Forces in Equilibrium - kerboodle Flashcards
6.1 Vectors and scalars
Representing a vector
- 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
- Some examples of vector quantities are displacement, which is the distance given a direction, Velocity, which is the speed given direction, and Force and acceleration
- Vectors are represented as an arrow, the length of the arrow represents the magnitude while the direction gives the direction of the vector
- Some scalar quantities are speed, distance, mass, but any quantity which does not have a direction is scalar
Addition of vectors using a scale diagram
- adding two vectors which are in opposite directions or even the same directions is easy, as you would just need to add or multiply them
- However the addition of vectors with directions which differ is much harder since you need to draw a triangle to calculate any missing angles/magnitudes
Resolving a vector into two perpendicular components
• This is the process of working out the components of a vector in two perpendicular directions from the magnitude and direction of the vector
• In general to resolve a vector into two perpendicular components, draw a diagram showing the two perpendicular directions and an arrow to represent the vector
• If the angle θ between the vector and one of the lines is known the component along that line = length of vector * cosθ and the component perpendicular to that line = length of vector * sinθ
• Thus a force F can be resolved into two perpendicular components:
- Fcosθ parallel to a line at angle θ to the line of action of the force and
- Fsinθ perpendicular to the line
6.2 Balanced forces
Equilibrium of a point object
- When two forces act on a point object the object is in equilibrium provided that the two forces are equal and opposite to each other
- When two forces are in equilibrium, it is said the two forces are balanced, and the net force acting on an object is zero
- When three forces act on a point object, their combined effect is zero if the resultant of two of the forces is equal and opposite to the third force
- To check if the combined effect of the three forces is zero resolve each force along the same parallel and perpendicular lines and balance the components along each line
6.3 The principle of moments
Turning effect
- Whenever you see a lever or a spanner, you are using a force to turn an object around a pivot.
- The effect of the force depends on how far it is applied from the wheel axle, the longer the spanner, the less force is needed to loosen the nut.
- However if the tension of the object to which force is being applied is too high, and the pivot is too long, the pivot cold snap
• The moment of a force about any point is defined as the force multiplied by the perpendicular distance from the line of action of the force to the point.
- The unit of the moment of a force is the newton metre (Nm)
- For a force F acting along a line of action at a perpendicular distance d from a certain point, the moment of the force = F * d
Notes
• The greater the distance d, the greater the moment
• The distance d is the perpendicular distance from the line of action of the force to the point.
The principle of moments
- An object that is not a point object is referred to as a body
- Any such object turns if a force is applied to it anywhere other than through its centre of mass
- If there is more than one force acting on a body and it is in equilibrium, the turning effects of the forces must balance out.
- In short, considering the moments of the forces about any point, for equilibrium the sum of the clockwise moments = the sum of the anticlockwise moments
- The above state is known as the principle of moments
Note
• if a third weight is suspended from the rule of the same side as the second weight at a different distance from the pivot, then the rule cna be rebalanced by increasing the first distance, this leads to the equation for the principle of moments looking like:
w1d1 = w2d2 + w3d3
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
Centre of mass tests
1. Balance a ruler at its centre on the end of your finger, the centre of mass of the rule is directly above the point of support. Tip the ruler too much and it falls off because the centre of mass is no longer above the point of support
2. To find the centre of mass of a triangular card, suspend the piece of card on a clamp stand. Draw pencil lines along the plumb line. The centre of mass is where the lines drawn on the card cross
Calculating the weight of a metre wall
- Locate the centre of mass of a metre rule
- Balance the metre rule off centre
- add a known weight to the short end of the metre rule, the weight of the known mass can be used as well as the principle of moments to calculate the weight of the metre rule when they are in equilibrium
6.4 More on moments
Support forces
Single support problems
• When an object in equilibrium is supported at one point only, the support force n the object is equal and opposite to the total downwards force
• The support force S on the rule balancing on a knife-edge must be equal to the total downward weight, Therefore.
S = w1 + w2 + wr where wr is the weight of the rule
Two support problems
- similar to one support problem, in two support problems you need to factor in the position of the support about the centre of mass to work out how much weight they support.
- If the centre of mass of the beam is midway between the two pillars, the weight of the beam is shared equally between the two supports
- If the centre of mass is at distance dx from pillar x and distance dy from pillar y, you would need to use the equation Sy * D = W * dx to get the equation Sy = (w * dx)/D to get the moments about where X is in contact with the beam
- To get where Y is in contact with beam you would instead use Sx = (W * dy)/D
Couples
- A couple is a pair of equal and opposite forces acting on a body, but not along the same line
- The moment of a couple = force * perpendicular distance between the lines of action of the forces
6.5 Stability
Stable and unstable equilibrium
- If a body in stable equilibrium is displaced and then released, it returns to its equilibrium position.
- the difference between being in stable equilibrium and unstable equilibrium is that, in stable equilibrium, the support force is directly below the centre of mass, while in unstable equilibrium, if the centre of mass is displaced slightly, it can no longer regain equilibrium.
Tilting and Toppling
- Tilting is where an object at rest on a surface is acted on by a force that raises it on one side
- Toppling is where a tilted object will fall because it is tilted too far, if an object on a flat surface is tilted more and more, to the point where the centre of mass is past where the pivot is, it will topple.
On a slope
- A tall object on a slope will topple if the angle of the slope is too great
- This will happen if the line of action of the wight lies outside the wheelbase of the vehicle.
6.6 Equilibrium Rules
Free body force diagrams
- When two objects interact, they always exert equal and opposite force on one another
- A diagram can be used to demonstrate the forces acting on an object, this type of diagram is called a free body force diagram
