Exercises: Mechanisms and simple machines

Explain what a lever is and name its three elements (fulcrum, effort, and load) using the example of a park seesaw.

What is a pulley used for? Explain its purpose and give two examples of machines or installations that use one.

The figure shows three different levers, with their effort F, load R, and fulcrum. Classify each one according to the class of lever (first, second, or third class) and name an everyday object that functions like each one.

Classify these objects as first-, second-, or third-class levers, reasoning where the fulcrum is in each one: a pair of scissors, a nutcracker, a pair of tweezers, and a seesaw.

Explain what a gear is and what its teeth are used for: why are two smooth touching wheels not simply used to transmit rotation?

Name two household machines or installations that use pulleys or belts (window blind, washing machine...) and explain what motion they transmit in each case.

Explain in your own words what the 'mechanical advantage' of a simple machine is and give an example using a lever.

A screw is said to be "an inclined plane wrapped around a cylinder". Explain this concept by comparing it to walking up a steep hill in a zig-zag pattern instead of in a straight line.

A lever has its fulcrum at $0.5\ \text{m}$ from the load and at $1.5\ \text{m}$ from the point where effort is applied. If the load is $60\ \text{N}$, apply the law of the lever, $F \times b_F = R \times b_R$, to calculate the required effort and state whether the lever provides a mechanical advantage.

In a pair of gears, the driver gear has 20 teeth and the driven gear has 40. If the driver rotates at $60\ \text{rpm}$, calculate the rotational speed of the driven gear and state whether the gear system multiplies force or speed.

Using the concept of gear ratio, explain why a bicycle uses a large chainring and a small sprocket to go fast on flat ground, and the opposite combination to go up a slope.

Using a $2\ \text{m}$ bar, you want to lift a load of $100\ \text{N}$ by placing the fulcrum at $0.5\ \text{m}$ from the load. Calculate the required effort by applying the law of the lever, $F \times b_F = R \times b_R$, and state the length of the effort arm remaining.

A winch winds up a rope to lift a bucket: the cylinder has a radius of $10\ \text{cm}$ and the crank handle describes a radius of $40\ \text{cm}$. Calculate the effort that must be applied to the crank to lift a load of $80\ \text{N}$, reasoning from the equilibrium of moments as in a lever.

The figure shows two gears in mesh: the driver gear has 12 teeth and rotates at $90\ \text{rpm}$ in the direction shown. Calculate the speed at which the driven gear of 36 teeth rotates, and determine its direction of rotation.

Explain what a reduction gearbox is and why almost all small electric motors have one built-in: explain the relationship between rotational speed and force (torque) at the input and output.

A bicycle has a 44-tooth chainring and an 11-tooth sprocket. Calculate how many turns the rear wheel makes for each full turn of the pedals and explain whether this gear ratio is better suited for speed or for climbing hills.

A robotics team has a motor with a torque of $0.3\ \text{N}\times\text{m}$ rotating at $1500\ \text{rpm}$. They need to lift a $1.5\ \text{kg}$ load hanging from the end of a $20\ \text{cm}$ arm attached directly to the motor shaft. A team member suggests: "To make the motor work less, we lengthen the arm to $40\ \text{cm}$ to have more leverage; and if it still struggles, we add a $1:5$ gear reducer, which will increase the force without changing the lifting speed." Refute both statements with technical arguments and calculations: calculate the initial and final load torque on the motor shaft (take $g = 10\ \text{N/kg}$), explain the conceptual error regarding the arm length, and justify which physical quantity will inevitably be sacrificed if the reducer is used.

You are designing a construction wheelbarrow with a total length of $1.4\ \text{m}$, where the wheel axle is located at one end ($0\ \text{m}$) and the handles at the other ($1.4\ \text{m}$). A load of $60\ \text{kg}$ will be transported (take $g = 10\ \text{N/kg}$). Due to occupational risk prevention regulations, the operator must not exert a vertical force greater than $150\ \text{N}$ when lifting it.

1. a) Model the wheelbarrow as a lever: classify it and calculate the maximum distance from the wheel axle at which the center of gravity of the load bucket must be located to comply with the standard.
2. b) If the operator shifts the load, placing it exactly directly above the wheel axle, how much vertical force will be needed to hold it? What mechanical and practical stability drawback arises from doing this?
3. c) Justify how the force supported by the operator varies as they lift the handles, tilting the wheelbarrow with respect to the horizontal.

A $60\ \text{kg}$ box needs to be lifted to a mezzanine located $3\ \text{m}$ high. The available person weighs $50\ \text{kg}$ ($500\ \text{N}$), so if they pull downwards on a rope, they can only apply at most their own weight before being lifted off the ground (take $g = 10\ \text{N/kg}$).

1. a) Design a pulley system indicating the minimum number of fixed and movable pulleys needed so that the person can raise the load by pulling vertically downwards on the rope comfortably.
2. b) Calculate the exact theoretical force they will exert, the meters of rope they will have to pull in to raise the box the $3\ \text{m}$, and the mechanical work performed.
3. c) Compare that work with the work they would perform if they lifted the box by hand, and explain why the simple machine does not save energy.

A block and tackle with four pulleys divides the effort by four. Calculate the force required to lift a load of $200\ \text{N}$ and explain the "cost" involved: how many meters of rope must be pulled to raise the load by $1\ \text{m}$.

You need to design the reduction gear transmission for the blades of a toy wind turbine. The micromotor rotates at $300\ \text{rpm}$ clockwise and the blade shaft must rotate at exactly $30\ \text{rpm}$, also clockwise. In your parts box you only have gears with between $10$ and $40$ teeth (no gear can have fewer than $10$ or more than $40$ teeth).

1. a) Demonstrate why it is impossible to solve this design with a single pair of meshed gears (analyse both the tooth count constraint and the direction of rotation).
2. b) Design a two-stage compound gear train that meets all conditions: specify the number of teeth of each gear ($Z_1, Z_2, Z_3, Z_4$), which gears are mounted on the same shaft, and justify why the final direction of rotation matches that of the motor.
3. c) Numerically verify using the overall transmission ratio that exactly the requested $30\ \text{rpm}$ are obtained.

A tower crane has a horizontal jib of $30\ \text{m}$ and a counter-jib of $10\ \text{m}$ with a concrete counterweight of $4500\ \text{kg}$. Neglect the self-weight of the metal structure and take $g = 10\ \text{N/kg}$.

1. a) Model the crane as a lever in equilibrium about the mast and calculate the maximum load it can support when the trolley is at the tip of the jib ($30\ \text{m}$) without the crane tipping forward.
2. b) If the trolley moves towards the center and is located only $6\ \text{m}$ from the mast, calculate the new theoretical admissible load. Interpret this result to explain why construction crane load charts allow lifting much more weight near the tower than at the tip.
3. c) The hook hangs from a hoist with a movable pulley that splits the load between two cable falls. If the winch motor spools cable at a speed of $0.8\ \text{m/s}$, determine the vertical speed at which the load rises and explain what mechanical advantage this arrangement provides to the motor.

The overhead door of a warehouse has a mass of $200\ \text{kg}$. To operate it, there is a debate between installing an ordinary spur gear reducer ($i = 1/40$) or a single-start worm gear mechanism with a $40$-tooth wheel ($i = 1/40$), both connected to a motor rotating at $1200\ \text{rpm}$.

1. a) Calculate the rotational speed of the door shaft with each system.
2. b) Imagine an unexpected power outage occurs while the door is stopped at mid-height. Predict and justify what will happen to the load with each of the two mechanisms based on the concept of mechanical irreversibility (self-locking).
3. c) Despite the safety advantage of the worm drive, explain why its efficiency is usually only around $50\,\%$ or $60\,\%$ compared to more than $90\,\%$ for spur gears, and what thermal problem would limit its use in a continuously operating machine.

A light vehicle manufacturer must choose the secondary transmission for two projects: an urban rental electric bicycle (exposed to rain and dirt, where users must not get stained with grease and maintenance must be minimal) and a racing electric motorcycle with very aggressive accelerations ($90\ \text{N}\times\text{m}$ of instantaneous torque).

1. a) Select and justify the most suitable system for each vehicle between roller chain and synchronous toothed belt, analysing lubrication requirements, cleanliness, system mass, and behavior under sudden stress peaks.
2. b) An operator states: "On rainy days, the toothed belt slips on the pulley just like an alternator flat belt, causing the transmission ratio to vary." Technically refute his statement by explaining the operating principle of synchronous belts versus friction belts.