Two forces 4N and 10N act on a 3 kg block on a
frictionless floor as shown. Find the magnitude of the
acceleration of the block towards the direction pointing
right.
Answers
Answer:
(a) A grand piano is being lifted to a second-story apartment. (b) Arrows are used to represent all forces:
\[\overset{\to }{T}\]
is the tension in the rope above the piano,
\[{\overset{\to }{F}}_{\text{T}}\]
is the force that the piano exerts on the rope, and
\[\overset{\to }{w}\]
is the weight of the piano. All other forces, such as the nudge of a breeze, are assumed to be negligible. (c) Suppose we are given the piano’s mass and asked to find the tension in the rope. We then define the system of interest as shown and draw a free-body diagram. Now
\[{\overset{\to }{F}}_{\text{T}}\]
is no longer shown, because it is not a force acting on the system of interest; rather,
\[{\overset{\to }{F}}_{\text{T}}\]
acts on the outside world. (d) Showing only the arrows, the head-to-tail method of addition is used. It is apparent that if the piano is stationary,
\[\overset{\to }{T}=\text{−}\overset{\to }{w}\]
.
As with most problems, we next need to identify what needs to be determined and what is known or can be inferred from the problem as stated, that is, make a list of knowns and unknowns. It is particularly crucial to identify the system of interest, since Newton’s second law involves only external forces. We can then determine which forces are external and which are internal, a necessary step to employ Newton’s second law. (See (Figure)(c).) Newton’s third law may be used to identify whether forces are exerted between components of a system (internal) or between the system and something outside (external). As illustrated in Newton’s Laws of Motion, the system of interest depends on the question we need to answer. Only forces are shown in free-body diagrams, not acceleration or velocity. We have drawn several free-body diagrams in previous worked examples. (Figure)(c) shows a free-body diagram for the system of interest. Note that no internal forces are shown in a free-body diagram.
Once a free-body diagram is drawn, we apply Newton’s second law. This is done in (Figure)(d) for a particular situation. In general, once external forces are clearly identified in free-body diagrams, it should be a straightforward task to put them into equation form and solve for the unknown, as done in all previous examples. If the problem is one-dimensional—that is, if all forces are parallel—then the forces can be handled algebraically. If the problem is two-dimensional, then it must be broken down into a pair of one-dimensional problems. We do this by projecting the force vectors onto a set of axes chosen for convenience. As seen in previous examples, the choice of axes can simplify the problem. For example, when an incline is involved, a set of axes with one axis parallel to the incline and one perpendicular to it is most convenient. It is almost always convenient to make one axis parallel to the direction of motion, if this is known. Generally, just write Newton’s second law in components along the different directions. Then, you have the following equations:
\[\sum {F}_{x}=m{a}_{x},\enspace\sum {F}_{y}=m{a}_{y}.\]
(If, for example, the system is accelerating horizontally, then you can then set
\[{a}_{y}=0.\]
) We need this information to determine unknown forces acting on a system.
As always, we must check the solution. In some cases, it is easy to tell whether the solution is reasonable. For example, it is reasonable to find that friction causes an object to slide down an incline more slowly than when no friction exists. In practice, intuition develops gradually through problem solving; with experience, it becomes progressively easier to judge whether an answer is reasonable. Another way to check a solution is to check the units. If we are solving for force and end up with units of millimeters per second, then we have made a mistake.
There are many interesting applications of Newton’s laws of motion, a few more of which are presented in this section. These serve also to illustrate some further subtleties of physics and to help build problem-solving skills. We look first at problems involving particle equilibrium, which make use of Newton’s first law, and then consider particle acceleration, which involves Newton’s second law.
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