4.1 Forces: Turning Effects & Equilibrium

In a double-decker bus there are two passenger decks, stacked one above the other.

The dimensions of an older, 'Routemaster' bus are:

Height = 4.38 m

Width = 2.44 m

Mass = 7.47 tons

To ensure safety, buses are tested for stability using a tilting platform as shown in Fig. 1.1. 

Weights are attached to the underside of the bus, then the angle the bus makes with the horizontal is gradually increased until the bus begins to topple to the left.

The bus is first tested completely empty, and is found to tilt to an angle of 40° from the horizontal before it starts to tip.

Determine the height above the ground of the centre of mass of the bus when it is empty and normally parked.

To be considered safe to use on the road the bus must be safe when carrying passengers.

To test this the top deck is weighted with iron blocks or sandbags to the equivalent of 60 passengers, while the lower deck is left empty.

The average passenger is assumed to have a mass of 65 kg and their centres of mass can be considered to act at a height of 3.28 m above the road.

1 ton = 1000 kg

To pass the safety test the bus must be able to tilt to 28° when loaded as in part (c).

In the 1st century CE Greek-Egyptian engineer Hero of Alexandria described a simple steam engine, which was at least 100 years old when he wrote about it.

Now commonly called a Hero's engine, the machine is a steam turbine which can be made to rotate by boiling water in a sealed vessel beneath it. Two pipes channel the steam produced through nozzles which project outwards.

Fig. 1.1. shows Hero's engine in operation. 

Fig. 1.1. 

Explain why the vessel rotates around its axis.

A version of Hero's engine can be made for demonstration purposes by puncturing the sides of an empty soda can, adding water then suspending it over a Bunsen burner, as shown in Fig. 1.2.

Fig. 1.2.

The holes are made by sticking a pin directly into the metal, then pulling the pin sideways to create an angled vent. This causes the jet of steam to leave the can tangentially. as shown in Fig. 1.3.

Fig. 1.3.

Explain why simply puncturing holes straight into the can will not produce the rotation seen in the original Hero's engine.

A student tries to make a Soda Can Hero's Engine with two pairs of steam holes rather than one.

The dimensions of the can are

height = 115 mm

diameter = 66 mm

Each of the four jets exerts a force of 0.60 N and the holes create jets of steam which are at 45° to the surface of the can.

Calculate the torque experienced by the can.

Without adding any further holes, discuss an adjustment that could be made to the can to make it spin more quickly.

A student investigates the principle of moments using the apparatus shown in Fig. 1.1.

Fig. 1.1

The scale on the forcemeter reads from 0 N to 10 N.

A 3.0 N weight is positioned so that it hangs straight downwards from the 70 cm mark on the ruler.

Describe what the student could do to ensure that the ruler is horizontal.

The student intends to investigate moments around the pivot.

The student positions the forcemeter at the 15 cm mark and uses it to hold the ruler horizontal.

He expects the reading on the forcemeter to be 14 N.

The actual reading is 10 N.

The teacher asks the student and his friend to move a table with some books on it. They carry the table as shown in Fig. 1.2.

Fig. 1.2

Use ideas about moments to explain why student A feels more force on their hands than student B.

A pole PR has a length 6.0 m and a weight W N. The pole is held in a horizontal position in equilibrium by two vertical ropes attached to the pole as shown in Fig. 1.1.

Fig. 1.1.

A load of weight 10 N is attached to the rod at R.

The tension in the rope attached at Q is six times as large as the tension in the rope attached at P. 

The pole is uniform and the ropes are light inextensible strings.

Calculate the value of W.

Q is now moved to a point further along the pole until the length QR is 0.8 m , as shown in Fig. 1.2.

Fig. 1.2

The pole remains in equilibrium. The load of 10 N is now moved to a point x metres from P.

The beam remains in equilibrium in a horizontal position.

The rope at Q will break if its tension exceeds 32 N. The rope at P cannot break.

Find the range of possible positions on the beam where the load can be attached without the rope at Q breaking.

State what is meant by the centre of gravity of an object.

A uniform plank of wood of mass 40 kg and length 5.0 m is used by a boy to help him cross a ditch.

In the ditch is a rock, which is used to support the plank horizontally 1.1 m from one end, as shown in Fig. 1.1. The other end of the plank is supported by the bank. 

Fig. 1.1

Calculate the vertical supporting force from the rock when the plank is placed in position as shown in Fig 1.1.

The boy begins to walk across the plank. The position of the boy is shown in Fig. 1.2.

Fig. 1.2

On Fig. 1.2, draw arrows to show the forces acting on the plank when the boy has crossed some of the plank. 

The boy has a mass of 50 kg.

Determine whether the boy can walk to the far end of the plank without it tipping. Support your answer with a calculation.

It is said that Archimedes used huge levers to sink Roman ships invading the city of Syracuse.

A possible system is shown in  Fig. 1.1 where a rope is hooked onto the front of the ship and the lever is pulled by several men. 

Fig. 1.1

Calculate the mass of the ship if its weight is 4.6 × 10⁴ N. Give your answer to an appropriate number of significant figures.

Calculate the moment of the ship’s weight about point P. State an appropriate unit for your answer.

Calculate the minimum vertical force T required to start raising the front of the ship. Assume the ship pivots about point P.

Calculate the minimum force F that must be exerted to start to raise the front of the ship.