Dynamics: Pulleys and Tension

If the 1 kg mass accelerates upwards with an acceleration of a, the movable pulley P also accelerates upwards with the same acceleration a.

If the tension in the string connecting the 2 kg and 3 kg masses is T, then the tension in the string connected to the fixed support above pulley P is also T.

The relative acceleration, denoted as $a_r$, is equal for both the 2 kg and 3 kg blocks, when measured relative to the movable pulley P.

If the 2 kg block's net acceleration relative to ground is $a + a_r$, then the 3 kg block's net acceleration must also be $a + a_r$, relative to the ground.

If the force applied on the upper string is 20 N, and assuming the pulley to be massless, the tension in the string connecting the 2 kg and 3 kg masses will be 15 N.

Lami's theorem is applicable for analyzing systems involving four concurrent forces in equilibrium.

If in the given example, the length of the string AC was 1m and BC remained as 0.3m, the value of $cos \theta$ would be 0.3.

In the example provided, if force F was acting vertically downwards instead of horizontally, the system could still be in equilibrium with appropriate adjustments to tension T and the 8N weight.

Lami's theorem states that for three concurrent forces in equilibrium, the magnitude of each force is inversely proportional to the sine of the angle between the other two forces.

If angle $\beta$ between $F_1$ and $F_3$ is 90 degrees, and angle $\gamma$ between $F_2$ and $F_3$ is also 90 degrees, then force $F_3$ must be zero for the system to be in equilibrium.

In the pulley system described, if $x_1 + x_2 = l$ where $l$ is the length of the string, then $x_1$ and $x_2$ must always represent the displacements from the center of the pulley.

Given the constraint equation $x_1 + x_2 = l$, where $l$ is constant, differentiating this equation with respect to time will yield $a_1 - a_2 = 0$, where $a_1$ and $a_2$ are the accelerations of the respective objects.

If the constraint equation yields $a_1 = -a_2$, then object 1 and object 2 always move in precisely opposite directions.

In using constraint methods, the length of the string segment draped over the pulley must be included in the constraint equation for accurate results.

When using constraint equations, only the movable points in the system need to be considered when formulating the equations.

In the given system, if the mass of the block on the horizontal surface is doubled while keeping the hanging mass constant, the acceleration of the system will be halved.

If the string connecting the masses is cut, the 4 kg block will experience free fall with an acceleration of approximately 10 m/s², while the 2 kg block will remain stationary.

If the surface is not frictionless, and the coefficient of kinetic friction is $\mu_k = 0.2$, the net pulling force would decrease, and the acceleration of the system would increase.

Assuming ideal conditions (massless string, frictionless pulley), the tension throughout the string connecting the 2 kg and 4 kg blocks will remain constant, regardless of the acceleration of the system.

If the 4 kg block is placed on an inclined plane with an angle of 30 degrees instead of hanging vertically, the net pulling force will remain the same.

If the mass of the 2 kg block is doubled, the tension in the string will increase, assuming the 4 kg mass remains constant and is hanging vertically.

If the 4 kg block is replaced with a 6 kg block, the new acceleration of the system will be exactly 1.5 times the original acceleration.

If both the 2 kg and 4 kg masses are doubled, the acceleration of the system will remain unchanged, assuming $g$ remains constant.

If block-1 experiences a tension force of 2T and block-2 experiences a tension force of T, block-2's acceleration will be twice that of block-1, assuming equal masses.

In a system with multiple blocks connected by strings and pulleys, if block-1 has an upward acceleration of a, then block-2 necessarily has a downward acceleration of 3a.

The constraint relation between multiple blocks connected by strings remains constant even if the strings are not perfectly inextensible.

If two blocks connected by a string have upward velocities of 1 m/s, a third block connected to the same string must also have an upward velocity of 1 m/s at that same moment.

If a 1 kg block and a 2 kg block are connected by a string over a light pulley, and all surfaces are smooth, the tension in the string will be greater than 10 N but less than 20 N.

In a system of blocks and pulleys, if the pulley and string are light, and all surfaces are smooth, increasing the mass of one of the blocks will always increase the tension in the string.

If a mass M remains at rest in a system of pulleys and strings with 3 kg and 2 kg blocks, and friction is absent, then M must be equal to the sum of the other two masses.

With a frictionless setup with 3kg and 2kg blocks connected by a string over pulleys, the heavier block will always accelerate downwards at g.

In a scenario with two blocks (m1 and m2) connected by a cord over a pulley, and pulled by a force F, increasing the mass of m1 while keeping F constant will always increase the tension in the cord connecting m2 and the pulley.

If block 3 has a downward acceleration of $6 m/s^2$ and block 2 has an upward acceleration of $4 m/s^2$, then block 1 must have an acceleration less than $4 m/s^2$.

When analyzing the acceleration of a block of mass M on a frictionless inclined plane connected to another mass (2M), the angle of the incline is the only factor determining its acceleration.

An inertial frame of reference is always at rest.

A frame of reference rotating at a constant angular speed can be considered an inertial frame.

The Earth is technically a non-inertial frame of reference, but for most everyday calculations, it can be treated as an inertial frame without introducing significant errors.

Pseudo forces are real physical forces that arise due to the acceleration of an object.

If a child is standing in an elevator accelerating upwards, an observer in the elevator (a non-inertial frame) will perceive the child as being at rest after applying the concept of pseudo force, assuming no other external forces act on the child.

Lami's Theorem

Equilibrium

Tension Force

Concurrent Forces

sin (θ)

Same String Constraint

Pulley Acceleration

Net Acceleration

Relative acceleration (a_r)

Massless Pulley Net Force

Constraint Method

Constraint Equation

Velocity (v)

Acceleration (a)

Meaning of a Negative Sign (Constraint Equations)

System Acceleration

String Tension

Free Body Diagram (FBD)

Net Pulling Force

Newton's Second Law

Total Mass

Perpendicular Weight

Confirming values.

Inertial Frame

Non-inertial Frame

Is Earth an Inertial Frame?

Pseudo Force

Why use Pseudo Force?

Frame of Reference

Non-inertial Effects

Condition Change

Constraint Relation

Acceleration Relationship (Tension)

Tension

Kinematic Constraints

Ideal Pulley System

Static Equilibrium (Mass M)

Block-String System

Smooth Surfaces & Light Pulleys