Engineering Mechanics Statics - Examples Part 1 pdf

Determine the magnitude of the resultant force and its direction measured counterclockwise from the positive x axis... Determine the magnitude of the resultant of these forces and its di

d

( ) km⋅μN = 1×10−3m N

km⋅μN = 1 mm N⋅

m

=

Mgmm

Ggm

10−6s

3Ns

s

=

mNμs

kNs

kNm

Represent each of the following to three significant figures and express each answer in SI units

using an appropriate prefix: (a) x, (b) y, and (c) z.

Convert: (a) S 1 to N m⋅ , (b) S 2 to kN/m3, (c) S 3 to mm/s Express the result to three

significant figures Use an appropriate prefix

What is the weight in newtons of an object that has a mass of: (a) m 1 , (b) m 2 , (c) m 3? Express

the result to three significant figures Use an appropriate prefix

( ) W = m 2 g

W = 4.90 mNc

( ) W = m 3 g

W = 44.1 kN

The specific weight (wt./vol.) of brass is ρ Determine its density (mass/vol.) in SI units.

Use an appropriate prefix

A concrete column has diameter d and length L If the density (mass/volume) of concrete is ρ,

determine the weight of the column in pounds

Problem 1-16

Two particles have masses m 1 and m 2 , respectively If they are a distance d apart, determine

the force of gravity acting between them Compare this result with the weight of each particle

Using the base units of the SI system, show that F = G(m 1 m 2 )/r2 is a dimensionally

homogeneous equation which gives F in newtons Compute the gravitational force acting

between two identical spheres that are touching each other The mass of each sphere is m 1, and

Evaluate each of the following to three significant figures and express each answer in SI units

using an appropriate prefix: (a) x, (b) y, (c) z.

Problem 1-19

Evaluate each of the following to three significant figures and express each answer in SI units

using an appropriate prefix: (a) a 1 /b 1 , (b) a 2 b 2 /c 2 , (c) a 3 b 3

=

c

( ) a 3 b 3 = 1.14 km kg⋅

Problem 1-20

Evaluate each of the following to three significant figures and express each answer in SI units

using an appropriate prefix: (a) a 1 /b 12 (b) a 22b 23

Units Used:

Mm = 106 m

Problem 2-1

Determine the magnitude of the resultant force FR = F 1 + F 2 and its direction, measured

counterclockwise from the positive x axis.

Determine the magnitude of the resultant force and its direction measured counterclockwise

from the positive x axis.

Given:

F 1 = 80 lb

F 2 = 60 lb

Determine the magnitude of the resultant force F R = F 1 + F 2 and its direction, measured

counterclockwise from the positive x axis

Angle measured ccw from x axis

360 deg−φ +β =353 deg

Problem 2-4

Determine the magnitude of the resultant force F R = F 1 + F 2 and its direction, measured

counterclockwise from the positive u axis.

Resolve the force F 1 into components acting along the u and v axes and determine the

magnitudes of the components

Given:

F 1 = 300 N α = 30 deg

Resolve the force F 2 into components acting along the u and v axes and determine the

magnitudes of the components

Determine the magnitude of the resultant force F R = F 1 + F 2 and its direction measured

counterclockwise from the positive u axis.

Resolve the force F 2 into components acting along

the u and v axes and determine the components.

Given:

F 1 = 25 lb

F 2 = 50 lb

θ1 = 30 deg

The component of force F acting along line aa is required to be F a Determine the magnitude

of F and its component along line bb.

A resultant force F is necessary to hold the ballon in place Resolve this force into components

along the tether lines AB and AC, and compute the magnitude of each component.

The post is to be pulled out of the ground using two ropes A and B Rope A is subjected to force

F 1 and is directed at angle θ1 from the horizontal If the resultant force acting on the post is to be

F R , vertically upward, determine the force T in rope B and the corresponding angle θ

Determine the magnitude and direction of the resultant force F R Express the result in terms of

the magnitudes of the components F 1 and F 2 and the angle φ

Solution:

F R2 = F 12 +F 22 −2F 1 F 2 cos 180 deg( −φ)

Since cos 180 deg( −φ) =−cos( )φ ,

If the tension in the cable is F 1, determine the magnitude and direction of the resultant force acting

on the pulley This angle defines the same angle θ of line AB on the tailboard block.

Problem 2-19

The riveted bracket supports two forces Determine the angle θ so that the resultant force

is directed along the negative x axis What is the magnitude of this resultant force?

The plate is subjected to the forces acting on members A and B as shown Determine the magnitude

of the resultant of these forces and its direction measured clockwise from the positive x axis.

Given:

F A = 400 lb

F B = 500 lb

Determine the angle θ for connecting member B to the plate so that the resultant of FA and F B

is directed along the positive x axis What is the magnitude of the resultant force?

Given:

F A = 400 lb

F B = 500 lb

θ1 = 30 deg

Determine the magnitude and direction of the resultant F R = F 1 + F 2 + F 3 of the three forces

by first finding the resultant F' = F 1 + F 2 and then forming F R = F' + F 3

Determine the magnitude and direction of the resultant F R = F 1 + F 2 + F 3 of the three forces by

first finding the resultant F' = F 2 + F 3 and then forming F R = F' + F 1

Given:

F 1 = 30 N

F 2 = 20 N

sin( )β = sin 90 deg( F' −θ)

β asin F 2 sin 90 deg( −θ)

F y' = 40.8 lb

Problem 2-25

The boat is to be pulled onto the shore using two ropes Determine the magnitudes of forces

T and P acting in each rope in order to develop a resultant force F 1, directed along the keel

Problem 2-26

The boat is to be pulled onto the shore using two ropes If the resultant force is to be F 1, directed

along the keel aa as shown, determine the magnitudes of forces T and P acting in each rope and

the angle θ of P so that the magnitude of P is a minimum T acts at θ from the keel as shown

Problem 2-27

The beam is to be hoisted using two chains Determine the magnitudes of forces F A and F B

acting on each chain in order to develop a resultant force T directed along the positive y axis.

=

F B = 311 N

The beam is to be hoisted using two chains If the resultant force is to be F, directed along the

positive y axis, determine the magnitudes of forces FA and F B acting on each chain and the

orientation θ of FB so that the magnitude of F B is a minimum.

Three chains act on the bracket such that they create a resultant force having magnitude F R If

two of the chains are subjected to known forces, as shown, determine the orientation θ of the

third chain,measured clockwise from the positive x axis, so that the magnitude of force F in this

chain is a minimum All forces lie in the x-y plane.What is the magnitude of F? Hint: First find

the resultant of the two known forces Force F acts in this direction.

Problem 2-28

Three cables pull on the pipe such that they create a resultant force having magnitude F R If two

of the cables are subjected to known forces, as shown in the figure, determine the direction θ of

the third cable so that the magnitude of force F in this cable is a minimum All forces lie in the

Determine the magnitude of the resultant force and its direction, measured clockwise from

the positive x axis

Determine the magnitude of the resultant force and its direction measured counterclockwise

from the positive x axis.

θ = 360 deg−θ'

θ = 347 deg

Problem 2-34

Determine the magnitude of the resultant force and its direction , measured counterclockwise

from the positive x axis

Three forces act on the bracket Determine the magnitude and direction θ of F 1 so that the

resultant force is directed along the positive x' axis and has a magnitude of FR

→ F Rx = SF x; F Rcos( )β = F 3+F 2cos( )α +F 1cos(θ β+ )

+↑ F Ry = SF y; −F Rsin( )β =F 2sin( )α −F 1sin(θ β+ )

F 1cos(θ β+ ) = F Rcos( )β −F 3−F 2cos( )α

F 1sin(θ β+ ) =F 2sin( )α +F Rsin( )β

θ atan F 2sin( )α +F Rsin( )β

=

F 1 = 889 N

Problem 2-36

Determine the magnitude and direction, measured counterclockwise from the x' axis, of the

resultant force of the three forces acting on the bracket

Determine the magnitude of the resultant force and its direction, measured counterclockwise

from the positive x axis.

Determine the magnitude of the resultant force and its direction, measured counterclockwise

from the positive x axis.

Problem 2-39

Determine the magnitude of the resultant force and its direction measured counterclockwise

from the positive x axis.

Problem 2-40

Determine the magnitude of the resultant force FR = F 1 + F 2 and its direction, measured

counterclockwise from the positive x axis by summing the rectangular or x, y components

of the forces to obtain the resultant force

Determine the magnitude and direction of the resultant F R = F 1 + F 2 + F 3 of the three forces

by summing the rectangular or x, y components of the forces to obtain the resultant force.

Given:

F 1 = 30 N

F 2 = 20 N

Determine the magnitude and orientation, measured counterclockwise from the positive y axis, of the

resultant force acting on the bracket

Given:

F A = 700 N

The directional angle θ measured counterclockwise

from the positive x axis is

Determine the magnitude and direction, measured counterclockwise from the positive x'

axis, of the resultant force of the three forces acting on the bracket

Given:

F 1 = 300 N

Determine the magnitude of the resultant force and its direction, measured counterclockwise

from the positive x axis.

Determine the magnitude of the resultant force and its direction measured clockwise from

the positive x axis.

Three forces act on the bracket Determine the magnitude and direction θ of F 1 so that the

resultant force is directed along the positive x' axis and has magnitude FR

Determine the magnitude and direction, measured counterclockwise from the positive x'

axis, of the resultant force acting on the bracket

=

Express each of the three forces acting on the column in Cartesian vector form and compute

the magnitude of the resultant force

Determine the magnitude of force F so that the resultant F R of the three forces is as small

as possible What is the minimum magnitude of F R?

The magnitude of the resultant force F R is

F R = F Rx2+F Ry2 (F 1−F sin( )θ )2

F cos( )θ −F 2

( )2+

=

Problem 2-52

Express each of the three forces acting on the bracket in Cartesian vector form with respect

to the x and y axes Determine the magnitude and direction θ of F 1 so that the resultant force

is directed along the positive x' axis and has magnitude F R

The initial guesses: F 1 = 20 N θ = 10 deg

The three concurrent forces acting on the post produce a resultant force F R = 0 If F 2 = (1/2)F 1,

and F 1 is to be 90° from F 2 as shown, determine the required magnitude F 3 expressed in terms

of F 1 and the angle θ

Solution: Use the primed coordiates

Problem 2-54

Three forces act on the bracket Determine the magnitude and orientation θ of F2 so that the

resultant force is directed along the positive u axis and has magnitude F R

Problem 2-55

Determine the magnitude and orientation, measured clockwise from the positive x axis, of

the resultant force of the three forces acting on the bracket

Three forces act on the ring Determine the range of values for the magnitude of P so that the

magnitude of the resultant force does not exceed F Force P is always directed to the right.

Since P > 0 we conclude that 0 <= P <= P max =1.6 kN

−0

=

The stock S mounted on the lathe is subjected to a

force F, which is caused by the die D Determine the

coordinate direction angle β and express the force as

cos2( )α +cos2( )β +cos2( )γ =1

β = acos 1( −cos( )α 2−cos( )γ 2) β = 90 deg

cos( )α

−cos( )β

−cos( )γ

− sin( )θsin( )φ

−40

=

− sin( )θsin( )φ

−40

The mast is subjected to the three forces shown Determine the coordinate angles α1, β1, γ1

of F 1 so that the resultant force acting on the mast is F Ri.

Given:

F R = 350 N

The mast is subjected to the three forces shown Determine the coordinate angles α1, β1, γ1

of F 1 so that the resultant force acting on the mast is zero

Given F 1

cos( )αcos( )βcos( )γ

−0

The shaft S exerts three force

components on the die D Find the

magnitude and direction of the resultant

force Force F 2 acts within the octant

β2 acos⎛ 1−cos( )α2 2 −cos( )γ2 2

The beam is subjected to the two forces shown Express each force in Cartesian vector form

and determine the magnitude and coordinate direction angles of the resultant force

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