# Pulleys  (Page 2/5)

 Page 2 / 5

There is a very useful technique to simplify the solution involving "mass-less" string and pulley. As string has no mass, the motion of the block-string system can be considered to be the motion of a system comprising of two blocks, which are pulled down by a net force in the direction of acceleration.

Let us consider two blocks of mass " ${m}_{1}$ " and " ${m}_{2}$ " connected by a string as in the previous example. Let us also consider that ${m}_{2}>{m}_{1}$ so that block of mass " ${m}_{2}$ " is pulled down and block of mass " ${m}_{1}$ " is pulled up. Let "a" be the acceleration of the two block system.

Now the force pulling the system in the direction of acceleration is :

$F={m}_{2}g-{m}_{1}g=\left({m}_{2}-{m}_{1}\right)g$

The total mass of the system is :

$m={m}_{1}+{m}_{2}$

Applying law of motion, the acceleration of the system is :

$a=\frac{F}{m}=\frac{\left({m}_{2}-{m}_{1}\right)g}{\left({m}_{1}+{m}_{2}\right)}$

Clearly, this method to find acceleration is valid when the block - string system can be combined i.e. accelerations of the constituents of the system are same. We can check the efficacy of this technique, using the data of previous example. Here,

${m}_{1}=10\phantom{\rule{1em}{0ex}}kg;\phantom{\rule{1em}{0ex}}{m}_{2}=20\phantom{\rule{1em}{0ex}}kg$

The acceleration of the block is :

$a=\frac{\left({m}_{2}-{m}_{1}\right)g}{\left({m}_{1}+{m}_{2}\right)}=\frac{\left(20-10\right)X10}{\left(10+20\right)}$

$a=\frac{10}{3}=3.33\phantom{\rule{1em}{0ex}}m/{s}^{2}$

## Moving pulley

Moving pulley differs to static pulley in one important respect. The displacements of pulley and block, which is attached to the string passing over it, may not be same. As such, we need to verify this aspect while applying force law. The point is brought out with clarity in the illustration explained here. Here, we consider a block attached to a string, which passes over a mass-less pulley. The string is fixed at one end and the Pulley is pulled by a force in horizontal direction as shown in the figure.

In order to understand the relation of displacements, we visualize that pulley has moved a distance “x” to its right. The new positions of pulley and block are as shown in the figure. To analyze the situation, we use the fact that the length of string remains same in two situations. Now,

Length of the string, L, in two situations are given as :

$L=\mathrm{AB}+\mathrm{BC}=\mathrm{AB}+\mathrm{BD}+\mathrm{ED}$

$⇒\mathrm{BC}=\mathrm{BD}+\mathrm{ED}$

$⇒\mathrm{ED}=\mathrm{BC}-\mathrm{BD}$

Now, displacement of the block is :

$⇒\mathrm{CE}=\mathrm{CD}-\mathrm{ED}=\mathrm{CD}-\mathrm{BC}+\mathrm{BD}=\mathrm{BD}+\mathrm{BD}=2\mathrm{BD}=2x$

Note that for every displacement “x” of pulley, the displacement of block is 2x. We can appreciate this fact pictorially as shown in the figure below :

Further, as acceleration is second derivative of displacement with respect to time, the relation between acceleration of the block ( ${a}_{B}$ ) and pulley ( ${a}_{P}$ ) is :

$\begin{array}{l}⇒{a}_{B}=2{a}_{P}\end{array}$

This is an important result that needs some explanation. It had always been emphasized that the acceleration of a taut string is always same through out its body. Each point of a string is expected to have same velocity and acceleration! What happened here ? One end is fixed, while other end is moving with acceleration. There is, in fact, no anomaly. Simply, the acceleration of the pulley is also reflected in the motion of the loose end of the string as they are in contact and that the motion of the string is affected by the motion of the pulley.

#### Questions & Answers

what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
n=a+b/T² find the linear express
Quiklyyy
Moment of inertia of a bar in terms of perpendicular axis theorem
How should i know when to add/subtract the velocities and when to use the Pythagoras theorem?
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
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
work done by frictional force formula
Torque
Why are we takingspherical surface area in case of solid sphere
In all situatuons, what can I generalize?
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
Explain it ?Fy=?sN?mg=0?N=mg?s