Linear Momentum & Energy - Mechanics & General Properties of Matter - Physics

Linear Momentum

In Newtonian mechanics, linear momentum, translational momentum, or simply momentum is the product of the mass m and velocityof an object.
It is a vector quantity, possessing a
In SI units, momentum is measured in kilogram-meter per second (kg-m/s).

Conservation of Momentum 

In the last section we found that the total external force F acting on a system is related to the total momentum P of the system by
Consider the implications of this for an isolated system, that is, a system which does not interact with its surroundings. In this case The total momentum is constant; no matter how strong the interactions among an isolated system of particles, and no matter how complicated the motions, the total momentum of an isolated system are constant. This is the law of conservation of momentum. If external force is zero then momentum of system is conserve and center of mass of system is not changing.
The Center of Mass Frame
Consider a inertial frame S and another inertial frame S' that at constant velocitywith respect to S .system of particle is moving with respect to S' is. Same system of particles have velocity with respect to S is
From Galilean transformation 
Let us consider the unique frame in which the total momentum of a particles is zero. This is called the center of mass frame or C .M frame If Total momentum from is from S frame then the center of mass is S ' that moves  with velocityis total mass
So from S ' or center of mass frame the momentum is
 
So center of mass frame is known as zero momentum frame.

Example 1: A mass m with speed v approaches a stationary particle of mass M
(a) Find the velocity of center of mass with respect to laboratory
(b) Find the velocity of particle with respect to center of mass before collision.
(c) Find the velocity of particles after collision such that center of mass has zero momentum i.e. with respect to center of mass.
(d) Find the velocity of particles just after collision with respect to laboratory.

(a)  m₁ = m m₂ = M v₁ = v and  v₁ = 0


(b) V_m,cm = v1 - v_cm = v

(c) If momentum of center of mass is zero then velocity of mass m with respect to center of mass is
 
If momentum of center of mass is zero then velocity of mass M with respect to center of mass is
 
(d) Then velocity of particle m after collision with respect
Then velocity of particle m after collision with respect

Example 2: A block of mass m  slides down frictionless wedge of mass M, when block will reach the bottom how much horizontal distance wedge will move.

Since there is not any force in horizontal direction, then momentum in horizontal direction is conserved. Therefore, center of mass in horizontal distance will not change. x distance of mass M move towards left with respect to surface. Same time m will move h cot θ - x with respect to surface towards right.
Let us assume center of mass is at origin so
put x₁ = -x, x₂ = h cot θ - x so

Example 3: All the surface shown in figure are smooth wedge of mass M is free to move .Block of mass m is given a horizontal velocity v0 as shown in figure. Find the maximum height h attained by m.

Since there is not any force in horizontal direction so momentum in horizontal direction is conserved.
From conservation of momentum  mv₀ = (m + M) v ⇒ v = mv₀/m +M

From conservation of energy

Example 4: A loaded spring gun, initially at rest on a horizontal frictionless surface, fires a marble at angle of elevation  θ. The mass of the gun is M, the mass of the marble is m and the muzzle velocity of the marble is v₀. What is the final motion of the gun?

Take the physical system to be the gun and marble, Gravity and the normal force of the table act on the system. Both these forces are vertical. Since, there are no horizontal external forces.
The x component of the vector equation F = dP/dt is
0 = dp_x/dt (1)
According to equation (1) P_x is conserved:
P_x,initial = P_x,final   (2)

Let the initial time be prior to firing the gun. Then P_x,initial = 0. Since the system is initially at rest. After the marble has left the muzzle, the gun recoils with some speed V_f, and its final horizontal momentum is MV_f, to the left. Finding the final velocity of the marble in volves a subtle point, however. Physically, the marble’s acceleration is due to the force of the gun, and the gun’s recoil is due t the reaction force of the marble. The gun stops accelerating once the marble leaves the barrel, so that at the instant the marble and the gun part company, the gun has its final speed V_f. At that same instant the speed of the marble relative to the gun is v₀. Hence, the final horizontal speed of the marble relative to the table is v₀ cos θ - V_f. By conservation of horizontal momentum. We have, 0 = m (v₀cos θ) - mV_f  - MV_f ⇒ v_f =

Example 5: Two identical buggies 1 and 2 with one man in each, move without friction due to inertia along the parallel rails towards each other. When the buggies get opposite to each other, the men exchange their places by jumping in the direction perpendicular to the motion direction. As a consequence, buggy 1 stops and buggy 2 keeps moving in the same direction, with its velocity becoming equal to v. Find the initial velocities of the buggies v₁ and v₂ if the mass of each buggy (without a man) equals M and the mass of each man is m.

(i) Initial condition of the buggies:
 
(ii) Status of buggies during jump:

(iii) Status of buggies after jump

During this exchange momentum will be conserved because there is no force is horizontal direction. Conservation of momentum for buggy (1) Mv₁ - mv₂ = 0 …(i)
Conservation of momentum for buggy (2) Mv₂ -  mv₁ = (m + M)v …(ii)
From (i) and (ii)

But in term of vector: v₂ has opposite direction as v₁.
Then

Energy

In Physics, Energy is the quantitative property that must be transferred to an object in order to perform work on, or to transfer heat to the object Energy is a conserved quantity; the law of conservation of energy states that energy can be converted in form, but not created or destroyed. The SI unit of energy is the joule, which is the energy transferred to an object by the work of moving it a distance of 1 metre against a force of 1 Newton.
Conservation of Energy: The energy can neither will created nor destroy, it can be transform from one form to another.

Different Form of Energy used in Mechanics

Kinetic energy

The energy which is responsible for motion of the particle. If particle of mass m moving with velocity then kinetic energy is given by
Kinetic energy in Cartesian coordinate
Kinetic energy in cylindrical coordinate
Kinetic energy in spherical coordinate
Work Done by Force Work done by the constant force is given as W == FS cosθ, where forceis force and making θ angle with displacement vector In general work done is a area under the  force F and  displacement S, which is for the variable force is defined as  W =

Work Done by a Variable Force
So far we have considered the work done by a force which is constant both in magnitude and direction. Let us now consider a force which acts always in one direction but whose magnitude may keep on varying. We can choose the direction of the force as x - axis. Further, let us assume that the magnitude of the force is also a function of x or say F(x) is known to us. Now we are interested in finding the work done by this force in moving a body from x₁ to x₂.
 
Work done in a small displacement from x and x + dx will be
dW = F.dx
Now, the total work can be obtained by integration of the above elemental work from x₁ to x₂ or It is important to note thatis also the area under F -x graph between x = x₁ to x = x₂.

Potential energy (U)
The energy which is required to perform the work is known as potential energy U. Hence force is defined as F = -(∂U/∂r. Then potential energy Ub - Uα = For the conservative force one can say potential energy is negative integral of the force .one can say in another way change in potential energy with respect to position is cause of force F.
There is different type of potential energy. For example, electrostatic potential energy, gravitation potential energy, stored energy in spring, mass energy in relativistic mechanics.
Total energy E is sum of kinetic energy K and potential energy U. So total energy E = K + U

Conservative and Non Conservative Force
Field We considered the forces which were although variable but always directed in one dimension. However, the most general expression for work done is

= initial position vector and = final position vector
A conservative force is a force with the property that the total work done in moving a particle between two points is independent of the taken path. Equivalently, if a particle travels in a closed loop, the total work done (the sum of the force acting along the path multiplied by the displacement) by a conservative force is zero.
For example Gravitational force is an example of a conservative force, while frictional force is an example of a non-conservative force.

Mathematical Interpretation of Conservative Force
A Force fielddefined everywhere in space (or within a simply connected volume of space), is called a conservative force or conservative if it meets any of these three equivalent conditions:
(1) The curl ofis the zero vector:
(2) There is zero net work (W) done by the force when moving a particle through a trajectory that starts and ends in the same place
(3) The  force can be written as the negative gradient of a potential,

Example 6: A body is displaced from A = ( 2m, 4 m, -6 m) to m under a constant forceN. Find the work done.

Example 7: A force F = - k/x² (x ≠ 0) acts on a particle in x - direction. Find the work done by this force in displacing the particle from x = + α to x = +2a . Here, k is a positive constant.

Work done by area under F - S or F - x Graph

Example 8: The field is given aswhere k is positive constant, then check whether the field is conservative or non conservative.

For conservative field

which justify the following force is conservative in nature.

Work Energy Theorem

The kinetic energy of a particle of mass m, moving with a speed v, is defined as  T = 1/2 mv². Let us consider a particle that moves from point 1 to point 2 under the action of a force. The total work done on the particle by the force as the particle moves from 1 to 2 is, by definition, the line integralis the displacement vector along the particle’s trajectory. If the particle under-goes an infinitesimal displacementunder the action of force, the scalar product dW = is the infinitesimal work done by the forceas the particle undergoes the displacementalong the particle’s trajectory. We use the Newton’s second law of motion,in the equation  to obtain an expression for the infinitesimal work

The scalar quantity 1/2 mv² is the kinetic energy of the particle, it follows that dW = dT . This Equation  dW = dT in  the differential form of the work-energy theorem. It states that the differential work of the resultant of forces acting on a particle is equal, at any time, to the differential change in the kinetic energy of the particle. Integrating equation between point 1 and point 2, corresponding to the velocities
v₁ and v₂ of the particle, we get W₁₂ =
This is the work-energy theorem, which states that the work done by the resultant forceacting on a particle as it move from point 1 to point 2 along its trajectory is equal to the change in the kinetic energy (T₂ - T₁) of the particle during the given displacement. When the body is accelerated by the resultant force, the work done on the body can be considered a transfer of energy to the body, where it is stored as kinetic energy.

Energy Conservation Theorem

If there exists a scalar function ∅(x,y,z,t), so that we could write. We shall say that the vector fieldis a potential field. The scalar function ∅(x,y,z,t), is then called the potential function of the field. The vector fieldis called conservative if ∅ does not explicitly depend on time. The potential function ∅(x,y,z,t), in this case, is called the force potential.
It is easy to show that if the force field is conservative the work done in moving the particle from 1 to 2 is independent of the path connecting 1 and 2 . The total work done on the particle by the force as it moves from 1 to 2 is given by
For a conservative force filed
Thus, the total work done is equal to the difference in force potential no matter how the particle moves from 1 to 2 following differential relation dW
If we now write θ(x,y, z) = -U (x,y, z) (inserting a minus sign for reasons of convention) and express the force as then the scalar function U is called the potential energy of the particle.
When is expressed as in the above equation, the work done becomes W₁₂ = U₁-U₂ It may be noted that the line integral of the fieldalong a closed curve (called circulation) is zero as shown below
it can be concluded that T₁ + U₁ = T₂ + U₂.
It says that the quantity T + U remains a constant as the particle moves from point 1 to point 2. Since 1 and 2 are arbitrary points, we have obtained the statement of conservation of total mechanical energy E = T+ U = constant.
Thus, the energy conservation theorem states that the total energy of a particle in a conservative force field is constant. It is instructive to note that equation (6) does not uniquely determine the function ∅. We could as well definewhere c is any constant. Hence, the choice for the zero level of ∅, and consequently U, is arbitrary.
We can verify directly from equation (11) that the total energy in a conservative field is a constant of the motion.
We have
The kinetic energy term can be written as
The potential energy U depends on time only through the changing position of the particle: U = U= U (x (t), y (t), z (t)). Thus, we have
 If follows that dE/dt =
Thus, the total energy of the particle moving in a conservative force field is a constant during the motion.

Gravitation

Force Let us consider a conservative forcethen we have
Therefore, we have the following relations:

This shows that the partial derivative of force potential in a given direction gives the force in that direction. An example of a force that derives from a potential is gravitational force

where the gravitational acceleration vector(gx, gy, gz). It follows that the negative of partial derivative of potential energy in a given direction gives the gravitational force in that direction. If gravitational acceleration vector is given by g(0, 0,-1)
then we have
Integrating the last of the above equation to obtain  U = mgz + f(x, y) Setting f(x, y) = 0, the potential energy of the particle in a gravitational field is given by U = mgz whereacts in the negative z direction. The total mechanical energy E is conserved when a particle moves under the action of the gravitational field.

Spring Force

An important example of the above idea is a spring that obeys. Hooke's Law. Consider the situation shown in figure. One end of a spring is attached to a fixed vertical support and the other end to a block which can move on a horizontal table. Let x = 0 denote the position of the block when the spring is in its natural length.

When the block is displaced by an amount x (either compressed or elongated) a restoring force (F) is applied by the spring on the block. The direction of this force F is always towards its mean position (x = 0) and the magnitude is directly proportional to x o


so potential energy of particle which is stored energy in spring is equivalent to u = 1/2kx²

Example 9: A mass m is shot vertically upward from the surface of the earth with initial speed v₀. Assuming that the only force is gravity, find its maximum altitude and the minimum value of v₀ for the mass to escape the earth completely.

The force on m is
The problem is one dimensional in the variable r and it is simple to find the kinetic energy at distance r by the work-energy theorem. Let the particle start at r = R, with initial velocity v₀.
 
We can immediately find the maximum height of m . At the highest point, v (r) = 0 and we have

It is a good idea to introduce known familiar constants whenever possible. For example 

Since,
The escape velocity from the earth is the initial velocity needed to move r_max to infinity.
Then escape velocity, V_escape  =

Example 10: Consider a mass M attached to a spring with spring constant k . Using the coordinate x measured from the equilibrium point x₀ solve the Equation of motion for simple harmonic motion with the help of work energy theorem.

Consider a mass M attached to a spring. Using the coordinate x measured from the equilibrium point, the spring force is F = -kx

In order to find x and v, we must know their values at some time t₀. Let us consider the case where at t = 0 the mass is released from rest, v₀ = 0, at a distance x₀ from the origin. Then
 

Separating the variables gives

Example 11: A chain of length πr/r, mass per unit length ρ is released from rest at θ = 0⁰. On a smooth surface. Find velocity of chain as it leaves the surface.

Taking a small element of chain taking angle dθ at point O, its mass = rdθ × ρ
P.E. of element = dm × g × h = rdθ × ρ × g × r cos θ

U₁ = ρgr²
Final P.E. = mass of chain × g × distance moved by centre of gravity of chain

or gross in K.E. = Loss in P.E.