The rigid body and the moment of a force

To open a door, you push perpendicularly to it with a force of $25\ \text{N}$ applied at $0.8\ \text{m}$ from the hinges, as shown in the figure (top view).

Calculate the moment of the force about the axis of the hinges, $M = F \times d$, and state its unit in the International System of Units.

A child pushes a merry-go-round by directing his force straight towards the central axis, and no matter how hard he pushes, he cannot get it to rotate. The situation is shown in the figure (top view).

1. a) Explain, using the definition of the moment of a force, why the moment he produces is zero.
2. b) State how he should direct his push to obtain maximum moment with the same force.

The wind pushes a portable solar panel with a force of $50\ \text{N}$ applied at $0.9\ \text{m}$ from its pivot axis and making an angle of $30^\circ$ with the arm of the panel, as shown in the figure.

When the force is not perpendicular to the arm, the moment is given by $M = F \times d \times \sen \alpha$ (use $\sen 30^\circ = 0.5$).

1. a) Calculate the moment of the wind force about the axis.
2. b) Reason for which angle of incidence the moment would be maximum with the same force.

Two children play with a hotel revolving door. One pushes it clockwise producing a moment of $12\ \text{N}\times\text{m}$ and the other pushes it anticlockwise producing a moment of $12\ \text{N}\times\text{m}$.

1. a) Taking the anticlockwise direction as positive, assign a sign to each moment.
2. b) Calculate the total moment on the door and state whether it will start to rotate (the door was initially at rest).

Two workers rotate the handwheel of an industrial valve by pushing in opposite directions: one exerts $40\ \text{N}$ at $0.5\ \text{m}$ from the axis anticlockwise and the other exerts $50\ \text{N}$ at $0.3\ \text{m}$ from the axis clockwise.

1. a) Calculate the moment of each force, assigning a positive sign to the anticlockwise direction.
2. b) Calculate the resultant moment on the handwheel and state which direction it will rotate in.

To open the gate of an industrial warehouse, a moment of $24\ \text{N}\times\text{m}$ about its hinges is required, and the handle is at $0.8\ \text{m}$ from the axis. Use $\sen 30^\circ = 0.5$.

1. a) Calculate the minimum force required if pulling perpendicularly to the gate.
2. b) Calculate the required force if pulling at an angle of $30^\circ$ to the gate (the moment is then $M = F \times d \times \sen \alpha$).
3. c) Reason what would happen if pulling parallel to the gate and conclude what factor, in addition to the magnitude of the force and the distance, determines the moment.

Two equal and opposite forces act on a car's steering wheel, one at each end of a diameter: the sum of the forces is zero, yet the steering wheel turns.

1. a) Explain why the condition $\sum \vec{F} = 0$ is not sufficient to guarantee the equilibrium of a rigid body.
2. b) State the two conditions that a rigid body must satisfy to be in complete equilibrium.

A uniform walkway of $5\ \text{m}$ and $80\ \text{kg}$ is horizontally supported on two supports at its ends. A person of $60\ \text{kg}$ is standing on the walkway, at $1\ \text{m}$ from the left-hand end, as shown in the figure.

Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the weight of the footbridge and that of the person, and state at which point each acts.
2. b) Applying the condition of moment equilibrium about the left support, calculate the force exerted by the right support.
3. c) Applying $\sum \vec{F} = 0$, calculate the force exerted by the left support.
4. d) Check the result by verifying the condition of moments about the right support.

In a garden wheelbarrow loaded with soil:

1. a) Identify where the fulcrum, effort and load are located.
2. b) State which element lies between the other two and deduce which class of lever it is.

A builder's wheelbarrow carries a load of $480\ \text{N}$ located at $0.5\ \text{m}$ from the wheel axle, and the operator holds the handles at $1.5\ \text{m}$ from the axle, as shown in the figure.

1. a) Calculate the vertical force that the operator must exert to maintain the wheelbarrow in equilibrium.
2. b) Compare this force with the weight of the load and explain what mechanical advantage this lever offers.

Unit challenge. Design and calibrate a homemade steelyard balance to determine the unknown mass of a key ring by applying the principles of statics.

1. Data collection: Rest a $30\text{ cm}$ ruler on a pencil that acts as a fulcrum. Hang the key ring at five different positions, measuring each distance to the fulcrum ($d \pm 0.1\text{ cm}$). Balance the ruler in each case by placing groups of €2 coins (official mass according to the Central Bank of Ireland: $8.50\text{ g}$) and measuring their distance to the fulcrum.

2. Graphical processing: Construct a table with your measurements and uncertainties. Plot the torque of the force exerted by the coins versus the distance $d$ and obtain, from the slope of the line of best fit, the experimental value of the key ring's mass.

3. Vector analysis: Draw the free-body diagram identifying all concurrent and non-concurrent vector forces (weights and the normal reaction force of the support). Prove quantitatively that at equilibrium $\sum \vec{F} = \vec{0}$ and $\sum \vec{M} = \vec{0}$ are simultaneously satisfied.

4. Report: Write a reasoned proposal comparing your result with the mass obtained using a kitchen scale, justifying any discrepancies.

The following will be assessed:

• Rigour in taking experimental measurements, expressing uncertainties, and graphical fitting.

• Correct vector formulation and resolution of the two equilibrium conditions for a rigid body.

• Critical capacity when comparing theoretical and experimental results.

1. a) Where is the centre of gravity of a uniform ruler approximately located? Describe how you could verify this by balancing it on a finger.
2. b) Explain what happens to an object resting on the ground when the vertical line through its centre of gravity falls outside its base of support, and give an everyday example.

1. a) Explain in your own words what the centre of gravity of a body is.
2. b) A delivery van travels with several heavy packages tied to the roof rack. Explain, using the moment of the weight and the position of the centre of gravity, why it is less stable when taking bends or driving on side slopes than if the load were on the floor of the cabin.
3. c) In a rugby maul, players bend their legs and spread their feet apart. Interpret this posture in terms of centre of gravity and base of support.

A decorative mobile, like those by Calder, has a horizontal bar of $60\ \text{cm}$ with a figure of $100\ \text{g}$ hanging from the left end and another of $150\ \text{g}$ from the right end. Take $g \approx 10\ \text{m/s}^2$ and ignore the weight of the bar.

1. a) Calculate the weight of each figure.
2. b) Set up the condition of moment equilibrium and calculate at which point on the bar the string supporting it from the ceiling must be tied so that it remains horizontal.
3. c) State where the centre of gravity of the system is relative to the attachment point and justify why the mobile hangs stably.