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W E + W F = Δ K + Δ U

We are required to find minimum force. We need to understand the implication of this phrase. It is clear that we can apply external force in such a manner that block of “2 kg” acquires kinetic energy by the time block of “1 kg” is initiated in motion. Alternatively as a base case, we can apply external force gradually and slowly in increasing magnitude till the block of “1 kg” is initiated in motion. In this case, the block of “2 kg” does not acquire kinetic energy. This mode of application of external force represents the situation when minimum external force will be required to initiate block of “1 kg”. Hence,

Δ K = 0

Motion of block is constrained in horizontal direction. There is no change of vertical elevation. Hence, there is no change in gravitational potential energy. However, spring is stretched from its neutral state. As a consequence, there is change in elastic potential energy of the spring. Let “x” be the extension in the spring by the time block of “1 kg” is initiated in motion. Then, the total change in potential energy is :

Δ U = 1 2 k x 2

The work by friction is done only on the right block of “2 kg” :

W F = - μ m 2 g x

On the other hand, the work by external force is :

W E = F x

Putting all these values in the equation of conservation of energy :

F x μ m 2 g x = 1 2 k x 2

F μ m 2 g = 1 2 k x

Clearly, we can not solve this equation as there are two unknowns, “F” and “x”. We, therefore, make use of the fact that spring force on the block of mass “1 kg” is equal to maximum static friction,

Forces on the block

The spring force is equal to maximum static friction.

k x = μ m 1 g

Combining two equations, we have :

F μ m 2 g = 1 2 μ m 1 g

F = μ g m 1 + m 2 2 = 0.5 X 10 1 + 2 2 X 10 = 10 N

It is interesting to note that force required to initiate left block is independent of spring constant.

Mechanical process without external force on the system

Since no external force operates on the system, there is no work by external force. The system under consideration is, therefore, an isolated system. The form of conservation law for general mechanical process is further reduced as :

W F = Δ K + Δ U

In words, we can put the conservation of energy for mechanical process under given condition as :

“Work by friction within an isolated system is equal to the change in potential and kinetic energy of the system.”

Example

Problem 2: In the arrangement shown, the block of mass 10 kg descends through a height of 1 m after being released. The coefficient of friction between block and the horizontal table is 0.3, whereas pulley is friction-less. Considering string and pulley to be “mass-less”, find the speed of the blocks.

Block pulley system

Blocks move by 1 m.

Solution : Here, friction is present as internal force to the system. Hence, we use the form of conservation law as :

W F = Δ K + Δ U

Let us denote 6 kg and 10 kg blocks with subscript “1” and “2”. There is no change of height for the block on the table. Only change in gravitational potential energy is due to change in the height of the block hanging on the other end of the string. Thus, change in potential energy is :

Δ U = m 1 g h + m 2 g h = 0 10 x 10 x 1 = - 100 J

Two blocks are constrained by a taut string. It means that both blocks move with same speed. Let the speed of the blocks after traveling “1 m” be "v". Now, initial kinetic energy of the system is zero. Therefore, the change in kinetic energy is given by :

Δ K = 1 2 m 1 v 2 + 1 2 m 2 v 2

Δ K = 1 2 X 6 X v 2 + 1 2 X 10 X v 2

Δ K = 8 v 2

Friction works only on the block lying on the table. Here, work by friction is given as :

W F = - μ N x = - μ m 1 g x = - 0.3 X 6 X 10 X 1 = - 18 J

Putting in the equation of conservation of energy,

- 18 = 8 v 2 100

v 2 = 82 8

v = 3.2 m / s

Questions & Answers

what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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can anyone tell who founded equations of motion !?
Ztechy Reply
n=a+b/T² find the linear express
Donsmart Reply
Quiklyyy
Sultan Reply
Moment of inertia of a bar in terms of perpendicular axis theorem
Sultan Reply
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
Yara Reply
Centre of mass of two uniform rods of same length but made of different materials and kept at L-shape meeting point is origin of coordinate
Rama Reply
A balloon is released from the ground which rises vertically up with acceleration 1.4m/sec^2.a ball is released from the balloon 20 second after the balloon has left the ground. The maximum height reached by the ball from the ground is
Lucky Reply
work done by frictional force formula
Sudeer Reply
Torque
Misthu Reply
Why are we takingspherical surface area in case of solid sphere
Saswat Reply
In all situatuons, what can I generalize?
Cart Reply
the body travels the distance of d=( 14+- 0.2)m in t=( 4.0 +- 0.3) s calculate it's velocity with error limit find Percentage error
Clinton Reply
Explain it ?Fy=?sN?mg=0?N=mg?s
Admire Reply

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Source:  OpenStax, Physics for k-12. OpenStax CNX. Sep 07, 2009 Download for free at http://cnx.org/content/col10322/1.175
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