The mass of an object represents the amount of material in it. It is measured in kilograms, and is a scalar. The mass results in inertia, the degree in which a body opposes the change in velocity. Mass does not change.
Weight is a force measured in newtons. It is the force that is the result of gravity acting on a force. Like all forces, weight is a vector, which always acts vertically downwards. Weight depends on the acceleration due to gravity.
Weight (N) = Mass (kg) × acceleration due to gravity (m s-2)
W = mg
The acceleration due to gravity on Earth is 9.81 m s-2. It is also expressed as force per unit mass, 9.81 N kg-1. On the Moon the acceleration due to gravity is 1.6 m s-2.
If we have hinged or pivoted body, any force applied changes the rotation of that body about the pivot. The turning effect is called a moment.
The equation is:
= force × perpendicular distance
In Physics Code:
This strange looking symbol,
G, which looks like a gallows,
is “Gamma”, a Greek capital letter ‘G’.
Moments have a direction. As they are turning effects, we can talk of clockwise and anti-clockwise moments. By convention, clockwise is positive.
The spanner in the picture is 30 cm long and the nut in question has to be tightened to a torque (moment) of 85 N m. What force must the fitter apply?
Consider a trap door held by a piece of string, BC. P and Q are forces. The trap door is hinged about point O.
The perpendicular distance of the line of action of force Q is the length of the line OC.
Moment of P
about O = P × OA
Moment of Q
about O = Q × OC
Note that the Moment of Q about O
is [Q ×OC]. This is because OC is
the perpendicular distance of the
force Q from the hinge O.
If the trap door remains in
moment (Q × OC) = clockwise
moment (P ×OA)
This is the
Principle of Moments.
Since OC = OB sin q we can say that Q × (OB sin q) = P × OA
The trap door in the diagram has a mass of 12 kg. The centre of mass is at A. The weight is force P. What is the value of force P?
The angle q is 25 degrees. The trap door is 100 cm wide. What is the value of force Q?
(Use g = 9.8 m s-2)
Use the quantities in the diagram above to calculate the angle q.
Principle of Moments
The principle of moments is an important rule that says:
• The sum of clockwise moments about a pivot… (1 mark)
• …is equal to the sum of anti-clockwise moments… (1 mark)
• …for a system in equilibrium. (1 mark)
When asked to state the principle of moments, you must include these three points. A pivot is sometimes called a fulcrum.
Consider a beam of negligible mass, on which three forces in equilibrium are acting. The forces are arranged as in the diagram:
F1 is x m from the pivot;
F2 is y m from the pivot;
F3 is z m from the pivot.
Using the principle of moments we can write:
Clockwise moments = anticlockwise moments
Centre of Mass
We treat objects as point masses referring to a single point called the centre
In regular objects like a cube or a sphere, the centre of mass is in the middle. In some objects the centre of mass is outside the object.
of mass (or
centre of gravity) is the point through which the entire weight is said to act.
Objects with a very low centre of gravity tend to be very stable.
Some objects are so stable that they never fall over. Objects with a high centre of gravity are unstable.
Many questions involve the balancing of see-saw around pivots. Let us look at some situations:
This is the simplest case. The pivot is in the middle of a uniform bar. It means that the object is totally regular and the centre of mass is in the middle. Therefore we can ignore the mass of the bar. If the bar is balanced, we can say that:
anticlockwise moments = clockwise moments
Ax = By
Let us look at a case where we move the pivot P to one side. The centre of mass stays where it is, and is NOT above the pivot. We now have to take it into account. Mass is not a force, but weight is. So we need the weight, W.
W = mg
In this case, the line of action of the weight, W, is z metres from the pivot P. Applying the principle of moments we can say:
Ax = By + Wz
Some children are playing on a see-saw as shown in the diagram. The see-saw is a plank of wood 3.0 m long, with a pivot exactly in the middle. The centre of mass is directly above the pivot, so we can ignore it.
(a) What are the weights of the children?
(b) Child B is sitting 0.4 m from the pivot. Where should child C sit so that the see-saw remains level?
(c) Child C misses its footing and falls off the end. What will happen to the others?
Use g = 9.8 m s-2
If two forces act about a hinge in opposite directions, there is an obvious turning effect called a couple. The resulting linear force from a couple is zero.
The couple is given by the simple formula:
This strange looking symbol, G, is “gamma”, a Greek capital letter ‘G’. Couples are measured in Newton metres (Nm). The code t (tau) is also used.
The turning effect is often called the torque. It is a common measurement made on motors and engines, alongside the power. Racing engines may be quite powerful but not have a large amount of torque. This is why it would not make sense for a racing car to be hitched to a caravan, any more so than trying to win a Formula 1 race in a 4 x 4.
Centre of Mass and Stability