Friction

Three blocks are stacked on a horizontal table. From top to bottom, their masses are $m_1 = 5m$, $m_2 = 3m$, and $m_3 = m$. A horizontal force $F$ is applied to the middle block, $m_2$. The coefficient of friction between block 1 and 2 is $4\mu$; between block 2 and 3 is $2\mu$; and between block 3 and the table is $\mu$. Assume the coefficients of static and kinetic friction are equal.

A boat of mass $m$ is traveling at an initial speed $v_i$ when its engine is shut off. The drag force from the water is proportional to the boat's speed $v$, given by $F_d = bv$, where $b$ is a constant drag coefficient.

A slab of mass $m_1$ rests on a frictionless floor. A block of mass $m_2$ rests on the slab. The coefficients of static and kinetic friction between the block and slab are $\mu_s$ and $\mu_k$, respectively. A horizontal force of magnitude $F$ is applied to the block in the positive x-direction.

A block of mass $M$ rests on a smooth, frictionless horizontal floor. A man of mass $m$ stands on top of the block. A system of two pulleys is used to accelerate the block:

1. One pulley is fixed to the wall (left).
2. One pulley is attached to the front of the block.
3. A massless, inextensible rope is anchored to the wall, passes over the block's pulley, then back over the wall's pulley, and finally into the hands of the man.

The rope segments are all horizontal. The coefficient of static friction between the man's shoes and the top of the block is $\mu_s$. The man pulls the rope with a tension force $T$, causing the entire system (block + man) to accelerate towards the wall (to the left).

1. Part A: Ideal Pulleys: Assume the pulleys are massless and frictionless.
2. Part B: Real Pulleys: Assume both pulleys are solid disks with mass $M_p$, radius $R$, and moment of inertia $I$.