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# example problems of resultant force

From Wikipedia

Net force

In mechanics, the net force (also known as resultant force) is the overall force acting on an object when all the individual forces acting on the object are added together.

## Informal introduction

A force in mechanics is a concept that has both a size and a direction. The net force acting on an object is the sum of the forces onto the object, taking into account both their sizes, and their directions. For example, if an object has two forces acting on it, with equal sizes but in opposite directions, then the net force will be zero (technically: a null vector). If instead the forces are of equal size and in the same direction, then the net force is equal to twice either force. The net force can be seen as a hypothetical force that, acting on an object, has the same effect as all the actual forces combined.

## Definition

The net force Fnet = F1 + F2 + â€¦ is a vector produced when two or more forces { F1, F2, â€¦ } act upon a single object. It is calculated by vector addition of the force vectors acting upon the object.

## Examples

When force A and force B act on an object in the same direction (parallel vectors), the net force (C) is equal to A + B, and points in the same direction as A and B.

When force A and force B act on an object in opposite directions (180 degrees between then - anti-parallel vectors), the net force (C) is equal to |A - B|, and points in the direction of whichever one has greater absolute value ("greater magnitude").

(Note: The illustration assumes that the object, in this case a square, has no center of mass and can be treated like apoint.)

When the angle between the forces is anything else, then the net force can be visualized using the parallelogram rule.

For example, see Figure 3. This construction has the same result as moving F2 so its tail coincides with the head of F1, and taking the net force as the vector joining the tail of F1 to the head of F2. This procedure can be repeated to add F3 to the resultant F1 + F2, and so forth. Figure 4 is an example.

Fictitious force

A fictitious force, also called a pseudo force, d'Alembert force or inertial force, is an apparent force that acts on all masses in a non-inertial frame of reference, such as a rotating reference frame.

The force F does not arise from any physical interaction but rather from the acceleration a of the non-inertial reference frame itself. As stated by Iro:

According to Newton's second law in the form F = ma, fictitious forces always are proportional to the mass m acted upon.

Four fictitious forces are defined in accelerated frames: one caused by any relative acceleration of the origin in a straight line (rectilinear acceleration), two caused by any rotation (centrifugal force and Coriolis force) and a fourth, called the Euler force, caused by a variable rate of rotation, should that occur.

## Background

The role of fictitious forces in Newtonian mechanics is described by Tonnelat:

## Fictitious forces on Earth

The surface of the Earth is a rotating reference frame. To solve classical mechanics problems exactly in an Earth-bound reference frame, three fictitious forces must be introduced, the Coriolis force, the centrifugal force (described below) and the Euler force. The Euler force is typically ignored because its magnitude is very small. Both of the other fictitious forces are weak compared to most typical forces in everyday life, but they can be detected under careful conditions. For example, LÃ©on Foucault was able to show the Coriolis force that results from the Earth's rotation using the Foucault pendulum. If the Earth were to rotate a thousand times faster (making each day only ~86 seconds long), people could easily get the impression that such fictitious forces are pulling on them, as on a spinning carousel.

## Detection of non-inertial reference frame

Observers inside a closed box that is moving with a constant velocity cannot detect their own motion; however, observers within an accelerating reference frame can detect that they are in a non-inertial reference frame from the fictitious forces that arise. For example, for straight-line acceleration:

Other accelerations also give rise to fictitious forces, as described mathematically below. The physical explanation of motions in an inertial frames is the simplest possible, requiring no fictitious forces: fictitious forces are zero, providing a means to distinguish inertial frames from others.

An example of the detection of a non-inertial, rotating reference frame is the precession of a Foucault pendulum. In the non-inertial frame of the Earth, the fictitious Coriolis force is necessary to explain observations. In an inertial frame outside the Earth, no such fictitious force is necessary.

## Examples of fictitious forces

### Acceleration in a straight line

Figure 1 (top) shows an accelerating car. When a car accelerates hard, the common human response is to feel "pushed back into the seat." In an inertial frame of reference attached to the road, there is no physical force moving the rider backward. However, in the rider's non-inertial reference frame attached to the accelerating car, there is a backward fictitious force. We mention two possible ways of analyzing the problem:

1. Figure 1, (center panel). From the viewpoint of an inertial reference frame with constant velocity matching the initial motion of the car, the car is accelerating. In order for the passenger to stay inside the car, a force must be exerted on the passenger. This force is exerted by the seat, which has started to move forward with the car and is compressed against the passenger until it transmits the full force to keep the passenger moving with the car. Thus, the passenger is accelerating in this frame due to the unbalanced force of the seat.
2. Figure 1, (bottom panel). From the point of view of the interior of the car, an accelerating reference frame, there is a fictitious force pushing the passenger backwards, with magnitude equal to the mass of the passenger times the acceleration of the car. This force pushes the passenger back into the seat, until the seat compresses and provides an equal and opposite force. Thereafter, the passenger is stationary in this frame, because the fictitious force and the (real) force of the seat are balanced.

How can the accelerating frame be discovered to be non-inertial? In the accelerating frame, everything appears to be subject to zero net force, and nothing moves. Nonetheless, compression of the seat is observed and is explained in the accelerating frame (and in an inertial frame) because the seat is subject to the force of acceleration from the car on one side, and the opposing force of reaction to acceleration by the passenger on the other. Identification of the accelerating frame as non-inertial cannot be based simply on the compression of the seat, which all observers can explain; rather it is based on the simplicity of the physical explanation for this compression.

The explanation of the seat compression in the accelerating frame requires not only the thrust from the axle of the car, but additional (fictitious) forces. In an inertial frame, only the thrust from the axle is necessary. Therefore, the inertial frame has a simpler physical explanation (not necessarily a simpler mathematical formulation, however), indicating the accelerating frame is a non-inertial frame of reference. In other words, in the inertial frame, fictitious forces are zero. See inertial frame for more detail.

This example illustrates how fictitious forces arise from switching from an inertial to a non-inertial reference frame. Calculations of physical quantities (compression of the seat, required force from the axle) made in any frame give the same ans

Algorithm examples

## An example: Algorithm specification of addition m+n

Choice of machine model:

There is no â€œbestâ€�, or â€œpreferredâ€� model. The Turing machine, while considered the standard, is notoriously awkward to use. And different problems seem to require different models to study them. Many researchers have observed these problems, for example:

â€œThe principal purpose of this paper is to offer a theory which is closely related to Turing's but is more economical in the basic operationsâ€� (Wang (1954) p. 63)
â€œCertain features of Turing machines have induced later workers to propose alternative devices as embodiments of what is to be meant by effective computability.... a Turing machine has a certain opacity, its workings are known rather than seen. Further a Turing machine is inflexible ... a Turing machine is slow in (hypothetical) operation and, usually complicated. This makes it rather hard to design it, and even harder to investigate such matters as time or storage optimization or a comparison between efficiency of two algorithms.â€� (Melzak (1961) p. 281)
Shepherdson-Sturgis (1963) proposed their register-machine model because â€œthese proofs [using Turing machines] are complicated and tedious to follow for two reasons: (1) A Turing machine has only one head... (2) It has only one tape....â€� They were in search of â€œa form of idealized computer which is sufficiently flexible for one to be able to convert an intuitive computational procedure into a program for such a machineâ€� (p. 218).
â€œI would prefer something along the lines of the random access computers of Angluin and Valiant [as opposed to the pointer machine of SchÃ¶nhage]â€� (Gurivich 1988 p. 6)
â€œShowing that a function is Turing computable directly...is rather laborious ... we introduce an ostensibly more flexible kind of idealized machine, an abacus machine...â€� (Boolos-Burgess-Jeffrey 2002 p.45).

About all that one can insist upon is that the algorithm-writer specify in exacting detail (i) the machine model to be used and (ii) its instruction set.

Atomization of the instruction set:

The Turing machine model is primitive, but not as primitive as it can be. As noted in the above quotes this is a source of concern when studying complexity and equivalence of algorithms. Although the observations quoted below concern the Random access machine model â€“ a Turing-machine equivalent â€“ the problem remains for any Turing-equivalent model:

â€œ...there hardly exists such a thing as an â€˜innocentâ€™ extension of the standard RAM model in the uniform time measure; either one only has additive arithmetic, or one might as well include all multiplicative and/or bitwise Boolean instructions on small operands....â€� (van Emde Boas (1992) p. 26)
â€œSince, however, the computational power of a RAM model seems to depend rather sensitively on the scope of its instruction set, we nevertheless will have to go into detail...
â€œOne important principle will be to admit only such instructions which can be said to be of an atomistic nature. We will describe two versions of the so-called successor RAM, with the successor function as the only arithmetic operation....the RAM0 version deserves special attention for its extreme simplicity; its instruction set consists of only a few one letter codes, without any (explicit) addressing.â€� (SchÃ¶nhage (1980) p.494)

Example #1: The most general (and original) Turing machine â€“ single-tape with left-end, multi-symbols, 5-tuple instruction format â€“ can be atomized into the Turing machine of Boolos-Burgess-Jeffrey (2002) â€“ single-tape with no ends, two "symbols" { B, | } (where B symbolizes "blank square" and | symbolizes "marked square"), and a 4-tuple instruction format. This model in turn can be further atomized into a Post-Turing machineâ€“ single-tape with no ends, two symbols { B, | }, and a 0- and 1-parameter instruction set ( e.g. { Left, Right, Mark, Erase, Jump-if-marked to instruction xxx, Jump-if-blank to instruction xxx, Halt } ).

Example #2: The RASP can be reduced to a RAM by moving its instructions off the tape and (perhaps with translation) into its finite-state machine â€œtableâ€� of instructions, the RAM stripped of its indirect instruction and reduced to a 2- and 3-operand â€œabacusâ€� register machine; the abacus in turn can be reduced to the 1- and 2-operand Minsky (1967)/Shepherdson-Sturgis (1963) counter machine, which can be further atomized into the 0- and 1-operand instructions of SchÃ¶nhage (and even a 0-operand SchÃ¶nhage-like instruction set is possible).

Cost of atomization:

Atomization comes at a (usually severe) cost: while the resulting instructions may be â€œsimplerâ€�, atomization (usually) creates more instructions and the need for more computational steps. As shown in the following example the increase in computation steps may be significant (i.e. orders of magnitude â€“ the following example is â€œtameâ€�), and atomization may (but not always, as in the case of the Post-Turing model) reduce the usability and readability of â€œthe machine codeâ€�. For more see Turing tarpit.

Example: The single register machine instruction "INC 3" â€“ increment the contents of register #3, i.e. increase its count by 1 â€“ can be atomized into the 0-parameter instruction set of SchÃ¶nhage, but with the equivalent number of steps to accomplish the task increasing to 7; this number is directly related to the register number â€œnâ€� i.e. 4+n):

More examples can be found at the pages Register machine and Random access machine where the addition of "convenience instructions" CLR h and COPY h1,h1 are shown to reduce the number of steps dramatically. Indirect addressing is the other significant example.

## Precise specification of Turing-machine algorithm m+n

As described in Algorithm characterizations per the specifications of Boolos-Burgess-Jeffrey (2002) and Sipser (2006), and with a nod to the other characterizations we proceed to specify:

(i) Number format: unary strings of marked squares (a "marked square" signfied by the symbol 1) separated by single blanks (signified by the symbol B) e.g. â€œ2,3â€� = B11B111B
(ii) Machine type: Turing machine: single-tape left-ended or no-ended, 2-symbol { B, 1 }, 4-tuple instruction format.
(iii) Head location: See more at â€œImplementation Descriptionâ€� below. A symbolic representation of the head's location in the tape's symbol string will put the current state to the right of the scanned symbol. Blank squares may be included in this protocol. The state's number will appear with brackets around it, or sub-scripted. The head is shown as

Contact force

In physics, a contact force is a force between two objects (or an object and a surface) that are in contact with each other. This is distinct from a non-contact force, or action-at-a-distance force (a force that acts over a distance), such as gravity or magnetic attraction/repulsion. Contact forces always exist in pairs of equal in magnitude but opposite in direction forces, by Newton's third law.

A contact force can be split into two components. The part of the force that lies within the plane of contact is friction, which must be overcome for the two objects to slide relative to one another along that plane. The part of the force that is perpendicular to the plane of contact is called the normal force. Friction is proportional to the normal contact force, and the constant of proportionality is denoted by the term Î¼ (mu).

Strictly speaking, contact forces are only a useful simplification for introductory physics classes and other applications of classical mechanics. Everyday objects on Earth do not actually touch each other; rather contact forces are the result of the interactions of the electrons at or near the surfaces of the objects (exchange interaction).

## Example

An example of contact force commonly encountered in college-level physics is the force between two masses A and B which are lying next to each other and a force F is being applied on one of the masses, for example A. In such a case, the contact force will be proportional to the mass of B.

Hence, we can see the many examples of contact forces in everyday life. Contact forces can act through a rigid connector or a non rigid connector .

For example when a boy pulls a cart through a rope he is connecting the force applied through a non rigid connector(the rope) He could also pull the cart through the handle of the cart hence transferring the force through the rigid connector (the handle)

But from case A the boy cannot push the cart(disadvantage of non rigid connector )

From this investigation we can prove that: a rigid connector can push or pull but a non rigid connector can only pull .

Question:The answer is 180 degrees but when I solved the problem I got 0 degrees so I don't understand what I did wrong. Can someone please explain this question?

Answers:Since a 12N force and a 7N force = 5N, the forces are going in opposite directions but in the same line. 12 N ----- 7 N <------- -------> = 5 N <------- The degree between them is 180 (a straight line). I haven't taken a physics class in 2 years but i think this is how you would get the answer.

Question:i need help with this problem please A coordinate system (in meters) is constructed on the surface of a pool table, and three objects are placed on the table as follows: a 5.0 kg object at the origin of the coordinate system, a 11.0 kg object at (0, 2.0), and a 12.0 kg object at (4.0, 0). Find the resultant gravitational force exerted by the other two objects on the object at the origin. i need to know the magnitude and direction of the resultant force..thanks

Answers:Use F = Gm1*m1/D First in the x direction, then in the y direction, giving Fx = ___ and Fy = ___ Add the 2 force vectors and you will have the answer. Fr = (Fx +Fy ) = arctan(Fy/Fx)

Question:I am a high school student and we are being taught gravitation at school. The gravitational force between two bodies is inversely proportional to the square of the distances between them. So that means if the distance between two bodies is decreasing, their force of gravity should increase (forget about 'by how much' it is increasing). Therefore, same should happen when a object is in free fall. The distance is constantly decreasing, hence the force of gravity should be gradually increasing. Now the textbook says, that an object falling under the influence of gravity and no external force is called free fall. It also says that in a free fall there is uniform acceleration. This means that the force of gravity between the object and the earth remains constant since there can be constant acceleration only when there is a constant unbalanced force acting on a body. But it cant be constant if distance is decreasing? Help!

Answers:The acceleration due to gravity actually DOES CHANGE with height. In such scenario, the distance is measured from the cenre of the earth. Radius of the earth is 6400km. So from centre to the surface, average distance is 6400km. For most practical purposes, the distance travelled by a freefalling object is too small as compared to the actual distance from the centre. Theerefore the change in gravitation is neglegible. For example, an object at 100m from surface whould be 6400km + 100m, ie 6400100m from the centre of earth, and if it's 50m, is 6400050m from the centre. And if you square that amount, the reduction in gravity is too small to cause significant error for the problems encountered in High School physics. If you study higher physics, then you'll also sudy the relativity model of gravity, which also takes into account time dilation and bending of space-time. Such accurate calculations are used in satellites, aircraf and spacecraft.

Question:I am postulating that inertial force created by the accelerated expansion of the universe creates what we observe as gravitation waves. If I could somehow relate the gravitational constant of the universe to the inertial force resulting from the accelerated expansion of the universe that could go a long way to proving or disproving my conjecture. The problem is I don t have nearly the level of mathematics needed to solve this question. Can you help me? I am hypothesizing that inertial force created by the accelerated expansion of the universe creates what we observe as gravitation waves. If I could somehow relate the gravitational constant of the universe to the inertial force resulting from the accelerated expansion of the universe that could go a long way to proving or disproving my conjecture. The problem is I don t have nearly the level of mathematics needed to solve this question. Can you help me? Please no silly answers mocking my grammer or lack of an advanced degree in Physics. That just bores everyone.

Answers:Some recent measurments have indicated that the rate of expansion of the universe is increasing, and that is where the asker got the concept of an accelerating universe. The force that causes this expansion is believed to be a result of the "cosmological constant" added to Einstein's equations of General Relativity. That results in gravity being counteracted by a repulsion force at very large distances. That repulsion force is the inertial force that causes the acceleration of the universe. I believe the value of the cosmological constant has not yet been determined.