(III) We usually neglect the mass of a spring if it is small compared to the truss attached to it. But in some applications, the mass of the spring must be taken into account. Consider a spring of unstretched length l and mass M S uniformly distributed along the length of the spring. A mass m is attached to the end of the spring. One end of the spring is fixed and the mass m is allowed to vibrate horizontally without friction (Fig. 7–30). Each point on the spring moves with a velocity proportional to the distance from that point to the fixed end. For example, if the mass on the end moves with speed ϑ 0 , the midpoint of the spring moves with speed ϑ 0 / 2 Show that the kinetic energy of the mass plus spring when the mass is moving with velocity ϑ is K = 1 2 M ϑ 2 where M = m 1 3 M S is the “effective mass” of the system. [ Hint : Let D be the total length of the stretched spring. Then the velocity of a mass d m of a spring of length d x located at x is ϑ ( x ) − ϑ 0 ( x / D ) . Note also d m = d x ( M S / D ) .] FIGURE 7-30 Problem 68.
(III) We usually neglect the mass of a spring if it is small compared to the truss attached to it. But in some applications, the mass of the spring must be taken into account. Consider a spring of unstretched length l and mass M S uniformly distributed along the length of the spring. A mass m is attached to the end of the spring. One end of the spring is fixed and the mass m is allowed to vibrate horizontally without friction (Fig. 7–30). Each point on the spring moves with a velocity proportional to the distance from that point to the fixed end. For example, if the mass on the end moves with speed ϑ 0 , the midpoint of the spring moves with speed ϑ 0 / 2 Show that the kinetic energy of the mass plus spring when the mass is moving with velocity ϑ is K = 1 2 M ϑ 2 where M = m 1 3 M S is the “effective mass” of the system. [ Hint : Let D be the total length of the stretched spring. Then the velocity of a mass d m of a spring of length d x located at x is ϑ ( x ) − ϑ 0 ( x / D ) . Note also d m = d x ( M S / D ) .] FIGURE 7-30 Problem 68.
(III) We usually neglect the mass of a spring if it is small compared to the truss attached to it. But in some applications, the mass of the spring must be taken into account. Consider a spring of unstretched length
l
and mass
M
S
uniformly distributed along the length of the spring. A mass
m
is attached to the end of the spring. One end of the spring is fixed and the mass m is allowed to vibrate horizontally without friction (Fig. 7–30). Each point on the spring moves with a velocity proportional to the distance from that point to the fixed end. For example, if the mass on the end moves with speed
ϑ
0
, the midpoint of the spring moves with speed
ϑ
0
/
2
Show that the kinetic energy of the mass plus spring when the mass is moving with velocity
ϑ
is
K
=
1
2
M
ϑ
2
where
M
=
m
1
3
M
S
is the “effective mass” of the system. [Hint: Let D be the total length of the stretched spring. Then the velocity of a mass
d
m
of a spring of length
d
x
located at
x
is
ϑ
(
x
)
−
ϑ
0
(
x
/
D
)
. Note also
d
m
=
d
x
(
M
S
/
D
)
.]
Consider a plot of the spring's compression ) as a function of the magnitude of the
applied force (F) for an ideal elastic spring. The slopeof the curve would be
none of the given choices
Othe mass of the object attacned tot
O the spring constam
O the acceleration due to gravity
the reciprocal of the sprng constant
When a body of mass 0.25 kg is attached to a vertical mass less spring, it is extended 5.0 cm from its unstretched length of 4.0 cm. The body and spring are placed on a horizontal frictionless surface and rotated about the held end of the spring at 2.0 rev/s. How far is the spring stretched?
A mathematical pendulum of mass 1.0 kg and length 1.0 m is moved by an angle 60° from the vertical axis and then let go. How fast does the wight move at the moment the rope encloses an angle of 30° with the vertical axis?
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