7/17/2019 Displacement

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First calculate all the displacements as below using the angles θ and  4tan x   θ =

 

B sin y x   β =

Substitute for x in the above equation

B   4 tan sin y   θ β =

Substitute the value of    in above equation

B   4 tan sin 30

2tan

 y   θ 

θ 

=

=

o

 Next find the distance between A and B

4

cos AB

θ =

 

hen x=0, the distance between A and B is given below

( )0

  4 m AB   =

!ence the extension of spring s is given below

( )0

44

cos

 s AB AB

θ 

= −

= −

  Next calculate the potential energ" of the spring

2

spring

#$

2 P E ks=

Substitute the values in the above equation2

spring

2

# 4$ % &N'm 4

2 cos

# 240 #

cos cos

 P E θ 

θ θ 

 = × × − ÷  

 = + − ÷  

 Next find the potential energ" of the cart

cart B$ (% &N P E y= ×

Substitute the value in the above equation

cart$ (% &N 2 tan

#%0tan

 P E    θ 

θ 

= ×

=

!ence total potential energ" of the s"stem is given below

2

# 240 # #%0 tan

cos cosV    θ 

θ θ 

 = + − − ÷  

 )sing the minimum potential energ" principal

0dV 

d θ =

!ence differentiate the above equation and equate it to *ero

3 2

2sin 2 sin40 #%0 tan 0

cos cos

## sin #$+(%

cos

θ θ θ 

θ θ 

θ θ 

 − − = ÷  

 − = ÷  

)se trial and error method to determine the angle

(0$4,θ   =o

Substitute the value of  in the expression for x

tan x   θ =

4 tan (0$4,

##$2( m

 x   =

=

o

!ence the value of x is as given below

##$2( m x   =