ME 370.3 Spring 18 Name(s)

Homework 5 Due: In class, Friday, Feb. 16

1. (10 pts) A rubber duck of mass 𝑚𝑚 = 0.3 kg is placed in a water tub with water density 𝜌𝜌 = 1000 kg/m3. The cross-sectional area of the floating part of the duck is 𝐴𝐴 = 0.03 m2. The buoyancy force the duck is subjected to is gAyFb ρ= , where g is the gravitational constant and y is the depth the duck is submerged in water as shown in the figure below.

(1) (2 pts) The static equilibrium depth in meter the duck is submerged in water is

a) 809.0=sty

b) 01.0=sty

c) 03.0=sty

d) 05.0=sty

(2) Suppose we press the duck down by a small amount from its equilibrium position and then release. The duck will then vibrate up and down. The frequency of this vibration in rad/s is a) 99.0=nω

b) 00.10=nω

c) 32.31=nω

d) 64.62=nω

e) 84.571=nω

f) 94.980=nω

2. (10 pts) For the system given below, assume the L-shaped bar is rigid and massless and neglect gravity.

The natural frequency of the system in rad/s is

a) mk

ba

n =ω

b) bmak

n =ω

c) mk

n =ω

d) mk

ab

n =ω

e) ambk

n =ω

3. (25 pts) The mass of a complex-shaped object shown below is 3.0 kg. When it is suspended like a pendulum at point A, it swings 30 cycles in one minute.

(1) (3pts) The natural frequency of the system in rad/s is

a) 30/πω =n

b) 2/πω =n

c) πω =n

d) πω 30=n

e) πω 60=n

(2) (7 pts) The moment of inertia of the object about its center of mass in N-m-s2/rad is about equal to

a) 009.0=cgJ

b) 046.0=cgJ

c) 187.0=cgJ

d) 476.0=cgJ

e) 796.0=cgJ

As worked out in class, the free vibration (ie. pendulum swing) is given by )sin()( φωθ += tAt n . Suppose the pendulum is displaced by an angle of 𝜃𝜃𝑜𝑜 = 0.1 rad and is released with a velocity of �̇�𝜃𝑜𝑜 =−0.5 rad/s.

(3) (2pts) The amplitude of the pendulum swing in rad is about equal to

a) 035.0=A

b) 050.0=A

c) 1.0=A

d) 188.0=A

e) 213.0=A

(4) (3pts) The phase angle in rad of the pendulum swing is about equal to

a) 581.2=φ

b) 897.1=φ

c) 4/πφ =

d) πφ =

e) 0=φ

(5) (10pts) Write a simple matlab program to calculate θ vs. t and plot the pendulum swing for two periods of pendulum swing. Make sure your plot shows a smooth curve. Also, make sure your plot matches the given ICs to be correct. Label your plot including units. Submit your matlab program along with results. (the class website has a simple matlab guide)

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