Levers and pulleys

There are three classes of lever depending on the relative positions of the fulcrum, the effort (force) and the load (resistance). Describe them and give an example of an everyday object for each class.

Explain what the mechanical advantage of a lever is and how it relates to the length of the effort arm and the load arm (which is greater than which when the lever 'multiplies' the force).

The principle of the lever states that $F \cdot d = R \cdot r$, where $F$ is the applied force (effort), $d$ the effort arm, $R$ the resistance (load) and $r$ the load arm. A lever has a load of $200\ \text{N}$ placed at $0{,}4\ \text{m}$ from the fulcrum and the effort is applied at $1{,}6\ \text{m}$ from the fulcrum. Calculate the value of the effort $F$ required to balance it, stating the unit of the result.

What is a fixed pulley? Describe two real situations where it is used (a well, a flagpole, a roller blind) and what it provides in each, even if it does not reduce the required force.

Explain the difference between a fixed pulley and a movable pulley. Why is it said that a fixed pulley does not reduce the required force, yet is equally useful for lifting loads? Give an example of how each is used.

To lift a weight of $200\ \text{N}$, a movable pulley supported by two rope segments is used. Calculate the force required at the free end (ignoring friction) and explain how much rope must be pulled in for the load to rise $1\ \text{m}$.

The inclined plane is another simple machine that 'multiplies' force: the force required to move a load up it (without friction) is $F = P \cdot \frac{h}{L}$, where $P$ is the weight, $h$ the height and $L$ the length of the plane. To move a box of $200\ \text{N}$ to a height of $0{,}5\ \text{m}$ up a ramp of length $2\ \text{m}$: (a) calculate the force required. (b) Compare this with lifting it straight up and explain what 'price' is paid in return (travelling a greater distance).

To lift a $250\ \text{kg}$ piano to a first floor located at a height of $4\ \text{m}$ ($g = 9{,}8\ \text{m/s}^2$), two technical options are compared: an $8\ \text{m}$ long wooden ramp with a dynamic friction coefficient $\mu = 0{,}20$, or a 4-strand block and tackle installed on the balcony with a mechanical efficiency of $80\,\%$ due to friction. You have two operators who can each exert, at most, a sustained force of $350\ \text{N}$. (a) Calculate the total force required to raise the piano at a constant speed in each system. (b) Determine whether the two operators can carry out the maneuver directly by themselves in each option. (c) Analyze the geometric and safety feasibility of the ramp (space occupied on the sidewalk and angle) versus the mechanical risk of overloading the balcony anchor, and provide a justified defense of which solution you would choose.

Two workers need to lift a rock that exerts a resistance of $600\ \text{N}$ to a height of $15\ \text{cm}$ using a uniform steel bar of $2\ \text{m}$ and $12\ \text{kg}$ mass ($g = 9{,}8\ \text{m/s}^2$). They have a fulcrum whose distance $x$ to the rock can be adjusted. One of them claims: "The closer we place the fulcrum to the rock, the better, because the less force we will exert and the less mechanical work it will cost us." (a) Model the static equilibrium taking into account the bar's own weight and obtain the expression for the effort force $F(x)$ at the opposite end as a function of the distance $x$ from the fulcrum to the rock. (b) Assess whether there is any distance $x$ at which the bar's own weight prevents reducing the required force. (c) Technically refute his partner's claim by calculating the displacement of the end and comparing the mechanical work they must perform to lift the rock $15\ \text{cm}$ when $x = 0{,}4\ \text{m}$ and when $x = 0{,}2\ \text{m}$.

A maintenance company needs to hoist a $600\ \text{N}$ load to a height of $6\ \text{m}$. An operator can only exert a maximum pulling force of $190\ \text{N}$ and has a spool with $28\ \text{m}$ of usable rope. Two commercial 4-rope fall systems supporting the load are evaluated: a block and tackle (two fixed pulleys and two movable pulleys) and a Spanish tackle (one fixed pulley and two independent movable pulleys). In both cases, each movable pulley has a mass of $2\ \text{kg}$ ($g = 9{,}8\ \text{m/s}^2$) and the mechanical efficiency due to friction in the axles is $85\,\%$. (a) Analytically derive the theoretical mechanical advantage and the length of rope that must be pulled in each system. (b) Calculate the actual force required in each option taking into account the weight of the movable pulleys and friction losses. (c) Technically justify which of the two systems is viable given the constraints of available force and rope length, and rule out the non-viable one.

In a real Spanish tackle (runner tackle), each additional movable pulley adds a mass of $3\ \text{kg}$ ($g = 9{,}8\ \text{m/s}^2$) and has an axle friction efficiency of $90\,\%$, such that the input tension to each stage is related to the previous one by $T_k = \frac{T_{k-1} + P_{\text{pulley}}}{2 \cdot 0{,}90}$. A load $R = 800\ \text{N}$ needs to be lifted. (a) Set up the successive equilibrium equations and analytically derive the required effort force $F$ with $n = 1$, $n = 2$, $n = 3$, and $n = 4$ movable pulleys. (b) Numerically calculate this force for each value of $n$. (c) Interpret the trend of the results and predict beyond what number of movable pulleys the system ceases to provide an advantage and becomes counterproductive, justifying the physical limit of the design.

A $1200\ \text{kg}$ vehicle has become stuck in the mud and is estimated to oppose a resistance force to motion of $4800\ \text{N}$. A rescuer has an inextensible rope, a pulley, and a tree located $15\ \text{m}$ directly in front of the car, but can only exert a maximum pulling force of $400\ \text{N}$. (a) Analyze two possible setups with the pulley (Setup A: fixed pulley anchored to the tree; Setup B: movable pulley anchored to the car's bumper) and calculate the required effort force and the maximum tension the rope would withstand in each case. (b) Upon verifying that direct pulling is insufficient, the rescuer firmly tightens the rope between the car and the tree and pushes laterally at the midpoint with their $400\ \text{N}$, deflecting the center by $y = 0{,}4\ \text{m}$ relative to the original alignment. Using force resolution, calculate the tension transmitted to the car's hitch and determine whether they manage to move it. (c) Justify what kinematic drawback this method presents as the vehicle moves forward and how it must be operated.