6 - Forces in Equilibrium (6.3 The principle of moments (Turning effects…
6 - Forces in Equilibrium
6.1 Vectors and Scalars
Representing a vector
An example of a vector is
, due to it having a magnitude and direction, which is the direct distance from the starting position
A vector is any physical quantity that has a direction as well as a magnitude
A scalar is any physical quantity that is not directional
Examples of scalars include distance, as it takes no account of direction, mass, density, volume and energy
Any vector can be presented as an arrow where the length of the arrow represents the magnitude. The direction of arrow gives direction
is speed in a given direction,
Force and acceleration
are both vector quantities
Addition of vectors using a scale diagram
Any two vectors can be added together using a scale diagram
6.2 Balanced forces
Equilibrium of a point object
When two forces act on a point object
, the object is in equilibrium only if the two forces are equal and opposite to each other. The resultant is zero and forces are
When three forces act on a point object
, their combined effect is zero is two of the forces are equal and opposite to the third. Check combined effect by resolving each force on the same parallel and perp lines and balancing the components along each line
Testing three forces in equilibrium
The three forces F1, F2 and F3 acting on P are in equilibrium so any two should give a resultant equal and opposite to the third force.
6.3 The principle of moments
Whenever you use a lever or a spanner, you are using a force to turn an object about a pivot
The moment of a force about any point is defined as the force x the perpendicular distance from the line of action of the force to the point
. Unit of the moment of a force is the newton metre(Nm)
The moment of the force = F x d
The principle of moments
the sum of the clockwise moments = the sum of the anticlockwise moments
An object that is not a point object is refereed to as a body. Any such objects turns if a force is applied to it anywhere other than the centre of mass. If there is more one force and it is in equilibrium, the turning effect must balance out
Centre of mass
The centre of mass of a body is the point through which a single force on the body has no turning effect
Balance a ruler at its centre on the end of your finger. The centre of mass of the ruler is directly above the point of support.
To find centre of mass of a triangular card, suspend the piece of card on a clamp stand. Draw pencil lines along the plumb line. Centre of mass is where the lines drawn cross.
Calculating the weight of a metre rule
1- locate the centre of mass of a metre rule by balancing it on a knife-edge. Not the position of the centre of mass. The rule is
if the centre of mass is exactly in the middle.
2- Balance the metre rule off-centre on a knife-edge, using a known weight. The position of the known weight needs to be adjusted gradually until the rule is exactly horizontal.
6.4 More on moments
When an object in equilibrium is supported by only one point, the only support force on the object is equal and opposite to the total downward force
S=W1+w2+W0, where W0 is the weight of the rule
A uniform beam supported on two pillars X and Y, which are at distance D apart. The weight of the beam is shared between the two pillars according to how far the beam's centre of mass is from each pillar
If the centre of mass is midway between pillars the weight is equally shared
If the centre of mass is at a different distance from pillar X to pillar Y then take moments about X and Y
is a pair of equal and opposite forces acting on a body, but not along the same line
The moment of a couple = force x perpendicular distance between the lines of action of the forces
Stable n unstable equilibrium
If a body in
is displaced then released, it returns to its equilibrium. For example, a coat hanger hanging on a support will return to its original position if slightly displaced
It returns to the same position as the centre of mass of the object is directly below the point of support when the object is at rest. The weight acts as its centre of mass which means the support force and weight are directly equal and opposite to each other when the object is in equilibrium. When it is displaced, at the instant of release, the line of action of the weight no longer passes through the point of support, so the weight returns the object to equilibrium
A plank balanced on a drum is in
. If it is displaced slightly it will not return to equilibrium and roll off the drum. This is because the centre of mass is no longer above the point of support. Weight therefore turns the plank further from equilibrium position
Where an object at rest on a surface is acted on by a force that raises it up on one side.
If horizontal force is applied to the top of a free standing bookcase, the force can make it tilt.
A tilted object will topple over if it is tilted to far due to the line of action of its weight passes closer and closer to the pivot
If the line action of its weight passes beyond the pivot, the object will topple over if allowed to. If the line of action passes the pivot it will topple over.
On a slope
A tall object on a slope will topple over if the line of action of the weight lies outside the pivot point. Such as a vehicle on a road.
6.6 Equilibrium rules
The triangle of forces
For a point object acted on by 3 forces to be in equilibrium must mean the resultant force is 0. Therefore
their vector sum F1+f2+F3=0
. This shows any sum of two forces is equal and opposite to the third force.
In equilibrium this applies but their lines of action must intersect at the same point otherwise the object cannot be in equilibrium as the forces will have a net turning effect.
For example for a rectangular block on a rough slope there is weight, support and friction which can create a triangle
The sine rule can be used to find angles or forces
Conditions for equilibrium
The resultant force must be zero
The principle of motion must apply