111. A rod of length 1 m is sliding in a corner as shown in figure. At an instant when the rod makes an angle of 60 degrees with the horizontal plane., the velocity of point A on the rod is 1m/ s. The angular velocity of the rod at this instant is
     

1. 1. 2 rad/s

   
   
   

   2. 1.5 rad/s

   
   
   

   3. 0.5 rad/s

   
   
   

   4. 0.75 rad/s

   
   
   

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1. View Hint View Answer Discuss in Forum

   
   VA = O'A × ω
   1 = 1 cos 60° × ω
   ω = 2 rad/s

   Correct Option: A

   
   VA = O'A × ω
   1 = 1 cos 60° × ω
   ω = 2 rad/s

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112. Figure shows a quick return mechanism. The cranks OA rotates clockwise uniformly. OA = 2 cm, OO' = 4 cm. The ratio of time for forward motion to that for return motion is
     

1. 1. 0.5

   
   
   

   2. 2.0

   
   
   

   3. √2

   
   
   

   4. 1

   
   
   

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1. View Hint View Answer Discuss in Forum

   Ratio of time of forward motion to return motion
   

   =	180 + 2α	=	180 + 60	=	240	= 2
   	180 − 2α		180 − 60		120

   
   (Given OD = 2 cm, OO ' = 4 cm, sin α = 2/4 = 0.5 ⇒ α = 30°)

   Correct Option: B

   Ratio of time of forward motion to return motion
   

   =	180 + 2α	=	180 + 60	=	240	= 2
   	180 − 2α		180 − 60		120

   
   (Given OD = 2 cm, OO ' = 4 cm, sin α = 2/4 = 0.5 ⇒ α = 30°)

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113. A slender uniform rigid bar of mass m is hinged at O and supported by two spr i ngs, wit h stiffnesses 3k and k, and a damper with damping coefficient c, as shown in the figure. For the system to be critically damped, t he ratio c / √km should be
     

1. 1. 2
      

   
   
   

   2. 2 √7
      

   
   
   

   3. 4
      

   
   
   

   4. 4 √7

   
   
   

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1. View Hint View Answer Discuss in Forum

   
   

   I0 =	m L2	+ m		L		2
   	12			4

   
   

   =	mL2	+	mL2	=	7	mL2
   	12		16		48

   
   Torque about hinge ‘O’
   

   =	7mL2	θ̇̇ + 3k.		L2		θ + k.	9L2	θ + C.	L2	θ̇ = 0
   	48			16			16		16

   
   

   ⇒	7mL2	θ̇̇ +	12kL2	θ +	CL2	θ̇ = 0
   	48		16		16

   
   
   
   

   2ξωn =	C × 48	=	3C
   	16 × 7 m		7m

   
   

   ⇒ 2√km =	3	×	√7 × C
   	7		6

   
   

   	C	= 4√7
   	√km

   Correct Option: D

   
   

   I0 =	m L2	+ m		L		2
   	12			4

   
   

   =	mL2	+	mL2	=	7	mL2
   	12		16		48

   
   Torque about hinge ‘O’
   

   =	7mL2	θ̇̇ + 3k.		L2		θ + k.	9L2	θ + C.	L2	θ̇ = 0
   	48			16			16		16

   
   

   ⇒	7mL2	θ̇̇ +	12kL2	θ +	CL2	θ̇ = 0
   	48		16		16

   
   
   
   

   2ξωn =	C × 48	=	3C
   	16 × 7 m		7m

   
   

   ⇒ 2√km =	3	×	√7 × C
   	7		6

   
   

   	C	= 4√7
   	√km

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114. A uniform thin disk of mass 1 kg and radius 0.1 m is kept on a surface as shown in the figure. The spring of stiffness k₁ = 400 N/m is connected to the disk center A and another spring of stiffness k₂ = 100 N/m is connected at point B just above point A on the circumference of the disk. Initially, both the springs are unstretched. A assume pure rolling of the disk. For small disturbance from the equilibrium, the natural frequency of vibration of the system is _____rad/ s (round off to one decimal place).
     

1. 1. 23.1 rad / sec

   
   
   

   2. 123.1 rad / sec

   
   
   

   3. 213.1 rad / sec

   
   
   

   4. 231.1 rad / sec

   
   
   

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1. View Hint View Answer Discuss in Forum

   Given, Disc mass m = 1 kg, R = 0.1 m
   K₁ = 400 N/m, K₂ = 100 N/m
   
   By assuming pure rolling of the disc, M_c = 0 at I.C. at point (C)
   I_cθ̇̇ + K₂(2R)θ × 2R + K₁Rθ × R = 0
   
   

   Ic = Idisc + mR2 =	mR2	+ mR2
   	2

   
   

   =	3	mR2 =	3	× 0.12 = 1.5 × 10-2
   	2		2

   
   
   ⇒ ω_n = √533.3rad / sec
   ω_n = 23.094 rad / sec
   ω_n = 23.1 rad / sec

   Correct Option: A

   Given, Disc mass m = 1 kg, R = 0.1 m
   K₁ = 400 N/m, K₂ = 100 N/m
   
   By assuming pure rolling of the disc, M_c = 0 at I.C. at point (C)
   I_cθ̇̇ + K₂(2R)θ × 2R + K₁Rθ × R = 0
   
   

   Ic = Idisc + mR2 =	mR2	+ mR2
   	2

   
   

   =	3	mR2 =	3	× 0.12 = 1.5 × 10-2
   	2		2

   
   
   ⇒ ω_n = √533.3rad / sec
   ω_n = 23.094 rad / sec
   ω_n = 23.1 rad / sec

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115. The natural frequencies corresponding t o the spring-mass system I and II are ω_I and ω_II, respectively.
     

     The ratio	ωⅠ	is
     	ωⅡ

     
     
     

1. 1. 4
      

   
   
   

   2. 2
      

   
   
   

   3. 1

      4

      
      

   
   
   

   4. 1

      2

   
   
   

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1. View Hint View Answer Discuss in Forum

   
   

   Now ,	1	=	1	+	1	(For series connection)
   	{Keq}		K1		K2

   
   

   ⇒		Keq1 =	K1K2		=	K
   			K1 + K2			2

   
   (K_eq)₂ = K₁ + K₂ = K (For parallel connection)
   

   Correct Option: D

   
   

   Now ,	1	=	1	+	1	(For series connection)
   	{Keq}		K1		K2

   
   

   ⇒		Keq1 =	K1K2		=	K
   			K1 + K2			2

   
   (K_eq)₂ = K₁ + K₂ = K (For parallel connection)
   

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