ASSIGMENT ON MECHANICS

Submitted by: Submitted to:

Name: Md Jahid Hasan Name: Jebun Naher Sikta

ID: 152-15-558 Dep: of Natural Science

Sec: A Daffodil International Univercity

Q: 1 . Show that projectile motion is a parabolic motion.

Projectile motion is a form of motion in which an object or particle (called a projectile) is thrown near the earth's surface, and it moves along a curved path under the action of gravity only. The only force of significance that acts on the object is gravity, which acts downward to cause a downward acceleration.

Let a projectile begain its flight from a point v0x with initial velocity v0 and making an angel θ with

the horizontal direction .Taking v0 x as origin lt the horizontal component of initial velocity,

Vx0 = v0 cosθ…………………….(i)

And the vertical component of velocity,

Vy0 = V0 sinθ………………………...(ii)

At a later time when t = t , let the objects velocity is V(

Now horizontal component of V( , is given as ,

Vx = vx0 + ax t

=) Vx = Vx0 [ since ax = 0 horzontaley ]

Vx =V0 COSθ [using can (i)……….(iii)

Horizontal displacement ,

X=0 M=V0 cosθ t [since vx=xt

t =X

Vocosθ………………( iv) =) x = vx t ]

Now , he vertical component of the velocity

Vy = vy0 + ay tx

Vy=V0 sinθ – gt [ay = - g]

So the vertical displacement at t = t

y = y0 + vy0t +12

ayt 2

or, y = 0 + v0 sinθ + t - 12

gt 2

= v0 sinθ t - 12

gt 2

y = v0 sinθ . x

V 0 cosθ-

12

g (X

V 0 cosθ¿2

= x tanθ - 12

g x2/v02 cos2θ

= x tanθ – gx2/ 2v02 cos2θ

tanθ = b and g

2v 02cos2θ = c as constants

now can be written as

y = bx - cx2

this is an equation of a parabola.hence the path of motion of a projectile is parabolic.

Q: 2 . statement and proof work-energey theorem

The energy associated with the work done by the net force does not disappear after the net force is removed (or becomes zero), it is transformed into the Kinetic Energy of the body. We call this the Work-Energy Theorem.

Proof :

Let a force f be appelied on an object f mass ‘m’ ad the velocity of the object changes from v1 to v2 when the object travels a distance s , then

V22 = V1

2 + 2as

S = V22

- V12/2a

So, the work done by the frc ,

W = F . s

= ma . V22

- V12/2a

=12

m (V22

- V12)

= 12

mV22 -

12

mV12

= Final kinetic energy – initial kinetic energy

= Increase of kinetic energy of the object

Hence , increase of kinetic eergy of an object is equal to the work done by the applied force.It is the work – energy theorem.

Q: 3. State zeroth, 1st, 2nd law of thermodynamics.

Zeroth law of thermodynamics:

The Zeroth Law of Thermodynamics states that if two bodies are each in thermal equilibrium with some third body, then they are also in equilibrium with each other. Thermal equilibrium means that when two bodies are brought into contact with each other and separated by a barrier that is permeable to heat, there will be no transfer of heat from one to the other.

The zeroth law of thermodynamics says that if system (or object) A is in thermal equilibrium with system (or object) B, and is also in thermal equilibrium with system (or object) C, then B and C must also be in thermal equilibrium with each other.

Perhaps that's confusing, so let's break it down. First of all, what is thermal equilibrium? Thermal equilibriumis when two systems or objects have no flow of heat between them despite being connected by a path permeable to heat. When does this happen in real life?

Well, heat always transfers spontaneously from hot places to cold places. That, as it happens, is one way of stating the 1st law of thermodynamics. But this means that for heat not to flow when it can, the two objects or systems must be the same temperature.

So we can restate the zeroth law of thermodynamics like this: if system (or object) A is the same temperature as system (or object) B, and is also the same temperature as system (or object) C, then B and C must also be the same temperature.

Written like that, suddenly the zeroth law becomes really obvious. Of course, if object A is the same temperature as B and C, then B and C are the same temperature as each other.

1st law of thermodynamics:

The first law of thermodynamics, also known as Law of

Conservation of Energy, states that energy can neither be created

nor destroyed; it can only be transferred or changed from one form

to another. For example, turning on a light would seem to produce

energy; however, it is electrical energy that is converted.A way of

expressing the first law of thermodynamics is that any change in

the internal energy (∆E) of a system is given by the sum of the heat

(q) that flows across its boundaries and the work (w) done on the

system by the surroundings: ΔE=q+w This law says that there

are two kinds of processes, heat and work, that can lead to a

change in the internal energy of a system. Since both heat and

work can be measured and quantified, this is the same as saying

that any change in the energy of a system must result in a

corresponding change in the energy of the world outside the

system. In other words, energy cannot be created or destroyed. If

heat flows into a system or the surroundings to do work on it, the

internal energy increases and the sign of q or w is positive.

Conversely, heat flow out of the system or work done by the system

will be at the expense of the internal energy, and will therefore be

negative.

2nd law of thermodynamics:

The second law of thermodynamics says that the entropy of any

isolated system not in thermal equilibrium almost always increases.

Isolated systems spontaneously evolve towards thermal equilibrium

—the state of maximum entropy of the system. More simply put: the

entropy of the world only increases and never decreases.A simple

application of the second law of thermodynamics is that a room, if

not cleaned and tidied, will invariably become more messy and

disorderly with time - regardless of how careful one is to keep it

clean. When the room is cleaned, its entropy decreases, but the

effort to clean it has resulted in an increase in entropy outside the

room that exceeds the entropy lost.

Q: 4. Distinction between reversible and irreversible process.

When a system undergoes from one state to another state, the change of state take place into two processes:-

1. Reversible process

2. Irreversible process

Distinction between reversible and irreversible process are given below:

Reversible Process Irreversible Process1. It takes place in infinite number of infinitesimally small steps and it would take finite time to occur.

1. It takes place infinite time.

2. It is imaginary as it assumes the presence of frictionless and weight less piston.

2. It is real and can be performed actually.

3. It is in equilibrium state at all stage of the operation.

It is in equilibrium state only at the initial and final stage of the operation.

4. All changes are reversed when the process is carried out in reversible direction.

4. After this type of process has occurred all changes do not return to the initial stage by themselves.

5. It is extremely slow. 5. It proceeds at measureable speed.6. Work done by a reversible process is greater than the corresponding irreversible process.

6. Work done by a irreversible process is smaller than the corresponding reversible process.

Q: 5.Define moment of inertia and calculate moment of inertia for

1. Linear bar.

2. Circular disc.

Moment of inertia:

The moment of inertia of an object is a calculated quantity for a rigid body that is undergoing rotational motion around a fixed axis. It is calculated based upon the distribution of mass within the object and the position of the axis, so the same object can have very different moment of

inertia values depending upon the location and orientation of the axis of rotation. It is given by, I =

∑ mr2

Linear bar:

a quantity expressing a body's tendency to resist angular acceleration, which is the sum of the products of the mass of each particle in the body with the square of its

distance from the axis of rotation. The parallel axis theorem the moment of inertia of a body

about any axis is equal to the sum of the moment of inertia of the body about a parallel axis passing throw the center of the mass and product of the mass of the body and the square of perpendicular distance between the two parallel axes.

Circular disc:

The moment of inertia of a thin circular disk is the same as that for a solid cylinder of any length, but it deserves special consideration because it is often used as an element for building up the moment of inertia expression for other geometries, such as the sphere or the cylinder about an end diameter. The moment of inertia about a diameter is the classic example of the perpendicular axis theorem for a planar object: