Forces, Density and Pressure

Centre of gravity and the moment of a force

Every object is pulled towards the Earth by its weight, and that force really acts on every particle of the object. For calculation you do not treat the particles separately: the weight of an object may be taken as acting at a single point called its centre of gravity, the point at which the whole weight of the object may be taken to act. For a uniform object of regular shape that point is at the geometric centre, so the centre of gravity of a uniform metre rule is at the 50 cm mark. For an irregular lamina it is found by hanging the object freely from two different points and drawing a plumb line downwards from each; the two lines cross at the centre of gravity.

A force also turns things. Push a door near its handle and it swings open easily; push it with the same force close to the hinge and it hardly moves. The turning effect is the moment of a force, defined as the product of the force and the perpendicular distance from the pivot to the line of action of the force, so moment = F x d. The unit is the newton metre, N m. Do not write J for a moment: a moment is not a transfer of energy, even though the unit looks the same on paper.

The distance in that definition is the perpendicular distance, the shortest distance from the pivot to the line along which the force acts. If the force acts at an angle theta to the object, the perpendicular distance is d sin theta, where d is the distance from the pivot to the point of application. A worked example: a wheel nut is tightened with a spanner 0.30 m long, and a force of 250 N is applied at right angles to the spanner, so the moment is 250 x 0.30 = 75 N m. If the same 250 N is applied at 60 degrees to the spanner instead, the perpendicular distance falls to 0.30 x sin 60 degrees = 0.26 m and the moment falls to 250 x 0.26 = 65 N m. That is why a longer spanner, or a force applied at right angles, makes a stiff nut far easier to turn.

• Moment of a force = force x perpendicular distance from the pivot to the line of action of the force; unit N m.
• Centre of gravity: the single point at which the whole weight of the object may be taken to act.
• For a uniform metre rule the centre of gravity is at the 50 cm mark; for an irregular lamina, hang it from two points and draw plumb lines.
• A force at an angle theta to the spanner has a perpendicular distance d sin theta, so its moment is smaller than F x d.

Couples and the torque of a couple

A couple is a pair of forces that acts to produce rotation only. The two forces are equal in magnitude and opposite in direction, but their lines of action do not coincide, so they are not in line with each other. Because the two forces are equal and opposite, the resultant force is zero: a couple cannot accelerate the centre of gravity of the object, and can only turn it. Turning a steering wheel with both hands, or a wheel brace with one push and one pull, are everyday examples.

The turning effect of a couple is the torque of a couple, defined as the product of the magnitude of one of the forces and the perpendicular distance between the lines of action of the two forces, so torque = F x d, again in N m. The perpendicular distance between the lines of action is the separation of the two forces, not the distance of either force from some chosen pivot. A useful property follows from this: the torque of a couple is the same about any point you choose to take moments about, which is why a couple does not need a pivot to be identified.

A worked example. Both hands grip a steering wheel of diameter 0.34 m, each hand pushing with a force of 20 N in opposite directions. The two forces are equal and opposite and their lines of action are 0.34 m apart, so the torque is 20 x 0.34 = 6.8 N m.

Compare this with the single moment of a force. A single force turns the object and also produces a resultant force that acts at the pivot or the centre of gravity, so a system acted on by a single force cannot be in equilibrium unless something else pushes back. For a couple, the resultant force is already zero, so the only effect is rotation.

• A couple is a pair of equal and opposite forces whose lines of action do not coincide.
• A couple produces rotation only, because the resultant force is zero.
• Torque of a couple = F x d, where d is the perpendicular distance between the lines of action.
• The torque of a couple has the same value about every point, so no pivot has to be chosen.

The principle of moments and equilibrium

The principle of moments states that, for a body in equilibrium, the sum of the clockwise moments about any point equals the sum of the anticlockwise moments about the same point. Take moments about the pivot whenever you can, because the force at the pivot then has zero moment and drops out of the equation. If you take moments about some other point you must include the moment of the pivot force as well.

A system is in equilibrium when there is no resultant force and no resultant torque. Both conditions are needed. A beam balanced on two supports has no resultant force, but it would tip if a heavy load were placed at one end, because the anticlockwise moment would then exceed the clockwise moment. A body can also have no resultant torque and still accelerate, which is why the two conditions are stated separately.

A worked example. A uniform metre rule is pivoted at its centre, the 50.0 cm mark. A weight of 6.0 N hangs at the 20.0 cm mark, which is 0.30 m to the left of the pivot, giving an anticlockwise moment of 6.0 x 0.30 = 1.8 N m. A weight of 9.0 N on the right-hand side balances the rule when 9.0 x d = 1.8 N m, so d = 0.20 m and the weight must hang at the 70.0 cm mark. The support at the pivot supplies an upward force of 6.0 + 9.0 = 15 N, so the resultant force is zero as well.

If the rule is pivoted away from its centre, the weight of the rule matters too: a uniform metre rule of weight 1.6 N pivoted at the 30.0 cm mark behaves as if that 1.6 N hangs at the 50.0 cm mark, which is 0.20 m to the right of the pivot and so gives a clockwise moment of 1.6 x 0.20 = 0.32 N m. Forgetting the weight of the rule itself is one of the commonest lost marks in moment questions.

Coplanar forces in equilibrium and the vector triangle

Forces are vectors, so they are added by drawing them tip to tail. Two forces acting at a point are represented by two arrows drawn to scale; the resultant is the single arrow drawn from the tail of the first to the tip of the second. If you add the forces tip to tail and the last arrow finishes exactly where the first one started, the vector diagram closes and the resultant is zero. For three coplanar forces in equilibrium the closed shape is a triangle of forces.

A picture hanging from two strings is the standard case. Three coplanar forces act on the picture: the weight, acting vertically downwards, and the two tensions, each along its own string. The picture does not move, so the three forces must have zero resultant, and their lines of action must all pass through the centre of gravity of the picture. Drawn tip to tail, the three arrows form a closed triangle, and the same closed triangle works for any three coplanar forces in equilibrium.

A worked example. A ring is pulled by a horizontal force of 6.0 N and a vertical force of 8.0 N. The resultant of those two is found from Pythagoras' theorem: the square root of (6.0 squared + 8.0 squared) is the square root of 100, which is 10.0 N. For the ring to be in equilibrium a third force of 10.0 N must act in the opposite direction to this resultant, so that the three arrows close into a triangle.

You can always check equilibrium by resolving instead of drawing. Choose two perpendicular directions, then the sum of the components in the first direction must be zero and the sum of the components in the second direction must be zero. The vector triangle and the resolving method are two ways of writing the same condition, and the resolving method is faster when forces act at awkward angles.

• Add coplanar forces tip to tail to scale; a closed triangle means the resultant is zero.
• Three coplanar forces in equilibrium have lines of action that pass through a single point.
• 6.0 N and 8.0 N at right angles have a resultant of 10.0 N, so a third force of 10.0 N in the opposite direction gives equilibrium.
• Resolving into two perpendicular directions and setting each sum of components to zero is an equivalent check.

Density

Density is defined as the mass per unit volume, so rho = m / V. Mass is measured in kg and volume in m3, so the unit of density is kg m-3. Density is a property of the material, not of the particular block you happen to have: a large block of aluminium and a small cube of aluminium have exactly the same density, and a material can often be identified by measuring its density.

Typical values are worth remembering, because they let you judge whether an answer is sensible. Water has a density of 1.0 x 10^3 kg m-3, which means that one cubic metre of water has a mass of 1000 kg. Ice is slightly less dense than water, which is why ice floats; lead is more than eleven times as dense as water.

Measuring density. For a regularly shaped solid, measure the mass with a top-pan balance and the dimensions with vernier callipers or a micrometer, then work out the volume from the dimensions. For an irregular solid, such as a stone, lower it into a displacement can (a eureka can) filled to the spout and collect the water that overflows; the volume of water displaced equals the volume of the stone. For a liquid, pour it into a measuring cylinder, take the mass of the cylinder and liquid together, subtract the mass of the empty cylinder, and use the reading on the cylinder as the volume.

Watch the units. Densities are often quoted in g cm-3, and 1 g cm-3 = 1000 kg m-3, so a density of 2.7 g cm-3 is 2.7 x 10^3 kg m-3. Volumes often come in cm3, and 1 cm3 = 1 x 10-6 m3, so a volume of 32.0 cm3 must be written as 3.20 x 10-5 m3 before it is substituted into rho = m / V.

Pressure

Pressure is defined as the normal force per unit area, so p = F / A, where the force F acts at right angles to the area A. Force is in newtons and area in square metres, so the unit of pressure is the newton per square metre, N m-2, which is given the name pascal, Pa. The force must be perpendicular to the surface; a force acting along the surface is a shear force and produces no pressure.

Pressure explains why sharp things work. A stiletto heel transmits a force of 120 N through an area of only 3.0 x 10-4 m2, so the pressure is 120 / (3.0 x 10-4) = 4.0 x 10^5 Pa. A flat shoe transmitting the same 120 N through 1.2 x 10-2 m2 gives only 1.0 x 10^4 Pa, forty times less. The same reasoning gives you the sharp edge of a knife, the point of a nail, and studs on a football boot, all of which increase the pressure by reducing the area. Wide tyres, caterpillar tracks and snowshoes do the opposite, spreading the same weight over a larger area to reduce the pressure.

In a fluid, pressure acts in all directions at a point. A small sensor lowered into water can be turned in any direction at a given depth and reads the same value, and the force that the fluid exerts on a surface is always perpendicular to that surface. This is the idea that turns into upthrust: because the pressure is greater deeper down, a submerged object is pushed up more strongly on its underside than it is pushed down on its top.

• p = F / A, with F perpendicular to the area; the unit is Pa, which is the same as N m-2.
• 120 N on 3.0 x 10-4 m2 gives 4.0 x 10^5 Pa; the same force on 1.2 x 10-2 m2 gives 1.0 x 10^4 Pa.
• Small area means large pressure: knife edges, nail points, stiletto heels.
• Large area means small pressure: caterpillar tracks, snowshoes, wide tyres.
• Pressure in a fluid acts in all directions at a point and is perpendicular to any surface it touches.

Hydrostatic pressure and the upthrust it produces

Deriving delta-p = rho g delta-h. Take a column of fluid of density rho, base area A and vertical height delta-h. The volume of the column is A x delta-h, so its mass is density x volume = rho A delta-h. Its weight is mass x g = rho A delta-h g, and this weight presses on the base area A. Pressure is force per unit area, so the pressure difference caused by the column is weight / area = rho A delta-h g / A. The area A cancels, leaving delta-p = rho g delta-h. Check the units: kg m-3 x m s-2 x m gives kg m-1 s-2, which is N m-2, a pressure, so the equation is consistent.

The cancellation of the area is the important part of the result. The pressure difference between two depths in a fluid depends only on the density of the fluid and on the vertical separation of the two points, not on the shape or width of the container. Water 0.50 m deep in a wide tank and water 0.50 m deep in a narrow tube have the same pressure at the bottom: 1.0 x 10^3 x 9.81 x 0.50 = 4.9 x 10^3 Pa. The equation gives the difference in pressure; if you need the total pressure at a depth, add the pressure of the atmosphere at the surface.

Upthrust. A submerged object senses a pressure difference between its top and its bottom. The bottom face is deeper by the height h of the object, so the pressure there is greater by rho g h. The forces on the four vertical faces cancel in pairs, so the only unbalanced effect is upwards: the resultant force is the extra pressure multiplied by the area of a horizontal face, rho g h x A. But h x A is the volume V of the object, so the upthrust is F = rho g V, where rho is the density of the fluid and V is the volume of fluid displaced. This is Archimedes' principle: the upthrust on a body in a fluid equals the weight of the fluid it displaces.

Floating and sinking. An object floats in equilibrium when the upthrust equals its weight, so rho-fluid x g x V-submerged = m x g. A body floats higher in a dense liquid than in a less dense one, which is what a hydrometer measures and why a swimmer floats much higher in the very salty water of the Dead Sea. A submarine controls its depth by taking water into ballast tanks, which increases its weight without changing its volume, so the upthrust stays the same while the net force changes from upwards to downwards.