We have seen that we can represent the forces acting on a particle by lines with arrows, the lengths of which represented the relative magnitude of the forces. Such diagrams are a useful way to represent the forces acting on a body or particle and are called free-body diagrams.
Figure 225 shows an object of mass M that is suspended vertically by a thread of negligible mass. It is then pulled to one side by a force of magnitude F and held in the position shown.
C
ORE
Mass = M
F
Figure 225 Forces acting on an object
The free-body diagram for the forces acting on the object in Figure 225 is shown in Figure 226
F
Weight = Mg
T
A
B
Figure 226 The Free-body Force Diagram for Fig. 225
The weight of the object is Mg and the magnitude of the tension in the thread is T.
The object is in equilibrium (see 2.2.6) and so the net force acting on it is zero. This means that the vertical components of the forces must be zero as must the horizontal components. Therefore, in the diagram the line A is equal in length to the arrow representing the force F and the line B is equal in length to the arrow representing the weight. The tension T, the “resultant” of A and B, is found by the using the dotted line constructions.
When producing a free-body diagram, there is no need to show these constructions. However, as well as being in the appropriate directions, the lengths of the arrows representing the forces should be approximately proportional to the magnitudes of the forces. The following example is left as an exercise for you.
Exercise 2.2 (c)
Draw a free-body force diagram of the forces acting on an aircraft which is flying horizontally with constant velocity.
2.2.3 DETERMINETHE
RESULTANT
FORCE
INDIFFERENT
SITUATIONS
Example
Two forces act on particle P as shown in the Figure below (N stands for ‘newton’ and is the SI unit of force as we shall see in the next section.)
Determine the magnitude of the net force acting in the horizontal direction and the magnitude of the net force acting in the vertical direction and hence determine the resultant force acting on P.
4.0 N
6.0 N
30°
P
The component of the 4.0 N in the horizontal direction is 4.0 cos 30 = 3.5 N.
Hence the magnitude of the force in the horizontal direction is 2.5 N.
2.0 N
2.5 N
R θ
The component of the 4.0 N force in the vertical direction is 4.0 sin 30 = 2.0 N and this is the magnitude of the force in the vertical direction.
The Figure above shows the vector addition of the horizontal and vertical components.
The resultant R has a magnitude = 3.2 N
C
ORE
2.2.4 State Newton’s first law of motion.
2.2.5 Describe examples of Newton’s first law.
2.2.6 State the condition for translational equilibrium.
2.2.7 Solve problems involving translational equilibrium.
© IBO 2007
2.2.4 STATE NEWTON’SFIRST
LAWOF
MOTION
At the beginning of this chapter we stated that the general mechanics problem is, that given certain initial conditions of a system, to predict the future behaviour of the system. The method that Newton devised to solve this problem is encompassed in his celebrated three laws of motion which he published in his Principia Mathematica circa 1660. Essentially Newton tells us to find out the forces acting on the system. If we know these then we should in principle able to predict the future behaviour of the system.
Newton’s First Law is essentially qualitative and is based on the work done by Galileo. Prior to Galileo’s work on mechanics, the Aristotelian understanding of motion was the accepted view, that a constant force is needed to produce constant motion. This seems to fit in with every day experience; if you stop pushing something then it will stop moving, to keep it moving you have to keep pushing it. Galileo’s brilliance was to recognise that the opposite is actually the case and his idea is summarised in the statement of Newton’s First Law: every object continues
in a state of rest of uniform motion in a straight line unless acted upon by an external force.
The Aristotelian view does not take into account that when you push something another force is usually acting on the body that you are pushing, namely the force of friction. In some situations, as we have seen, the frictional force, acting on a moving object is a function of the object’s velocity and in fact increases with velocity. Hence a greater engine power is required to move a car at high speed than at low speed. We shall return to this idea later.
2.2.5 DESCRIBE
EXAMPLES
OF
NEWTON’S
FIRST
LAW
If you eliminate friction and give an object a momentary push, it will continue moving in the direction of the push with constant velocity until it is acted upon by another force. This can be demonstrated to a certain degree using the linear air-track. It is to a limited degree since it is impossible to eliminate friction completely and the air track is not infinite in length. It is in this sense impossible to prove the first law with absolute certainty since sooner rather than later all objects will encounter a force of some kind or another.
In fact Arthur Eddington (1882- 1944), is reputed to have quoted Newton’s first law thus:
‘Every object continues in a state of rest or uniform motion in a straight line in so far as it doesn’t.’
By this he meant that nothing in the Universe is ever at rest and there is no such thing as straight line motion. However, we are inclined to believe that if a body is not acted upon by a force, then Galileo’s description of its motion is correct.
It is sometimes difficult to discard the Aristotelian view of motion particularly in respect of objects which are subject to a momentary force. Consider the example shown in Figure 229: gravity forward thrust ball gravity ball 1. 2.
Figure 229 Aristotelian and Galilean forces on a ball
A girl throws a ball towards another girl standing some metres away from her, it is tempting to think that, as Aristotle did, there must be a forward thrust to keep the ball moving through the air as shown in 1. However, if air resistance is neglected, the only force acting on the ball is gravity as shown in 2.
C
ORE
2.2.6, 2.2.7
STATE
THECONDITION
FOR
TRANSLATIONALEQUILIBRIUM
There are essentially two types of equilibrium, static and dynamic. If an object is at rest, then it is in static equilibrium and, if it is moving with constant velocity, then it is in dynamic equilibrium. It follows from Newton’s first law that the condition for both equilibriums is that the net force acting on the object is zero. We can express that mathematically as Σ F = 0. That is the vector sum of the forces acting on the object is zero.