The mechanical work It is defined as the change in the energy state of a system, caused by the action of external forces such as gravity or friction. The units of mechanical work in the International System (SI) are newton x meter or joules, abbreviated by J.
Mathematically it is defined as the scalar product of the force vector and the displacement vector. Yes F is the constant force and l is the displacement, both vectors, the work W is expressed as: W = F ● l
When the force is not constant, then we must analyze the work done when the displacements are very small or differential. In this case, if point A is considered as the starting point and B as the arrival point, the total work is obtained by adding all the contributions to it. This is equivalent to calculating the following integral:
Variation in energy of the system = Work done by external forces
ΔE = Wext
When energy is added to the system, W> 0 and when energy is subtracted W<0. Ahora bien, si ΔE = 0, puede significar que:
-The system is isolated and there are no external forces acting on it.
-There are external forces, but they are not doing work on the system.
Since the change in energy equals the work done by external forces, the SI unit of energy is also the joule. This includes any type of energy: kinetic, potential, thermal, chemical, and more..
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We have already seen that work is defined as a dot product. Let's take the definition of work done by a constant force and apply the concept of the dot product between two vectors:
W = F ● l = F.l.cos θ
Where F is the magnitude of the force, l is the magnitude of the displacement and θ is the angle between the force and the displacement. In figure 2 there is an example of an inclined external force acting on a block (the system), which produces a horizontal displacement.
Rewriting the work as follows:
W = (F. cos θ). l
We can state that only the component of the force parallel to the displacement: F. cos θ es able to do work. If θ = 90º then cos θ = 0 and the work would be zero.
Therefore it is concluded that the forces perpendicular to the displacement do not do mechanical work.
In the case of figure 2, neither the normal force N nor the weight P do work, since both are perpendicular to the displacement l.
As explained above, W It can be positive or negative. When cos θ> 0, the work done by the force is positive, since it has the same direction of motion.
Yes cos θ = 1, force and displacement are parallel and work is maximum.
In case cos θ < 1, la fuerza no está a favor del movimiento y el trabajo es negativo.
When cos θ = -1, force is completely opposite to displacement, such as kinetic friction, whose effect is to slow down the object on which it acts. So the work is minimal.
This agrees with what was said at the beginning: if the work is positive, energy is being added to the system, and if it is negative, it is subtracting.
Net work Wnet It is defined as the sum of the works done by all the forces acting on the system:
Wnet = ∑Wi
Then we can conclude that to guarantee the existence of net mechanical work it is necessary that:
-External forces act on the object.
-These forces are not all perpendicular to the displacement (cos θ ≠ 0).
-The jobs done by each force do not cancel each other out.
-There is a displacement.
-Whenever it is required to put an object in motion from rest, it is necessary to do mechanical work. For example pushing a refrigerator or a heavy trunk on a horizontal surface.
-Another example of a situation where you need to do mechanical work is changing the speed of a moving ball..
-Work is required to raise an object to a certain height above the floor.
Now, there are equally common situations in which not work is done, even if appearances indicate otherwise. We have said that to raise an object to a certain height you have to do work, so we carry the object, raise it above our head and hold it there. Are we doing work?
Apparently yes, because if the object is heavy, the arms will get tired in a short time, however, no matter how hard it is, no work is being done from the point of view of Physics. Why not? Well because the object is not moving.
Another case in which, despite having an external force, it does not perform mechanical work is when the particle has a uniform circular motion.
For example, a child spinning a stone tied to a string. The string tension is the centripetal force that allows the stone to rotate. But at all times this force is perpendicular to the displacement. Then he does not perform mechanical work, although it favors movement.
The kinetic energy of the system is that which it possesses by virtue of its movement. Yes m is the mass and v is the speed of motion, the kinetic energy is denoted by K and is given by:
K = ½ mvtwo
By definition, the kinetic energy of an object cannot be negative, since both the mass and the square of the velocity are always positive quantities. The kinetic energy can be 0, when the object is at rest.
To change the kinetic energy of a system, it is necessary to vary its speed - we will consider that the mass remains constant, although this is not always the case. This requires doing net work on the system, therefore:
Wnet = ΔK
This is the work - kinetic energy theorem. It states that:
Net work equals the change in kinetic energy of the system
Note that although K is always positive, ΔK can be positive or negative, since:
ΔK = Kfinal - K initial
Yes Kfinal >K initial the system has gained energy and ΔK> 0. On the contrary, if Kfinal < K initial, the system has yielded power.
When a spring is stretched (or compressed), work must be done. This work is stored in the spring, allowing this in turn to do work on, say, a block that is attached to one of its ends..
Hooke's law states that the force exerted by a spring is a restoring force - it is contrary to displacement - and also proportional to said displacement. The constant of proportionality depends on how the spring is: soft and easily deformable or rigid.
This force is given by:
Fr = -kx
In the expression, Fr is the force, k is the spring constant and x is the displacement. The negative sign indicates that the force exerted by the spring opposes the displacement.
If the spring is compressed (to the left in the figure), the block at its end will move to the right. And when the spring is stretched (to the right) the block will want to move to the left.
To compress or stretch the spring, some external agent must do the work, and since it is a variable force, to calculate said work, we must use the definition given at the beginning:
It is very important to note that this is the work done by the external agent (a person's hand, for example) to compress or stretch the spring. That is why the negative sign does not appear. And since the positions are squared, it doesn't matter if they are compressions or stretches..
The work that the spring will in turn do on the block is:
Wspring = -Wext
The block in figure 4 has mass M = 2 kg and slides down the inclined plane without friction, with α = 36.9º. Assuming that it is allowed to slide from rest from the top of the plane, whose height is h = 3 m, find the speed with which the block reaches the base of the plane, using the work-kinetic energy theorem.
The free-body diagram shows that the only force capable of doing work on the block is weight. More accurate: the weight component along the x-axis.
The distance traveled by the block on the plane is calculated using trigonometry:
d = 3 / (cos 36.9º) m = 3.75 m
Wweight = (Mg). d. cos (90-α) = 2 x 9.8 x 3.75 x cos 53.1 º J = 44.1 J
By work-kinetic energy theorem:
Wnet = ΔK
Wnet = Wweight
ΔK = ½ MvFtwo- ½ Mvortwo
Since it is released from rest, vor = 0, Thus:
Wnet = ½ MvFtwo
A horizontal spring, whose constant is k = 750 N / m, is fixed at one end to a wall. A person compresses the other end a distance of 5 cm. Calculate: a) The force exerted by the person, b) The work he did to compress the spring.
a) The magnitude of the force applied by the person is:
F = kx = 750 N / m. 5 x 10 -two m = 37.5 N.
b) If the end of the spring is originally at x1 = 0, to take it from there to the final position xtwo = 5 cm, it is necessary to do the following work, according to the result obtained in the previous section:
Wext = ½ k (xtwotwo - x1two) = 0.5 x 750 x (0.05two -0two) J = 0.9375 J.
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