117

Ch 5: Solutions to HW Problems P5.1 For the same force F, acting on different masses

!

F = m1a1 and

!

F = m2a2

(a)

!

m1

m2

=a2

a1

=1

3

(b)

!

F = m1 + m2( )a = 4m1a = m1 3.00 m s2( )

!

a = 0.750 m s2

P5.3

!

m = 3.00 kg

a = 2.00ˆ i + 5.00ˆ j ( ) m s2

F = ma = 6.00ˆ i + 15.0ˆ j ( ) N"

F" = 6.00( )2+ 15.0( )2 N = 16.2 N

P5.9

!

Fg = mg = 900 N ,

!

m =900 N

9.80 m s2 = 91.8 kg

!

Fg( )on Jupiter

= 91.8 kg 25.9 m s2( ) = 2.38 kN

P5.13 (a) You and the earth exert equal forces on each other:

!

myg = Me ae . If your mass is 70.0 kg,

!

ae =70.0 kg( ) 9.80 m s2( )

5.98 " 1024 kg= ~ 10#22 m s2 .

(b) You and the planet move for equal times intervals according to

!

x =1

2at

2 . If the seat is

50.0 cm high,

!

2xy

ay

=2xe

ae

!

xe =ae

ay

xy =my

me

xy =70.0 kg 0.500 m( )

5.98" 1024 kg~ 10#23 m .

118 The Laws of Motion

P5.21 (a) Isolate either mass

!

T + mg = ma = 0

T = mg .

The scale reads the tension T, so

!

T = mg = 5.00 kg 9.80 m s2( ) = 49.0 N . (b) Isolate the pulley

!

T2 + 2T1 = 0

T2 = 2T1 = 2mg = 98.0 N .

(c)

!

F" = n + T + mg = 0 Take the component along the incline

!

nx + Tx + mg x = 0 or

!

0 + T "mgsin 30.0° = 0

T = mgsin 30.0° =mg

2=

5.00 9.80( )2

= 24.5 N .

FIG. P5.21(a)

FIG. P5.21(b)

FIG. P5.21(c)

P5.22 The two forces acting on the block are the normal force, n, and the weight, mg. If the block is considered to be a point mass and the x-axis is chosen to be parallel to the plane, then the free body diagram will be as shown in the figure to the right. The angle θ is the angle of inclination of the plane. Applying Newton’s second law for the accelerating system (and taking the direction up the plane as the positive x direction) we have

!

Fy" = n #mg cos$ = 0 :

!

n = mgcos"

!

Fx" = #mgsin$ = ma :

!

a = "gsin#

FIG. P5.22

(a) When

!

" = 15.0°

!

a = "2.54 m s2

(b) Starting from rest

!

vf2 = vi

2 + 2a x f " xi( ) = 2ax f

v f = 2ax f = 2 "2.54 m s2( ) "2.00 m( ) = 3.18 m s

Chapter 5 119

*P5.24 First, consider the block moving along the horizontal. The only force in the direction of movement is T. Thus,

!

Fx" = ma

!

T = 5 kg( )a (1) Next consider the block that moves vertically. The forces on it

are the tension T and its weight, 88.2 N. We have

!

Fy" = ma

!

88.2 N "T = 9 kg( )a (2)

5 kg

n

T

49 N

+x

9 kg

T +y

Fg= 88.2 N

FIG. P5.24

Note that both blocks must have the same magnitude of acceleration. Equations (1) and (2) can be

added to give

!

88.2 N = 14 kg( )a . Then

!

a = 6.30 m s2

and T = 31.5 N .

P5.36 For equilibrium:

!

f = F and

!

n = Fg . Also,

!

f = µn i.e.,

!

µ =f

n=

F

Fg

µs =75.0 N

25.0 9.80( ) N= 0.306

and

!

µk =60.0 N

25.0 9.80( ) N= 0.245 .

FIG. P5.36

P5.41

!

m = 3.00 kg ,

!

" = 30.0° ,

!

x = 2.00 m ,

!

t = 1.50 s

(a)

!

x =1

2at

2 :

!

2.00 m =1

2a 1.50 s( )

2

a =4.00

1.50( )2

= 1.78 m s2

FIG. P5.41

!

F" = n + f + mg = ma :

!

Along x : 0 " f + mgsin 30.0° = ma

f = m gsin 30.0° " a( )Along y : n + 0 " mgcos 30.0° = 0

n = mgcos 30.0°

(b)

!

µk =f

n=

m gsin 30.0° " a( )mgcos 30.0°

,

!

µk = tan 30.0° "a

gcos 30.0°= 0.368

120 The Laws of Motion

(c)

!

f = m g sin30.0° " a( ) ,

!

f = 3.00 9.80 sin30.0° " 1.78( ) = 9.37 N (d)

!

vf2

= vi2

+ 2a x f " xi( ) where

!

x f " xi = 2.00 m

!

vf2

= 0 + 2 1.78( ) 2.00( ) = 7.11 m2

s2

vf = 7.11 m2

s2

= 2.67 m s