Mechanics

Edited by Jen Moreau

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Mechanics

Branch of physics which deals with the study of behavior of physical bodies under the action of different forces.

Its origins are from ancient Greek with the writings of Aristotle and Archimedes. In modern era scientists like Newton, Galileo, Khayyam and Kepler laid the foundation which is called classical mechanics.

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Sub-disciplines of Mechanics

There are two main disciplines of mechanics

  • Classical mechanics
  • Quantum mechanics

Classical mechanics

Classical mechanics is further divided into several branches. Here I mention those branches.

  • Newtonian mechanics
  • Analytical mechanics
  • Hamiltonian mechanics
  • Lagrangian mechanics
  • Astrodynamics
  • solid mechanics
  • Fracture mechanics
  • Acoustics
  • Statics
  • Fluid mechanics
  • Continuum mechanics
  • Hydraulics
  • Fluid statics
  • Applied mechanics
  • Biomechanics
  • Relativistic mechanics

Quantum mechanics

Following are categories of Quantum mechanics.

  • Schrödinger wave mechanics
  • Matrix mechanics
  • Quantum statistical mechanics
  • particle physics
  • Nuclear physics
  • POnsed matter physics

Motion in one Dimension

The motion of a body is called one dimensional if only one coordinate specify the object's position changes with respect to time.

Cycle-Motion-Bike-Riding-Fast-Moving-Bicycle-Bike-1149234.jpg

Examples

The motion of athletes running in straight line.

Distance

The total length of the path is the distance traveled by the particle.

Displacement

The displacement of a particle is defined as its change in position.When a particle moves from an initial position xi to a final position xf , its displacement is given by,

dispostion.PNG

It is a vector quantity.

The shortest distance between the initial and final position of the particle is the displacement.

SI Unit: meter (m)

A vector is a physical quantity that requires the specification of both direction and magnitude.

By contrast, a scalar is a quantity that has magnitude and no direction.

Velocity

The average velocity of a particle is defined as the particle's displacement �"x divided by the time interval �"t during which that displacement occurred.

Dispersion with group and phase velocity.png

The average speed of a particle, a scalar quantity, is defined as the total distance traveled divided by the total time it takes to travel that distance.

Unit :m/s

Instantaneous velocity vx equals the limiting value of the ratio �"x/�"t as �"x approaches zero.

The instantaneous speed of a particle is defined as the magnitude of its velocity.

Acceleration

The average acceleration of the particle is defined as the change in velocity �"vx divided by the time interval �"t during which that change occurred.

The instantaneous acceleration is the limit of the average acceleration as �"t approaches zero.

The instantaneous acceleration equals the derivative of the velocity with respect to time.

One dimensional motion with constant acceleration

When this is the case, the average acceleration over any time interval equals the instantaneous acceleration at any instant within the interval, and the velocity changes at the same rate throughout the motion.

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vxf = vxi + axt

Motion in a straight line:

The equations of kinematics for a particle moving along the x-axis with uniform acceleration ax (constant in magnitude and direction) are:

Freely Falling objects

A freely falling object is any object moving freely under the influence of gravity alone, regardless of its initial motion. Objects were thrown upward or downward and those released from rest are all falling freely once they are released. Any freely falling object experiences an acceleration directed downward, regardless of its initial motion.

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gravity.png

An object falling freely in the presence of the Earth's gravity experiences a free-fall acceleration directed toward the center of the Earth. If air resistance is neglected, if the motion occurs near the surface of the Earth, and if the range of the motion is small compared with the Earth's radius, then the free-fall acceleration g is constant over the range of motion, where g is equal to 9.80 m/S2.

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Force

Force is that which causes a body to accelerate.

When several forces act simultaneously on an object the net force acting on an object is defined as the vector sum of all forces acting on the object.

If the net force exerted on an object is zero, then the acceleration of the object is zero and its velocity remains constant.

When the velocity of an object is constant (including the case in which the object remains at rest), the object is said to be in equilibrium.

Newton first law of motion

In the absence of external forces, an object at rest remains at rest and an object in motion continues in motion with a constant velocity.

The tendency of an object to resist any attempt to change its velocity is called the inertia of the object.

An inertial frame of reference is one that is not accelerating.

Mass

Mass is that property of an object that specifies how much inertia the object has.

Mass is an inherent property of an object and is independent of the object's surroundings and of the method used to measure it. Also, mass is a scalar quantity.

Newton 2nd law of motion

The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass.

F=ma

Unit of Force: Newton -which is defined as the force that, when acting on a 1-kg mass, produces an acceleration of 1 m/S2.

1 N= 1 kg.m/S2

The attractive force exerted by the Earth on an object is called the force of gravity Fg. This force is directed toward the center of the Earth and its magnitude is called the weight of the object.

Applying Newton's second law , F=ma to a freely falling object of mass m, with a=g and F=Fg , we obtain

Fg =mg

Newton 3rd law of motion

If two objects interact, the force F12 exerted by object 1 on object 2 is equal in magnitude to and opposite in direction to the force F21 exerted by object 2 on an object.

F12 = -F21

Force of friction

When a body is in motion either on a surface or in a viscous medium such as air or water, there is resistance to the motion because the body interacts with its surroundings. We call such resistance a force of friction.

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Consider a book on a table.If we apply an external horizontal force F to the book, acting to the right, the book remains stationary if F is not too great. The force that counteracts F and keeps the book from moving acts to the left and is called the frictional force f. As long as the book is not moving, f= F. Because the book is stationary, we call this frictional force the force of static friction fs.

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Friction relative to normal force (diagram).png

If we increase the magnitude of F, as shown in Figure 5.17b, the magnitude offs increases along with it, keeping the book in place. The force fs cannot increase indefinitely, however. Eventually, the surfaces in contact can no longer supply sufficient frictional force to counteract F and the book accelerates. When it is on the verge of moving, fs is a maximum. When F exceeds fs, max, the book accelerates to the right. Once the book is in motion, the retarding frictional force becomes less than fs, max.When the book is in motion, we call the retarding force the force of kinetic friction fk.

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The direction of the force of static friction between any two surfaces in contact with each other is opposite the direction of relative motion and can have values where the dimensionless constant µs is called the coefficient of static friction and n is the magnitude of the normal force.

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The direction of the force of kinetic friction acting on an object is opposite the direction of the object's sliding motion relative to the surface applying the frictional force and is given by where µk is the coefficient of kinetic friction.

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Impulsive force

An impulsive force is a very great force acting for a very short time on a body, so that the change in the position of the body during the time the force acts on it may be neglected.

(e.g.) The blow of a hammer, the collision of two billiard balls etc.

Impulse of force

The impulse J of a constant force F acting for a time t is defined as the product of the force and time.

(i.e) Impulse = Force x time J = F x t

WORK, ENERGY, AND POWER

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WORK:

Work done by a force is defined as the product of that force times the parallel distance over which it acts.

Unit: In SI system, Newton-metre (N-m) or Joule(J).

One Joule is the work done by a force of IN when it displaces an object 1m in the direction of the force.

ENERGY:

It is the measure of the ability of a body or system to do work or produce a change.

Unit: Joule or kWh

Two types of energy:

1. Kinetic Energy(K.E): energy possessed by an object when it is in motion. If an object of mass m is moving with a speed v, it has translational K.E given by,

Units: m in kg, v in m/s

2. Potential Energy(P.E):energy possessed by an object due to gravitational interaction.

If the object is at a height h above the zero level, then its

Where, m=mass of the object, g-acceleration due to gravity

Units: 'm' in kg, 'g' in m/S2 and 'h' is in m.

WORK-ENERGY THEOREM:

When work is done on a point mass or a rigid body and there is no change in potential energy, the energy imparted can only appear as kinetic energy.

CONSERVATION OF ENERGY:

Energy can neither be created nor be destroyed, but only transformed from one form to another.

POWER:

Power is the rate of doing work.

Also power = force X speed

Unit: In SI system, Watt(W)

1W=1J/s

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