# How Do Pulley Motions Influence Particle Dynamics?

• f(m)
In summary, the conversation is about solving two questions regarding the motion of particles connected by an inelastic string over a smooth pulley. The first question asks to show that if a certain condition is met, one of the particles will not reach the table. The second question asks for an explanation of the motion and a proof for the given condition. The solution involves using the work energy theorem, which is essentially a manifestation of Newton's second law.
f(m)
I was helping my friend with some questions regarding motions of pulleys and i am came up with the following question which i couldn't think of an argument... please check

Two particles A and B of masses M and m respectively are connected by an inelastic string which passes over a smooth pulley. Initially A is at rest on a smooth horizontal table and B hangs at a height h above the table. B is then raised through an extra height H and allowed to fall.
i. Show that if (m2/(M2-m2 ))H<h, B does not reach the table.
ii. If (m2/(M2-m2 ))H≥h show that, during the motion, A rises to a maximum height m{2(M+m)h+mH}/(m+M)

can anyone explain how the motion happens and what is the condition i need start to prove the above questions

thanks...

the second part can also be proved:
after m hits the table ,the string becomes lose
find the velocity of M at that instant using work energy theorem
and then equate this K.E of M to its max potential energy(that corresponds to max height )

it did help but i am wondering is there any other way of proving it rather using conservation of energy theorem, can't we prove only using the Newton's laws of motion and with other required theorem

i think the work energy theorem itself is a manifestation of Newton's secomd law

we write

F(net)=ma

say the motion is 1 dimensional in the x-axis

F(x)=mvdv/dx
take the dx to the other side and integrate

integral(F(x)dx) =delta kinetic energy

so if u have done by work energy theorem ..u are indirectly doing by Newton's laws

## What are pulleys used for?

Pulleys are simple machines that are used to lift or move heavy objects by changing the direction of the force needed to move them. They are commonly used in various machines, such as cranes, elevators, and construction equipment.

## How do pulleys work?

Pulleys work by using a rope or belt to wrap around a wheel or drum. When one end of the rope is pulled, the other end moves in the opposite direction, allowing for the transfer of force to lift or move an object.

## What are the different types of pulleys?

There are three main types of pulleys: fixed pulleys, movable pulleys, and compound pulleys. Fixed pulleys are attached to a stationary object and change the direction of the force. Movable pulleys are attached to the object being moved and reduce the amount of force needed. Compound pulleys combine fixed and movable pulleys to increase the mechanical advantage.

## What is mechanical advantage in pulleys?

Mechanical advantage is a measure of how much the force needed to move an object is reduced by using a pulley. It is calculated by dividing the weight of the object being lifted by the force needed to lift it. The more pulleys used, the greater the mechanical advantage.

## What are some common problems with pulleys?

Some common problems with pulleys include friction, which can decrease the efficiency of the pulley system, and wear and tear on the rope or belt, which can cause it to break. It is important to regularly maintain pulley systems to ensure they are functioning properly.

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