Exercises: Rigid bodies 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.

Using the formula for moment $M = F \times d$, explain two everyday observations: why a doorknob is placed as far as possible from the hinges and why a longer spanner allows tighter nuts to be loosened with the same force.

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.

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).

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.

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.

1. a) Where is the center 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 center of gravity falls outside its base of support, and give an everyday example.

On a see-saw, a girl of $40\ \text{kg}$ sits at $1.5\ \text{m}$ from the pivot axis and another girl of $30\ \text{kg}$ sits at $2\ \text{m}$ from the pivot, on the opposite side, as shown in the figure.

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

1. a) Calculate the moment of each girl's weight about the pivot, stating the direction of rotation each produces.
2. b) Verify that the sum of moments is zero and deduce whether the see-saw is in equilibrium.

On a see-saw, a boy of $45\ \text{kg}$ sits at $1.6\ \text{m}$ from the pivot. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the moment of his weight about the pivot.
2. b) Calculate at what distance from the pivot a boy of $40\ \text{kg}$ must sit on the other side so that the see-saw is balanced.

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.

A nutcracker exerts a load force of $150\ \text{N}$ on a nut, applied at $0.03\ \text{m}$ from the fulcrum, and the hand squeezes at $0.12\ \text{m}$ from the fulcrum.

1. a) Calculate the moment exerted by the nut about the fulcrum, in $\text{N}\times\text{m}$.
2. b) Calculate the minimum force the hand must exert to crack the nut.

To loosen a wheel nut, a moment of $120\ \text{N}\times\text{m}$ is required.

1. a) Calculate the force that must be exerted using a spanner of length $0.3\ \text{m}$, pushing perpendicularly.
2. b) Calculate the required force if the spanner is extended with a pipe to $0.6\ \text{m}$.
3. c) In light of the results, explain why mechanics use extension handles.

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 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.

A uniform bar of $3\ \text{m}$ is used as a see-saw: a person of $60\ \text{kg}$ sits at the left-hand end and a person of $20\ \text{kg}$ sits at the right-hand end, as shown in the figure.

Take $g \approx 10\ \text{m/s}^2$ and ignore the weight of the bar.

1. a) Set up the equilibrium condition of moments about the pivot point, letting $x$ be the distance from this point to the left-hand end.
2. b) Solve the equation and calculate where the pivot must be placed.
3. c) Explain why the pivot point ends up closer to the heavier person.

A fisherman holds his fishing rod horizontally: the fish pulls at the end with a force of $25\ \text{N}$ at $2\ \text{m}$ from the hand acting as the fulcrum, and the other hand pushes upwards at $0.4\ \text{m}$ from the fulcrum.

1. a) Calculate the force that the hand must exert to keep the rod in equilibrium.
2. b) Compare this force with the force exerted by the fish, identify which class of lever it is and explain what is gained in exchange for exerting a larger force.

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.

A seesaw has its pivot at its center. On the left side sits a child of $30\ \text{kg}$ at $2\ \text{m}$ from the pivot; on the right side there is a child of $20\ \text{kg}$ at $2\ \text{m}$ from the pivot and a third child at $1\ \text{m}$ from the pivot. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the total moment produced by each side.
2. b) Calculate the mass of the third child so that the seesaw is in equilibrium.
3. c) Suggest another combination of mass and distance for the third child that also gives equilibrium, and explain what all solutions have in common.

A fishmonger's steelyard balance has the goods hook at $5\ \text{cm}$ from the fulcrum and uses a sliding weight of $0.5\ \text{kg}$, as shown in the diagram.

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

1. a) Calculate the weight of a fish of $2\ \text{kg}$ and that of the sliding weight.
2. b) Calculate at what distance from the fulcrum the weight must be placed to balance that fish.
3. c) Calculate the mark corresponding to a fish of $3.5\ \text{kg}$ and deduce, by comparing the parts, why the marks on the steelyard balance are equally spaced.

1. a) Explain in your own words what the center 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 center of gravity, why it is less stable when taking curves 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 center of gravity and base of support.

A person holds a dumbbell of $4\ \text{kg}$ with their forearm horizontal. The dumbbell is at $32\ \text{cm}$ from the elbow joint and the biceps pulls the forearm at $4\ \text{cm}$ from the elbow, as shown in the diagram.

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

1. a) Calculate the moment of the weight of the dumbbell about the elbow.
2. b) Calculate the force that the biceps must exert to maintain equilibrium.
3. c) Compare that force with the weight of the dumbbell, explain why the muscle exerts much more force than the supported weight, and what is gained in return with this lever.

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.

A construction tower crane lifts a load of $2000\ \text{kg}$ at $20\ \text{m}$ from the mast, and carries a counterweight of $5000\ \text{kg}$ on the other side of the mast, as shown in the diagram.

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

1. a) Calculate the moment of the load about the mast.
2. b) Calculate at what distance from the mast the counterweight must be placed to balance that moment.
3. c) If the load were doubled, calculate the new required distance for the counterweight and explain what would happen if it could not move far enough away.

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 center of gravity of the system is relative to the attachment point and justify why the mobile hangs stably.