MTH622 Final Term Notes

Important notes Mth-622 · Final term for MCQ’S · 1. Path Independence

• Independence of path is defined as that the work done by an object remains same between initial point and the final destination of the particle, it does not matter which path is taken by the object.
• “A force field is said to conservative if the total work done by the particle moving along a curve is independent of the path taken by the particle and depend upon the end points of the curve only.”

1. Necessary and sufficient conditions for a Conservative Force Field:-

• A force field F is conservative if and only if there exists a continuously differentiable scalar field V such that 𝐹 ⃗ =−∇𝑉 or, equivalently, if and only if
• A continuously differentiable force field F is conservative if and only if for any closed nonintersecting curve C (simple closed curve) 𝑊= c F dr =
• The total work done in moving a particle around any closed path is zero.

2. Examples of Conservative Forces

• Gravitational force , Elastic spring force is example of conservative force.
• The work done of a particle moving along a closed path is zero and the force which causes such motion is conservative.
• Forces that cannot be expressed in the term of a potential energy function are called non-conservative forces.
• The forces that do not store energy are called nonconservative or dissipative forces.

is called a non-conservative force field.

• Friction, the impulse (time dependent force) is also a non-conservative force.
• Simple Harmonic Motion (SHM) is a particular type of oscillation and periodic motion in which restoring force of an object is directly proportional to the displacement of the object acting in opposite direction of displacement.
• Mathematically, the restoring force 𝐹 is given by R where the subscript 𝑅 represents the restoring force and 𝑘 is the constant of proportionality often called the spring constant or modulus of elasticity.
• In Newtonian mechanics, by Newton's second law we have eq. of S.H.M d x dt
• This vibrating system is called a simple harmonic oscillator or linear harmonic oscillator.

• Mass on a spring:-

An object of mass m linked to a spring of spring constant k represents the simple harmonic motion in closed region.

• The equation representing the period for the mass attached on a spring
• 𝑻 = 2 m k
• The above equation expresses that the time period of oscillation is independent of amplitude as well as the acceleration.

• Simple pendulum:-

The movement of the mass attached to a simple pendulum is considered as simple harmonic motion.

• The time period of a mass m attached to a pendulum of length l with gravitational

acceleration g is given by T= 2 l g

• Amplitude:-
• Amplitude is defined as the maximum distance covered by the oscillating body in one oscillation or length of a wave measured from its mean position.
• The amplitude of a pendulum is one-half the distance that the mass covered in moving from one terminal to the other.
• The vibrating sources generate waves, whose amplitude is proportional to the amplitude of the vibrating source.
• Time period is minimum time required by a oscillation system to complete its one cycle of oscillation of the specific system.
• It is denoted by T and measured in seconds.
• The frequency (f) of an oscillatory system is the number of oscillations pass through a specific point in one second.
• It is measure in hertz (Hz).
• The frequency of S.H.M can be calculate by using the following relation 𝑓 = 1 T .
• If T is the kinetic energy, V the potential energy and 𝐸 = 𝑇 + 𝑉 the total energy of a

2 mv

and 𝑉 = 2 kx .

Then the total energy of S.H.M will be 𝐸 = 2 mv + 2 kx .

• The forces acting on a harmonic oscillator are called damping forces which tend to decrease the amplitude of the successive oscillations or simply force apposing the motion.
• The damping force is proportional to the velocity.
• where 𝑑 represents the damping force and 𝑏 is the damping coefficient.
• The negative sign shows that that direction of 𝐹𝑑 is opposite to the velocity 𝑣. b m = 2𝜁,

k m = 𝜔0

• The equation of damped harmonic oscillator can be written as ̈ +2𝜁𝑥̇ +  x = 0.
• The theorem, called Euler's theorem, is fundamental in the motion of rigid bodies.
• “A rotation of a rigid body about a fixed point of the body is equivalent to a rotation about a line which passes through the point.”
• The line referred to is called the instantaneous axis of rotation.
• Rotations can be considered as finite or infinitesimal.
• Finite rotations cannot be represented by vectors since the commutative law fails.
• Infinitesimal rotations can be represented by vectors.
• Chasle’s theorem states that the most general rigid body displacement can be produced by a translation along a line (called its screw axis) followed (or preceded) by a rotation about that line.
• A rigid body has six degrees of freedom.
• By Euler’s theorem, three of these are associated with pure rotation.
• The remaining three must be associated with translation.
• To describe the general motion of a rigid body, think of the general motion as translation of a fixed point 𝑂 in the body to a point 𝑂′ followed by the rotation about an axis through 𝑂′.
• Kinematics is the branch of mechanics deals with the moving objects without reference to the forces which cause the motion.
• Some features of rigid body motion are:-
• Displacement.
• Position.
• Velocity.
• Linear Velocity & Angular Velocity.
• Linear Acceleration & Angular Acceleration.
• Motion of a Rigid Body (Translation & Rotation).
• In Space this is closely related to the concepts of point, position, ' direction and displacement’. Measurement in space involves the concepts of length or distance, with which we assume familiarity.
• Units of length are feet, meters, miles, etc.
• In Time this concept is derived from our experience of having one event taking place after, before or simultaneous with another event.
• Units of time are seconds, hours, years, etc.