FORCE AND MOTION
Newtonian Mechanics
An interaction that can cause an acceleration of a body is called a Force.
The relationship between a force and the acceleration was first understood by Newton (1642-1727). The study of that relationship, as Newton Presented it, is called the Newtonian Mechanics.
Newton’s First Law
If no force acts on a body, the body’s velocity cannot change; that is, the body cannot accelerate.
Newton’s Second Law
The net force on a body is equal to the product of the body’s mass and its acceleration.
In equation form
Fnet = ma
Fnet α a
Newton’s Third Law
When two bodies interact, forces on the bodies from each other are equal in magnitude but opposite in direction.
Inertial Reference Frame
An inertial frame is one in which Newton’s laws hold.
For Example,
If we the ground as an inertial frame and neglect the earth rotation.If we slide the ball from north pole to south pole.
The Gravitational Force
A gravitational force Fg on a body is a pull that is directed towards a second body(earth).
Suppose a body of mass m is in free fall motion with an acceleration of magnitude g.
Neglecting the air friction,
Now from Newton’s Second Law
F = ma
If we take the vertical y-axis along the body path, with the positive direction upward
Then - Fg = m(-g)
Fg = mg
The weight w of a body is the magnitude of the net force required to prevent the body from freely fall as measured on the ground.
Two forces acting on the free fall body
1- A downward gravitational force Fg
2- A balancing upward force of magnitude w
From Newton’s Second Law
Fnet,y = may
As ay = 0
So it becomes
W-Fg = m(0)
Hence W = Fg
The Normal Force
A Normal force N is the force on the body from a surface against which the body presses.
From Newton’s Second Law
F = ma
Now
Fnet,y = may
N-Fg=may
Since ay = 0
So N- Fg = 0
N = Fg = mg
Friction
The resistance in the motion of a body is considered to be a single force F is called the friction force or frictional force.
When the applied force reaches a certain magnitude then the frictional force that opposes the motion is called kinetic coefficient of friction.
Properties of Friction
1- If the block does not move then N becomes the gravitational Force.
2- The magnitude of fs has a maximum value fs,max
fs = μs N
Where μs is the coefficient of static friction and N is the magnitude of the normal force.
3- If applied force increases the fs,max, the block begin to move then fs decreses to fk
fk = μk N
Where μk is coefficient of kinetic friction that opposes the motion.
A fluid is anything that can flow. For example, air, gas, liquid.
The body that experiences a drag force D that opposes the relative motion and points in the direction in which fluid flows relative to the body. The magnitude of D is
D = ½ cρAv²
Where c is the drag coefficient, ρ is the density of fluid, A is the cross-sectional area of the body and v is the speed.
When body falls through air,
From Newton’s second law
Fnet,y = may
D-Fg = may
When D = Fg then ay = 0
When ay = 0 then the body speed no longer increases. The body falls at constant speed is called Terminal Speed.
½ cρAvt² - Fg = 0 When D = Fg
Vt = √2Fg/cρA
Uniform Circular Motion
a = v²/r ----------(1)
v = speed of the particle
r = radius of the circle
Circumference of circle 2πr in time T is
T = 2πr/v
Where T is the period of revolution
Proof:
Now we shall prove a = v²/r
Figures Show
(a) Position of Particle and its component at a certain instant
(b) Velocity v and its components
(c) Acceleration a and its components
Now V= Vxi + Vyj ----------- (2)
From fig (b)
Cosα = Vy/V
Vy = Vcosα
And Sinα = -Vx/v
Vx = -Vsinα
Putting these values of Vx and Vy in equation (2)
V = -Vsinαi + Vcosαj ----------- (3)
From fig (b)
Cosα = xp/r
Sinα = yp/r
Using these values in equation (3), we get
V = -V/r ypi + V/r xpj
Differentiate w.r.t “t”
a = -V/r Vyi + V/r Vxj ------------ (4)
We know that from fig.(b)
Vx = -Vsinα
Vy = Vcosα
Putting these values in equation (4) and taking magnitude of acceleration, we get
a = V²/r
© COPYRIGHT materialengg.blogspot.com
Newtonian Mechanics
An interaction that can cause an acceleration of a body is called a Force.
The relationship between a force and the acceleration was first understood by Newton (1642-1727). The study of that relationship, as Newton Presented it, is called the Newtonian Mechanics.
Newton’s First Law
If no force acts on a body, the body’s velocity cannot change; that is, the body cannot accelerate.
Newton’s Second Law
The net force on a body is equal to the product of the body’s mass and its acceleration.
In equation form
Fnet = ma
Fnet α a
Newton’s Third Law
When two bodies interact, forces on the bodies from each other are equal in magnitude but opposite in direction.
Inertial Reference Frame
An inertial frame is one in which Newton’s laws hold.
For Example,
If we the ground as an inertial frame and neglect the earth rotation.If we slide the ball from north pole to south pole.
The Gravitational Force
A gravitational force Fg on a body is a pull that is directed towards a second body(earth).
Suppose a body of mass m is in free fall motion with an acceleration of magnitude g.
Neglecting the air friction,
Now from Newton’s Second Law
F = ma
If we take the vertical y-axis along the body path, with the positive direction upward
Then - Fg = m(-g)
Fg = mg
The weight w of a body is the magnitude of the net force required to prevent the body from freely fall as measured on the ground.
Two forces acting on the free fall body
1- A downward gravitational force Fg
2- A balancing upward force of magnitude w
From Newton’s Second Law
Fnet,y = may
As ay = 0
So it becomes
W-Fg = m(0)
Hence W = Fg
The Normal Force
A Normal force N is the force on the body from a surface against which the body presses.
From Newton’s Second Law
F = ma
Now
Fnet,y = may
N-Fg=may
Since ay = 0
So N- Fg = 0
N = Fg = mg
Friction
The resistance in the motion of a body is considered to be a single force F is called the friction force or frictional force.
When the applied force reaches a certain magnitude then the frictional force that opposes the motion is called kinetic coefficient of friction.
Properties of Friction
1- If the block does not move then N becomes the gravitational Force.
2- The magnitude of fs has a maximum value fs,max
fs = μs N
Where μs is the coefficient of static friction and N is the magnitude of the normal force.
3- If applied force increases the fs,max, the block begin to move then fs decreses to fk
fk = μk N
Where μk is coefficient of kinetic friction that opposes the motion.
A fluid is anything that can flow. For example, air, gas, liquid.
The body that experiences a drag force D that opposes the relative motion and points in the direction in which fluid flows relative to the body. The magnitude of D is
D = ½ cρAv²
Where c is the drag coefficient, ρ is the density of fluid, A is the cross-sectional area of the body and v is the speed.
When body falls through air,
From Newton’s second law
Fnet,y = may
D-Fg = may
When D = Fg then ay = 0
When ay = 0 then the body speed no longer increases. The body falls at constant speed is called Terminal Speed.
½ cρAvt² - Fg = 0 When D = Fg
Vt = √2Fg/cρA
Uniform Circular Motion
a = v²/r ----------(1)
v = speed of the particle
r = radius of the circle
Circumference of circle 2πr in time T is
T = 2πr/v
Where T is the period of revolution
Proof:
Now we shall prove a = v²/r
Figures Show
(a) Position of Particle and its component at a certain instant
(b) Velocity v and its components
(c) Acceleration a and its components
Now V= Vxi + Vyj ----------- (2)
From fig (b)
Cosα = Vy/V
Vy = Vcosα
And Sinα = -Vx/v
Vx = -Vsinα
Putting these values of Vx and Vy in equation (2)
V = -Vsinαi + Vcosαj ----------- (3)
From fig (b)
Cosα = xp/r
Sinα = yp/r
Using these values in equation (3), we get
V = -V/r ypi + V/r xpj
Differentiate w.r.t “t”
a = -V/r Vyi + V/r Vxj ------------ (4)
We know that from fig.(b)
Vx = -Vsinα
Vy = Vcosα
Putting these values in equation (4) and taking magnitude of acceleration, we get
a = V²/r
© COPYRIGHT materialengg.blogspot.com
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