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41 problems tagged with Friction

P0720 The "Self-Propelled" Man
Intermediate
Mechanics › Rotational Motion

P0720

Intermediate Mechanics › Rotational Motion

The "Self-Propelled" Man

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 $I$.
  1. Kinematic Constraint: If the man pulls a length $L$ of rope through his hands relative to his position on the block, what is the displacement $d$ of the block relative to the ground?
  2. Ideal Dynamics (Part A): Determine the acceleration $a$ of the system (man + block) in terms of $T$, $m$, and $M$.
  3. The "Slippery Shoes" Limit (Part A): Determine the maximum tension $T_{max}$ the man can apply before his shoes slip on the block. Discuss how the direction of the required friction force depends on the ratio of the masses $M$ and $m$.
  4. Kinetic Energy (Part A): What is final kinetic energy of the system after the block has moved a distance $d$ from rest?
  5. Non-Ideal Dynamics(Part B): Derive the new acceleration $a'$ of the system, taking into account the moment of inertia $I$ of both pulleys. Express your answer in terms of $T$, $m$, $M$, $I$, and $R$.
  6. Friction Comparison(Part B): Does the presence of heavy pulleys (non-zero $I$) increase or decrease the static friction force required between the man's shoes and the block to prevent slipping (compared to the ideal case in Q3)? Justify your answer physically.
Friction Newton's Law rigid body dynamics
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