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.

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.

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, $F \cdot b_F = R \cdot b_R$, to calculate the required effort and state whether the lever provides a mechanical advantage.

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

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 metres 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.

In a workshop, a tackle with two fixed pulleys and two movable pulleys is used to lift a $160\ \text{kg}$ motor to a height of $1,5\ \text{m}$ ($g = 10\ \text{N/kg}$). When performing the manoeuvre at a constant speed, a dynamometer placed in the rope measures an actual pulling force of $500\ \text{N}$.

1. a) Calculate the ideal theoretical force that should be applied and determine the mechanical efficiency ($\eta$) of the actual pulley system.
2. b) Justify physically what causes the difference between the theoretical and actual force, and indicate what the 'lost' mechanical energy is transformed into.
3. c) If the operator has a usable length of rope of $5\ \text{m}$ before bottoming out on the drum, deduce through calculations whether they will be able to lift the motor the required $1,5\ \text{m}$ continuously.

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

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.

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

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 robotics team has a motor with a torque of $0,3\ \text{N}\cdot\text{m}$ 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.

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, $F \cdot b_F = R \cdot b_R$, 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.

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.

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.