$\vec{F}$=m $\vec{a}$

$\vec{a}$ is as measured from an inertial reference frame

$\vec{F}$: Information is going to come from the free body diagram.

$\vec{a}$: Kinematics

For example: hanging pulley

Top pulley may be fixed, the bottom pulley, may be moving.

We could have complex cases like this example.

In general the accelerations of bodies 1, 2 and 3 may not be equal.

You may have to find a relation between $a_1$, $a_2$ & $a_3$.

The free body diagrams of bodies 1, 2, and 3 separately.

$\Sigma \vec{F}$=m $\vec{a}$

Rod is light, we are assuming mass is nearly 0.

$\vec{F_1}= \vec{F_2}$

Case:1 Rod is in compression.

Case:2 Rod is in tension.

Impending slip, $\rightarrow$ f=$\mu_s N$

Actual slip, $\rightarrow$ f=$\mu_k N$

Friction $\propto$ N

1) Assume no slip

When there is no movement acceleration of body is 0.

Find the value of f, from equation.

$\Sigma F =0$.

No slip then friction f is an unknown force.

Check:

is f <$\mu_s N$

f <$\mu_k N$

You will have to go to the y direction equations get the value of N.

$f<\mu_s N=$ Assumption

Else resolve the problem assume slip, $f=\mu_k N$

a $\rightarrow$ Unknown, solve for 'a'

Check:

Is f <$\mu_s N$

f <$\mu_k N$

You will have to go to the y direction equations get the value of N.

$f<\mu_s N=$ Assumption

Else resolve the problem assume slip, $f=\mu_k N$

Pulley $P_1$ (fixed) on end of string, ties to mass $m_1$, other end to mass $m_2$.

Assumed that the pulleys are frictionless and light.

String is inextensible which means the length of the string is constant.

Find acceleration $a_1$ and acceleration $a_2$ of masses $m_1$ and $m_2$.

FBD of body 2

$m_2 g-T=m_2 a_2$

FBD of body 1 :

$\Sigma F_y$: $N_1=m g$

$T=m_1 a_1$

$a_1 = a_2$

i) When a string passes over a fixed pulley then the accelerations (magnitudes) of the bodies tied on the two ends are equal.

ii) The tension in the string is equal.

$T_1 = T_2$.

$T =m_2(g-a)$

$T =m_1 a$

$m_1 a =m_2(g-a)$

$(m_1+m_2) a =m_2 g$

a =$\frac{m_2 g}{(m_1+m_2)}$

If we add friction on the table for mass 1

$m_2 g-T =m_2 a$

$N_1 =m_1 g$

$T-f =m_1 a$

Where: $f = \mu_k N_1$

Pulley $P_1$ (Fixed)

At the two ends the magnitude of accelerations will be equal.

The tension will be the same throughout the string.

We assume frictionless contacts at the slope as well as at the table.

Draw the free body diagram of body 1.

$N_1=m_1 g$

$T=m_1 a$

There are two unknowns, only one equation.

$N_2=m_2 g \cos \theta$

$m_2 g \sin \theta -T = m_2 a$

There are two unknowns T and a.

Accelerations of blocks connected to the ends of same string passing over a fixed pulley have the same magnitude.

Fixed or moving pulleys

The pulley is light and frictionless: $T_1=T_2$

Problems with multiple accelerating pulleys and string, set up equations relating accelerations of various blocks/ bodies connected by string.

Done by relating coordinate of moving blocks and pulleys to the length of the string which is fixed.

Using the length of the string is fixed, find the accelerations and relation between them.

Pulley is not fixed.

$m_1$, $m_2$ and F are given.

Frictionless contact

Coordinate of mass $m_1$ (reference should be a fixed )

$x_1$ = coordinate of $m_1$

$x_2$= centre of pulley.

What is the length of the string going on the pulley

Express length in terms of $x_1$ and $x_2$.

$L=2 x_2-x_1$

Differentiate with respect to time.

$0=2 \dot{x}_2-\dot{x}_1$

Differentiate $2^{\text {nd }}$ time

$0=2 \ddot{x}_2-\ddot{x}_1$

${x} _1=2 \ddot{x} _2$

$a_1=2 a_2$

Generalize: If one end of a string passing over a moving pulley is fixed, acceleration of the other end is twice the accelerator of pulley.

$F - T_2 = m_2a_2$

Free body diagram of body 1.

$T_1 = m_1a_1$

$2T_1 = T_2$

$a_1 = 2 a_2$

$T_1=m_1 a_1$

$F-T_2=m_2 a_2$

$a_1=2 a_2$

$2T_1 = T_2$

$T_1=m_1 a_1=2 m_1 a_2$

$F-2 (2 m_1 a_2)=m_2 a_2$

F=$(4 m_1+m_2) a_2$ $\longleftarrow$ $a_2$ & $a_1$

Draw free body diagram of bodies 2 and 3.

$m_{2} g-T_2=m_2 a_2$

$m_{2} g-T_2=m_3 a_3$

FBD of $P_2$

$T_1 = 2T_2$

FBD of $m_1$

$2 T_2=m_1 a_1$

$a_1 , a_2 , a_3$, and $T_2$

We have four unknowns and we have three equations.

Find relation between accelerations

x $\leftarrow$ Distance of mass 1 form right .

$x_1 + x_p = l_1$

Where: $l_1$ is length of the first string.