MECHANICS
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.
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:
Speed &Velocity
Net torque:
- 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.
- the total angular momentum of a rotating object remains constant if the
Resultant force/ torque = 0
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
Relationship Between G and g :
Weight(on earth) is the force on an object due to the gravitational attraction of the earth.
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.
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
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
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.
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
Potential energy is energy that is stored or conserved in an object or substance
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
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
- 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.
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
9.3 Constant Angular Acceleration
Instantaneous velocity=dx/dt
τ = rFsinθ
τ = Iα
-force need to make an object start moving
unit : Nm
vector quantity
Moment of Inertia :
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
Newton's Third Law of Motion
Every body continues to be at rest or of uniform motion unless acted by an external force.
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.
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
- For rotational motion, if the net torque on system is zero, then the dL = constant
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.
Rotational Kinetic Energy
Momentum
From Newton's second law, the force acting on the body is the rate of change of momentum of body.
W= τθ
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.
Elastic Collision
- Total momentum is conserved.
- Total kinetic energy is NOT conserved.
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
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
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
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 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.
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
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
Kinematics vector
Potential Energy Diagrams; Stable and Unstable Equilibrium
Stress
Strain
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
Bulk Modulus
Shear modulus
Relative velocity-the velocity of an object B in the rest frame of another object A.
tangential speed
Derived from centripetal force = gravitational attraction
v² inversely proportional to r
relate to centripetal force
Work Done By A Variable Force
Under shaded area of F vs x
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
Application of Newton's Laws
Friction
present when 2 surface slide along each other
-Static friction
-Kinetic friction
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
Non-Uniform
Centripetal acceleration : Acceleration of a body in circular motion, and the direction is towards the center of the circle.
Centripetal Force : Resultant force towards the center of the circle.
verticle circular motion
Horizontal Circular Motion
Lift