VECTOR MECHANICS FOR ENGINEERS W/CON >B
VECTOR MECHANICS FOR ENGINEERS W/CON >B
12th Edition
ISBN: 9781260804638
Author: BEER
Publisher: MCG
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Chapter 2.2, Problem 2.22P

Determine the x and y components of each of die forces shown.

Chapter 2.2, Problem 2.22P, Determine the x and y components of each of die forces shown. Fig. P2.22

Fig. P2.22

Expert Solution & Answer
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To determine

To determine the x and y components of the forces shown in figure P2.22.

Answer to Problem 2.22P

The x component of the 800N force is +640N_ and the y component of 800N force is +480N_.

The x component of the 424N force is 224N_ and the y component of 424N force is 360N_.

The x component of the 408N force is +192N_ and the y component of 408N force is 360N_

Explanation of Solution

Refer figure P2.22.

The simplified diagram of the figure P2.22 is shown in figure 1 below. Figure 1 shows all the three components of force and the corresponding position vectors and their magnitudes. Position vectors OA,OBandOC represents the resultant force components along X, Y and Z axis.

VECTOR MECHANICS FOR ENGINEERS W/CON >B, Chapter 2.2, Problem 2.22P

Let OA be the position vector corresponding to 800N force, let OB be the position vector corresponding to 424N and let OC be the position vector corresponding to 408N.

Write the equation to find the magnitude of position vector OA.

|OA|=x2+y2                                                                                                      …...(I)

Here, |OA| is the magnitude of position vector OA, x is the x component of position vector OA, and y is the y component of position vector OA.

Write the equation to find the magnitude of position vector OB.

|OB|=x2+y2 (II)

Here, |OB| is the magnitude of position vector OB, x is the x component of position vector OB, and y is the y component of position vector OB.

Write the equation to find the magnitude of position vector OC.

|OC|=x2+y2 (III)

Here, |OC| is the magnitude of position vector OC, x is the x component of position vector OC, and y is the y component of position vector OC.

Let F be the resultant force acting on the body. The force vector can be resolved into x and y components by taking the cos and sin of the resultant force.

Write the equation to find the x component of the 800N force.

Fx=F800cosa (IV)

Here, Fx is the x component of the 800N force, and F800 is the magnitude of the resultant force.

cosa is the cosine of the x and y components of 800N force which is equal to adjacentsidehypotenuse

Fx=F800cos[adjacentsidehypotenuse] (V)

Here, hypotenuse is otherwise the resultant force.

Similarly write the equation to find the y component of 800N force.

Fy=F800sina (VI)

Here, Fy is the y component of the 800N force, F800 is the resultant force, and sina is the sine of the x and y components of position vectors of 800N force which is equal to adjacentsidehypotenuse

Rewrite the equation of y component of 800N force.

Fy=F800sin[adjacentsidehypotenuse] (VII)

Here, hypotenuse is otherwise the resultant force.

Write the equation to find the x component of the 424N force.

Fx=F424cosa (VIII)

Here, Fx is the x component of the 424N force, F424 is the magnitude of resultant force, and

cosa is the cosine of the x and y components of 424N force which is equal to adjacentsidehypotenuse.

Fx=F424cos[adjacentsidehypotenuse] (IX)

Here, hypotenuse is otherwise the resultant force.

Similarly write the equation to find the y component of 424N force.

Fy=F424sina (X)

Here, Fy is the y component of the 424N force, F424 is the magnitude of resultant force, and sina is the sine of the x and y components of position vectors of 424N force which is equal to adjacentsidehypotenuse.

Fy=F424sin[adjacentsidehypotenuse] (XI)

Here, hypotenuse is otherwise the resultant force.

Write the equation to find the x component of the 408N force.

Fx=F408cosa (XII)

Here, Fx is the x component of the 408N force, F408 is the magnitude of resultant force, and

cosa is the cosine of the x and y components of 408N force which is equal to adjacentsidehypotenuse

Fx=F408cos[adjacentsidehypotenuse] (XIII)

Here, hypotenuse is otherwise the resultant force.

Similarly write the equation to find the y component of 408N force.

Fy=F408sina

Here, Fy is the y component of the 408N force, F408 is the magnitude of resultant force, and sina is the sine of the x and y components of position vectors of 408N force which is equal to adjacentsidehypotenuse.

Fy=F408sin[adjacentsidehypotenuse] (XIV)

Here, hypotenuse is otherwise the resultant magnitude of the force.

Conclusion:

Substitute 800mm for x and 600mm for y in the equation (I) to get |OA|.

|OA|=(800mm)2+(600mm)2=1000mm

Substitute 560mm for x and 900mm for y in the equation (II) to get |OB|.

|OB|=(900mm)2+(560mm)2=1060mm

Substitute 480mm for x and 900mm for y in the equation (III) to get |OC|.

|OA|=(900mm)2+(480mm)2=1020mm

Substitute 800N for Fx and 800mm for adjacentside, and 1000mm for hypotenuse in the equation (V) to get Fx.

Fx=800Ncos[800mm1000mm]=+640N

Substitute 800N for Fy and 600mm for adjacentside, and 1000mm for hypotenuse in the equation (VII) to get Fy.

Fx=800Ncos[600mm1000mm]=+480N

Substitute 424N for F424 and 560mm for adjacentside, and 1060mm for hypotenuse in the equation (IX) to get Fx.

Fx=(424N)cos[560mm1060mm]=224N

Substitute 424N for Fy and 900mm for adjacentside, and 1060mm for hypotenuse in the equation (XI) to get Fy.

Fx=424Ncos[900mm1060mm]=360N

Substitute 408N for F408 and 480mm for adjacentside, and 1020mm for hypotenuse in the equation (XIII) to get Fx.

Fx=(408N)cos[480mm1020mm]=192N

Substitute 408N for Fy and 900mm for adjacentside, and 1020mm for hypotenuse in the equation (XIV) to get Fy.

Fy=(408N)cos[900mm1020mm]=360N

Therefore, The x component of the 800N force is +640N_ and the y component of 800N force is +480N_.

The x component of the 424N force is 224N_ and the y component of 424N force is 360N_.

The x component of the 408N force is +192N_ and the y component of 408N force is 360N_

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Chapter 2 Solutions

VECTOR MECHANICS FOR ENGINEERS W/CON >B

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