Mechanical Engineering Principles

194 MECHANICAL ENGINEERING PRINCIPLES

3.4 m. Determine the earliest time of day

that the yacht is refloated.

[11h,11min,52s]

3. A mass of 2 kg is attached to a vertical
   spring. The initial state displacement of
   this mass is 74 mm. The mass is displaced
   downwards and then released. Determine
   (a) the stiffness of the spring, and (b) the
   frequency of oscillation of the mass.

[(a) 265.1 N/m (b) 1.83 Hz]

4. A particle of mass 4 kg rests on a smooth
   horizontal surface and is attached to a hor-
   izontal spring. The mass is then displaced
   horizontally outwards from the spring a
   distance of 26 mm and then released to
   vibrate. If the periodic time is 0.75 s,
   determine (a) the frequency f,(b)the
   force required to give the mass the dis-
   placement of 26 mm, (c) the time taken
   to move horizontally inwards for the first
   12 mm.
   [(a) 1.33 Hz (b) 7.30 N (c) 0.12 s]


5. A mass of 3 kg rests on a smooth hori-
   zontal surface, as shown in Figure 17.5.
   If the stiffness of each spring is 1 kN/m,
   determine the frequency of vibration of
   the mass. It may be assumed that initially,
   the springs are un-stretched. [4.11 Hz]

Figure 17.5

6. A helical spring, which has a mass of
   10 kg attached to its top. If the mass
   vibrates vertically with a frequency of
   1.5 Hz, determine the stiffness of the
   spring. [94.25 N/m]

17.4 The simple pendulum

A simple pendulum consists of a particle of mass m
attached to a mass-less string of length L, as shown
in Figure 17.6.

x

L

mg

P

q

P

Figure 17.6

From Section 13.4, page 148,

T=Ioα=−restoring couple

=−mg(Lsinθ)

But, Io=mL^2

hence,mL^2 α+mgLsinθ= 0

For small deflections, sinθ=θ

Hence, L^2 α+gLθ= 0

or α+

gθ

L

= 0

But α+ω^2 θ=0 (see Section 17.6)

Therefore, ω^2 =

g

L

and ω=

√

g

L

( 17. 12 )

Now T=

2 π

ω

= 2 π

√

L

g

( 17. 13 )

and f=

1

T

=

√

g

L

2 π

( 17. 14 )

Problem 2. If the simple pendulum of

Figure 17.6 were of length 2 m, determine its

frequency of vibration. Takeg= 9 .81 m/s^2.