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Grade 12th passIIT JEE Entrance Exam

A solid body rotates with deacceleration about a stationary axis with an angular deacceleration `a` directle proportional to (W)^(1/2) where W is its angular vel. If at the initial moment of time its angular vel was equal to w` then the mean angular vel of the body averaged over the whole time of rotation till it comes to rest is w`/n . Were n is-

Profile image of Shikhar srivastava
8 Years agoGrade 12th pass
Answers icon

1 Answer

Profile image of Arun
6 Years ago
In accordance with the problem, β 
z
​ 
 
 
Thus, − 
dt
dω
​ 
 =k 
ω
​ 
 , where k is proportionality constant
 
⇒−∫ 
ω 
0
​ 
 
​ 
 
ω
​ 
 
​ 
  
ω
​ 
 
dω
​ 
 =k∫ 
0
t
​ 
 dt or,  
ω
​ 
 = 
ω 
0
​ 
 
​ 
 − 
2
kt
​ 
 
 
when, ω=0, total time of rotation, t=τ= 
k
2 
ω 
0
​ 
 
​ 
 
​ 
 
 
Average angular velocity 〈ω〉= 
∫dt
∫ωdt
​ 
 = 
k
2 
ω 
0
​ 
 
​ 
 
​ 
 
∫ 
0
k
2 
ω 
0
​ 
 
​ 
 
​ 
 
​ 
 (ω 
0
​ 
 + 
4
k 
2
 t 
2
 
​ 
 −kt 
ω 
0
​ 
 
​ 
 )dt
​ 
 
 
Hence, 〈ω〉= 
2 
k
ω 
0
​ 
 
​ 
 
​ 
 
[ω 
0
​ 
 t+ 
12
k 
2
 t 
3
 
​ 
 − 
2
k
​ 
  
ω 
0
​ 
 
​ 
 t 
2
 ] 
0
k
2 
ω 
0
​ 
 
​ 
 
​ 
 
​ 
 
​ 
 = 
3
ω 
0
​ 
 
​ 
 
 
Therefore, x=3

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