MECHANICS

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Rotation of Rigid Bodies

Motion in Two dimensions

Equilibrium and Elasticity

Gravitation

Newton’s law of universal gravitation: Any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of their distance apart. image image image

Newton's Laws of Motion


Work and Kinetic Energy

Momentum, Impulse and Collisions

Newton's First Law of Motion

Potential Energy and Conservation of
Energy

Angular Momentum and its Conversation

Equilibrium

Elasticity

Work is dot scalar product of force and displacement
W = F.s

Angular momentum of particle:


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Speed &Velocity

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Net torque: image

  • If the net torque is zero, the vector angular momentum is conserved.
  • Newton’s second law is valid
    even if I is not constant
    • the total angular momentum of a rotating object remains constant if the
      net external torque acting on it is zero.

Resultant force/ torque = 0 unnamed (1)

Sum of the force in any direction = 0

Unit = Joule(J)

scalar quantity (no direction)

Force is being exerted at an angle θ to the displacement, the work done is W = fd cos θ

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Relationship Between G and g :
Weight(on earth) is the force on an object due to the gravitational attraction of the earth. image image
g depends on how far you are from this center.

Kinetic energy is the energy an object has because of its motion

WHAT KEEPS A SATELLITE IN ORBIT???

The satellite's speed or centripetal acceleration is caused by GRAVITATIONAL FORCE. dropped image link

if an object’s moment of inertia
changes, its angular speed changes

Acceleration

Weightlessness =condition experienced while in free-fall

KEPLER'S LAW

Energy is ability to do work
Work done = Energy transferred

FIRST LAW

SECOND LAW

THIRD LAW

Dynamics of Rotational motion dropped image link

Work Energy Theorem

speed- rate of change of distance

All planets move about the Sun in elliptical orbits, having the Sun as one of the foci.

velocity- rate of change of displacement

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A radius vector from a sun to a planet sweeps out equal areas in equal lengths of time.

Work done is equal to Kinetic Energy

If the net work is positive, kinetic energy will increase.
If the net work is negative, kinetic energy will decrease

The squares of the periods is directly proportional to the cubes of their average distances from the sun is the same for every one of the planets. keplers-third-law

Projectile Motion: Two dimensional motion where the horizontal motion is a constant velocity motion and the vertical motion is motion under gravity.

vector quantity

  • use right hand rule to determine the direction

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Potential energy is energy that is stored or conserved in an object or substance

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A conservative force is one for which work done by or against it depends only on the starting and ending points of a motion and not on the path taken

A non-conservative force is one for which work depends on the path taken

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A conservative

A non- conservative

Acceleration- rate of change of velocity

Angular Quantities
-angle, θ (in rad)= s/r
-angular displacement, Δθ = θ - θ
-angular velocity, ω=θ/t
-linear velocity, v=rω ( r↑ , v↑ )
-angular acceleration, α=ω/t
-tangential acceleration, a = rα
-centripetal acceleration, a = ω²r
-ω = 2πf
- T=1/f

Potential is energy stored in an object due to its position, state or shape

Gravitational Field image

  • In a closed system, angular momentum is conserved in all directions after a collision.
  • Since momentum is conserved, part of the momentum in a collision may become angular momentum as an object starts to spin after a collision. image

Conservation of Mechanical Energy

Average velocity=displacement/time elapsed

Average speed=distance/
time elapsed

Mechanical Energy is the sum of Potential energy and Kinetic energy

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Instantaneous velocity=dx/dt

τ = rFsinθ
τ = Iα
-force need to make an object start moving
unit : Nm
vector quantity

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Moment of Inertia : dropped image link

parallel axis-theorem

perpendicular axis-theorem

The Law of Conservation of Energy

The total energy of an isolated system remains constant

Instantaneous acceleration=dv/dt

Energy before = Energy After

Newton's Second Law of Motion

Constant acceleration
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Newton's Third Law of Motion

Every body continues to be at rest or of uniform motion unless acted by an external force. 509729784

Rate of change of momentum of a body is directly proportional to the resultant force on it and is in the same direction as the resultant force. image

Law of Conservation of Momentum

  • For translational motion, net force on system is zero, total linear momentum constant
    • For rotational motion, if the net torque on system is zero, then the dL = constant
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Free falling object-Near surface of the Earth, objects experience approximately the same acceleration due to gravity(9.80 m/s^2)

Gravitational Potential Energy and Escape Velocity

Energy can be transferred, but cannot created or destroyed

Every action has a direction which is of the same magnitude but opposite in direction. laws-of-motion

Rotational Kinetic Energy
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Momentum image

From Newton's second law, the force acting on the body is the rate of change of momentum of body. images

W= τθ
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Power = τω

Conservation of Momentum: The total momentum of a system is constant unless acted upon by an external force.

  • Total momentum is conserved.
  • Total kinetic energy is conserved.
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Elastic Collision

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  • Total momentum is conserved.
  • Total kinetic energy is NOT conserved. download (2)

Inelastic Collision-Two objects move together after collision.

Conservation of Energy :
Mgh = 1/2 Mv² + 1/2 Iω²

Impulse=F(average)t=change in momentum image

U = mgh

  • Moment of inertia is the name given to rotational inertia, the rotational analog of mass for linear motion.


  • Moment of inertia about any axis parallel to an axis that goes through the center of mass of an object


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Escape velocity is the speed at which the sum of an object's kinetic energy and its gravitational potential energy is equal to zero

Vectors &Scalors

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Addition of vector
-Tail-to-tip method
-Parallelogram method
-Pythagorean Theorem
Subtraction of vector
-negative vector
Multiplication by scalar
-same direction different magnitude

Center of Mass image Point where any uniform force on the objects acts

G is the universal gravitational constant (G ≈ 6.67×10−11 m3·kg−1·s−2)

Center of Gravity :point at which the gravitational force can be considered to act.

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POWER

Vectors - magnitude & direction
Scalars - magnitude only

Time rate of doing work & The rate at which energy is transferred

Unit : Watt

1 hp = 746 W

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Instantaneous Power
P = F.v

Unit vectors
-Scalar product(dot product)
-Vector product(cross product)

Average power is the work done divided by the time it takes to do work

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Kinematics vector
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Potential Energy Diagrams; Stable and Unstable Equilibrium
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Stress image

Strain image

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Stress is defined as the force per unit area

Strain is defined as the ratio of the change in length to the original length

Young Modulus image

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Bulk Modulus image

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Shear modulus image

Relative velocity-the velocity of an object B in the rest frame of another object A.

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2021-01-19 (3)

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Derived from centripetal force = gravitational attraction

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v² inversely proportional to r

relate to centripetal force

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Work Done By A Variable Force
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Work done by a spring force

Hookes Law

Principle
If only conservative forces are doing work, the total mechanical energy of a system neither increases nor decreases in any process . It stays constant and conserved

Object moves in the opposite direction of a conservative net force, the potential energy will increase; and if the speed (not the velocity) of the object changes, the kinetic energy of the object also changes

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Application of Newton's Laws

Friction
present when 2 surface slide along each other

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-Static friction
-Kinetic friction
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μ of kinetic < μ of static

Banked and Unbanked

Circular Motion

Motion of object in a circle at a constant speed with constant radius.

When an object rotates with variable speed, acceleration and force have tangential components.

Period (T) is the time taken by the object to move around a complete circle. uniform-circular-motion-3-638

Uniform

Non-Uniform

Centripetal acceleration : Acceleration of a body in circular motion, and the direction is towards the center of the circle. image

Centripetal Force : Resultant force towards the center of the circle. 111

verticle circular motion
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Horizontal Circular Motion
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Lift

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