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Inclined planes and levers are the two main groups of simple machines. Figures 129.1(a) and (b) illustrate a simple inclined plane and a wedge respectively.
(a) Explain why the length NQ is longer than the length NM.
(b) Write an expression for the effort that pushes the solid ball into the cart if the weight of the ball is W and the angle of inclination of the plane is θ:
(i) If friction is not at play.
(ii) If friction is taken into account.
(c) The wedge with sides EF = 1" and GH = 3" is used to split a log (figure 129.1(b). How wide does the force of the effort force the log apart?
(d) What is the mechanical advantage of the wedge?
(e) Write the PjProblemstrings of the work done in figures 129.1(a) and 129.1(b).
Pj Problem of Interest is of type force (push/pull).
(a) Let NQ = L and NM = l.
The work to be done is to lift solid ball into cart. This work is nonvariant.
How the work is done can vary. A very strong person may choose to exert a force F1 to lift the solid ball vertically via MN into the cart. A not too strong person with the help of a simple machine such as the inclined plane may exert a lesser force F2 to get the solid ball via QN into the cart.
So, work done = F1l = F2L .
F1/F2 = L/l > 1
So, L is longer than l.
In order to gain in effort one sacrifices some distance.
(b)i Effort = Wsinθ (neglecting friction)
(b)ii Effort = W(μcosθ + sinθ). Where μ is the coefficient of friction.
(c) The log will be forced apart a distance equal to the width of the broad end of the wedge. This width is 1" for the wedge illustrated in figure 129.1(b).
The wedge is useful in situations where other simple machines are inapplicable.
(d) Mechanical Advantage of wedge , M.A = 3/1 = 3.
(e)Consider figure 129.1(a), if work is done via MN, effort involves force-pull to lift solid ball and force-push to carry solid ball against the pull of gravity into cart in a linear direction.
Assuming a multi-matter-multi-dynamic space (S7) because of the dynamism of atoms of materials and the fact that there are several matter in the space.
So, the PjProblemStrings S7P3A31, S7P3A32 and S7P4A41 are implied.
If work is done via QN, the inclined plane, no lifting is required
So, only force-push is required to push solid ball into cart in a linear direction.
So, S7P3A32 and S7P4A41 are implied.
Static equilibrium of the plane NQ is required at N and Q.
So, the PjProblemString S7P7A71 is implied at N and Q.
Linearly directed Force-push resulting in linear motion, at play (effort and resistance) in figure 129.1(b).
So,the PjProblemStrings S7P3A32 and S7P4A41 are implied.
The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.
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