utilized in overcoming the internal friction of moving parts of the engine. The net output of the engine is B.H.P.
B.H.P.= I.H.P.- Losses
Mechanical Efficiency
$\displaystyle \small \bullet$ Power input is the power given to a machine to do work. Power output is what we get out of the machine.
$\displaystyle \small \bullet$ Power output is always less than power input due to friction in the machine.
$\displaystyle \small \bullet$ The ratio between power output to power input is efficiency of the machine and it is expressed in percentage
$\displaystyle \small Efficiency=\frac{Power\; output}{Power\; input}\times 100%$
$\displaystyle \small \bullet$ Mechanical efficiency is the ratio of B.H.P. to I.H.P.
$\displaystyle \small Mechanical\; Efficiency=\frac{BHP}{IHP}\times 100%$
Energy
$\displaystyle \small \bullet$ The capacity of a body to do work is called energy.
$\displaystyle \small \bullet$ Energy can neither be created nor destroyed and can only be converted from one form to another. If one form of energy disappears, it reappears in another form. This principle is known as law of conservation of energy.
$\displaystyle \small \bullet$ Types of energy
Kinetic energy
Kinetic energy is the energy a body possesses by virtue of its motion
$\displaystyle \small KE=\frac{1}{2}mv^{2}$ Joules
$\displaystyle \small KE=\frac{mv^{2}}{2g}$ kg meters
where,
m = mass of body (kg)
v = velocity of body (m/s)
g = acceleration due to gravity
Ex: moving train, flowing water, blowing wind, rotating wheels etc.
Potential energy
Potential energy is the energy a body possesses by virtue of its position
$\displaystyle \small PE=mgh$ Joules
$\displaystyle \small PE=mh$ kg meters
where,
m = mass of body (kg)
g = acceleration due to gravity
h = height (m)
Ex: a body placed at height, water in overhead tank, gas stored in overhead tank, springs of clock etc.
0 Comments