3.4.1.1 Scalars and vectors
Nature of scalars and vectors.
Examples should include:
velocity/speed, mass, force/weight, acceleration, displacement/distance.
Addition of vectors by calculation or scale drawing.
Calculations will be limited to two vectors at right angles.
Scale drawings may involve vectors at angles other than 90°.
Resolution of vectors into two components at right angles to each other.
Examples should include components of forces along and perpendicular to an inclined plane.
Problems may be solved either by the use of resolved forces or the use of a closed triangle.
Conditions for equilibrium for two or three coplanar forces acting at a point. Appreciation of the meaning of equilibrium in the context of an object at rest or moving with constant velocity.
Free-body Diagrams
When considering the forces acting on an object it is often very useful to draw what is called a free-body diagram. Free-body diagrams show two things:
• The size or magnitude of the force vector • The direction of the force vector
Free-body diagrams are schematic, this means that they are not lifelike drawings, but they merely represent a physical situation. The object is drawn as a box or a dot, and the force arrows are drawn from the edge of the object. The force arrows must be drawn to scale and be pointing in the correct direction. It is also useful to label the arrows so we know what is producing the force. (In class I will often draw the force from the point where it acts to help us visualise the situation better.)
If the free-body diagram has been drawn accurately it can be used to solve a number of problems, such as resolving forces into components, seeing whether a system is in equilibrium, finding the value of unknown forces and even identifying missing forces.
Forces in equilibrium
When two forces act on an object and the object is stationary or moving at constant velocity the forces are in equilibrium. If the forces do not cancel out but has a resultant the body will accelerate.
When we add forces in equilibrium the vector sum is zero.
This should be read as, “The sum of all the forces acting on the object equals zero.”
If the forces do not cancel out, there will be a resultant force acting on the body and it will accelerate.
The example on the left is a simple situation.
The block is not moving so it must be in equilibrium. The downward force, that of its weight is cancelled out by the support force provided by the floor.
When there are only two forces acting on a body they can only be in equilibrium if they are acting parallel to each other as they are in the example of above of the box on the floor. With situations when an object is moving and the forces are in equilibrium, such as a car travelling at a constant speed, the car’s engine will produce a driving force to overcome resistive forces such as air resistance, rolling resistance etc.
When there are three forces acting on an object the resultant can take any value from 0 (if the object is in equilibrium) to the absolute sum of the three forces if they are all acting in the same direction. If three forces act on an object they are in equilibrium only if the resultant of any two of the forces is equal and opposite to the third.
There are two ways to check whether the forces acting on an object are in equilibrium, and like we have previously seen when using vectors there is a graphical method and a mathematical method.
To determine graphically whether the object is in equilibrium we need to be able to draw a closed equilibrium triangle. This involves arranging the three forces without rotating them into a closed triangle. It is important that each of the vectors join from head to tail. If two arrowheads join then it is not an equilibrium situation. If it is not possible to draw the triangle then the forces are not in equilibrium.
In example iii above the three forces are parallel to each other so the clearly cannot be arranged into a triangle. Looking at the other two examples, as you can see below only example i can be arranged into an equilibrium triangle. Although example ii can be arranged into a closed triangle, there are two arrow heads meeting at the same corner so it cannot be in equilibrium.
The mathematical method involves resolving the forces into horizontal and vertical components. If the sum of the components of the forces equals 0 then the object is in equilibrium. This method has two steps:
- Resolve the forces along the same parallel and perpendicular axis.
- Balance the components along these lines.
In this example there are no angles given, but resolving vectors into components was explained in vectors section of this site.
A good example of three forces acting in equilibrium in an object resting on an inclined slope, as shown in the diagram below. This will be discussed fully in class, bit you should spend some time considering it beforehand.
- What do the three forces arrows represent?
- How can we tell the object is in equilibrium?
- Can you draw a free-body diagram for the object?
- If we knew the value of one of the forces, how could we calculate the size of the other two?
