Exercises: 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.

Indicate which class of lever each of these objects belongs to and explain where the fulcrum, effort and load are in each case: (a) a nutcracker, (b) tweezers, (c) a seesaw, and (d) a wheelbarrow.

The figure shows a lever with its elements indicated by letters: two arms (d and r), two forces (F and R) and the fulcrum, as shown in the figure.

State what each letter represents (d, r, F, R) and explain, looking at the diagram, why a large load can be lifted using a small effort with this lever.

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).

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.

Indicate which class of lever each of these objects belongs to and where the fulcrum is in each case: a bottle opener, a wheelbarrow, a peg and an oar of a boat.

The effect of a force on a lever depends on the moment (the force and its distance from the fulcrum). Using the example of a door, explain why it is much easier to open it by pushing far from the hinges than by pushing close to them.

Indicate where the fulcrum is located in a pair of scissors, pliers and a wheelbarrow, and explain what these three fulcrums have in common (how each object pivots when used).

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.

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.

The figure shows a lever in equilibrium with a load of $300\ \text{N}$ at $0{,}5\ \text{m}$ from the fulcrum and the effort applied at $1{,}5\ \text{m}$ from the fulcrum, as shown in the figure.

Applying the principle of the lever $F \cdot d = R \cdot r$, calculate the value of the effort $F$ that maintains equilibrium, with its unit, and explain why it is smaller than the load.

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).

A wheelbarrow carries a load of $360\ \text{N}$ centred $0{,}4\ \text{m}$ from the wheel axle (which acts as the pivot), and the handles are held $1{,}2\ \text{m}$ from the axle. Calculate the total force that the arms must exert to hold the load, and explain why the wheelbarrow makes the weight feel so much lighter.

On a seesaw, a child weighing $30\ \text{kg}$ sits $2\ \text{m}$ from the centre and balances an adult of $60\ \text{kg}$. Calculate how far from the centre the adult must sit (weights are proportional to masses, so you can work directly with kilograms in the principle of moments / law of the lever).

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 $1\ \text{m}$.

A classmate claims: "Energy is created with a lever because I move a large weight with a small force." Refute this using the concept of work (force times displacement): if the force is halved, what happens to the displacement of the point where you apply it?

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.

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.

To strip a concrete block that exerts a downward vertical resistance of $900\ \text{N}$, you have a rigid steel lever of $3\ \text{m}$ in length. Two operators argue about how to position it: operator A proposes using it as a first-class lever by placing a fulcrum on the ground at $0{,}5\ \text{m}$ from the block and pushing down on the opposite end; operator B proposes using it as a second-class lever by resting one end on the ground, placing the block's attachment at $0{,}5\ \text{m}$ from the fulcrum, and pulling upward from the opposite end. (a) Calculate the required force at the free end and the vertical reaction on the fulcrum in each of the two configurations. (b) If the ground is soft soil and gives way if the concentrated force on the fulcrum exceeds $1000\ \text{N}$, determine which configuration is unfeasible and why. (c) Predict how the mechanical advantage of each system would vary if the bar operates at an inclination of $25^\circ$ with respect to the horizontal.

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.

A robotic surgical micro-gripper uses a third-class lever to manipulate delicate tissues that suffer necrosis if the clamping force exceeds $0{,}8\ \text{N}$. The gripper measures $10\ \text{cm}$ from the pivot axis to the gripping tip. The piezoelectric actuator pushes perpendicularly at only $1\ \text{cm}$ from the pivot axis, with a linear advance resolution of $\pm 0{,}02\ \text{mm}$ and a force tolerance of $\pm 0{,}1\ \text{N}$. (a) Calculate the force that the actuator must apply when the gripper exerts exactly the safe limit of $0{,}8\ \text{N}$ on the tissue. (b) Determine the displacement transmission ratio (kinematic advantage) and calculate the minimum advance experienced by the tip of the gripper with each step of the actuator. (c) A novice designer suggests changing the micro-gripper to a second-class lever to "save motor power". Justify why their proposal would ruin the safety of the surgical procedure by analyzing force control sensitivity and kinematic precision.

The biomechanical system of the human forearm acts as a 3rd-class lever: the elbow joint is the fulcrum, the biceps tendon inserts at $4\ \text{cm}$ from the joint axis with a traction angle of $90^\circ$ when the forearm is horizontal, and the center of gravity of the forearm and hand ($m_a = 1{,}5\ \text{kg}$) is located at $14\ \text{cm}$ from the elbow. A person holds in their hand a load of $8\ \text{kg}$ located at $34\ \text{cm}$ from the elbow ($g = 9{,}8\ \text{m/s}^2$). (a) Model the rotational equilibrium and calculate the contraction force that the biceps must exert, as well as the joint compression force supported by the elbow cartilage. (b) If the biceps shortens at a constant speed of $4\ \text{cm/s}$, determine the upward linear speed of the load in the hand and the instantaneous mechanical power delivered by the muscle. (c) Biomechanically refute the statement that "the human body would be better designed if the biceps were anchored near the wrist so as not to have to exert such disproportionate muscular forces", analyzing the angular range, the distal velocity, and the required body volume.