Question

A bright physics student purchases a weather vane for her father’s garage. The vane consists of a rooster sitting on top of an arrow. The vane is fixed to a vertical shaft of radius 0.0293 m and mass 2.67 kg that is free to turn in its roof mount as shown in the figure. The student sets up an experiment to measure the rotational inertia of the rooster and arrow. String wound about the shaft passes over a pulley and is connected to a 2.71 kg mass hanging over the edge of the roof. When the 2.71 kg mass is released, the student determines the time t that the mass takes to fall through a distance 0.692 m. The acceleration of gravity is 9.8 m/s 2. Find the rotational inertia I of the rooster and arrow, if t = 25.5 s. Answer in units of kg m2 .

Answer 1

Let T be the tension in the string(assuming it to be massless i.e ideal).
now,

s= 1/2 a t² (as initial spped of block is 0).

a=2s/t² = 2*0.692 /(25.5*25.5)

a= 0.0021 m/s²

v=a*t

=0.0021*25.5

=0.0535m/s

ω_shaft=V/r

=0.0535/0.0293

=1.8259 rad/s

Now,

conserving the energy of the system,,

W_gravity=change in energy of shaft and rooster+change in energy of block

mgh=1/2 m V_final² + 1/2 I _roosterω² +1/2 I_{shaft *}ω²

=> 2.71*9.8*0.692 = 0.5 * 2.71* 0.0535² '+ 0.5 *I_rooster*1.826² + 0.5*2.67*0.0293*0.0293*1.826*1.826/2(I_shaft=mr²/2)

18.378=0.003878 + 1.667 I_rooster + 0.00191

|_rooster= 11.0209 Kgm²