36. A simple pendulum is suspended from the roof of a trolley which moves in a horizontal direction with an acceleration a, then the time period is given by T = 2π √l / g where g is equal to​​
    

1. 1. g

   
   
   

   2. g – a

   
   
   

   3. g + a

   
   
   

   4. √g² + a²

   
   
   

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1. View Hint View Answer Discuss in Forum

   The effective value of acceleration due to gravity is √(a² + g²)

   Correct Option: D

   The effective value of acceleration due to gravity is √(a² + g²)

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37. A body is executing S.H.M. When the displacements from the mean position are 4cm and 5 cm, the corresponding velocities of the body are 10 cm per sec and 8 cm per sec. Then the time period of the body is​
    

1. 1. 2π sec

   
   
   

   2. π/2 sec

   
   
   

   3. π sec

   
   
   

   4. (3π/2) sec

   
   
   

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1. View Hint View Answer Discuss in Forum

   For S.H.M., Velocity,
   v = ω √a² - x² at displacement x.
   ⇒ 10 = ω √a² - 16 ..........(1)
   and 8 = ω √a² - 25 ..........(2)
   

   Dividing ,	52	=	a2 - 16	=	25
   	42		a2 - 25		16

   
   or , 16a² - 256 = 25a² - 625 ⇒ 9a² = 369
   

   ∴ a2 =	369
   	9

   
   Putting this value in equation (2) mentioned above,
   

   10 = ω √{ (369 / 9) - 16 } ⇒ 10 = ω
   		225
   		9

   
   

   or , ω =	10 × 3	= 2 radian / sec
   	15

   
   

   Time period =	2π	=	2π	= π sec
   	ω		2

   
   

   Correct Option: C

   For S.H.M., Velocity,
   v = ω √a² - x² at displacement x.
   ⇒ 10 = ω √a² - 16 ..........(1)
   and 8 = ω √a² - 25 ..........(2)
   

   Dividing ,	52	=	a2 - 16	=	25
   	42		a2 - 25		16

   
   or , 16a² - 256 = 25a² - 625 ⇒ 9a² = 369
   

   ∴ a2 =	369
   	9

   
   Putting this value in equation (2) mentioned above,
   

   10 = ω √{ (369 / 9) - 16 } ⇒ 10 = ω
   		225
   		9

   
   

   or , ω =	10 × 3	= 2 radian / sec
   	15

   
   

   Time period =	2π	=	2π	= π sec
   	ω		2

   
   

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38. A simple harmonic oscillator has an amplitude A and time period T. The time required by it to travel from x = A to x = A/2 is​

1. 1. ​T/6

   
   
   

   2. T/4

   
   
   

   3. T/3

   
   
   

   4. T/2

   
   
   

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1. View Hint View Answer Discuss in Forum

   ​

   For S.H.M., x = A sin			2π	. t
   			T

   
   

   When x = A , A = A sin			2π	. t
   			T

   
   

   ∴ sin			2π	. t		= 1
   			T

   
   

   ⇒ sin

   2π

   . t

   = sin

   π

   ⇒ t = (T / 4)

   T

   2

   
   

   When x =	A	,	A	= A sin			2π	.t
   	2		2				T

   
   

   or , sin	π	= sin			2π	.t		or t = (T / 12)
   	6				T

   
   Now, time taken to travel from x = A to​x = A/2 is  (T/4 – T/12) = T/6

   Correct Option: A

   ​

   For S.H.M., x = A sin			2π	. t
   			T

   
   

   When x = A , A = A sin			2π	. t
   			T

   
   

   ∴ sin			2π	. t		= 1
   			T

   
   

   ⇒ sin

   2π

   . t

   = sin

   π

   ⇒ t = (T / 4)

   T

   2

   
   

   When x =	A	,	A	= A sin			2π	.t
   	2		2				T

   
   

   or , sin	π	= sin			2π	.t		or t = (T / 12)
   	6				T

   
   Now, time taken to travel from x = A to​x = A/2 is  (T/4 – T/12) = T/6

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39. ​If a simple harmonic oscillator has got a displacement of 0.02 m and acceleration equal to 2.0 m/s² at any time, the angular frequency of the oscillator is equal to​​
    

1. 1. 10 rad/s

   
   
   

   2. 0.1 rad/s​

   
   
   

   3. 100 rad/s

   
   
   

   4. 1 rad/s

   
   
   

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1. View Hint View Answer Discuss in Forum

   ω2 =	acceleration	=	2.0
   	displacement		0.02

   
   ω² = 100 or ω = 10 rad / s

   Correct Option: A

   ω2 =	acceleration	=	2.0
   	displacement		0.02

   
   ω² = 100 or ω = 10 rad / s

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40. A hollow sphere is filled with water. It is hung by a long thread. As the water flows out of a hole at the bottom, the period of oscillation will​​
    

1. 1. first increase and then decrease

   
   
   

   2. ​first decrease and then increase

   
   
   

   3. go on increasing

   
   
   

   4. go on decreasing

   
   
   

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1. View Hint View Answer Discuss in Forum

   Time period of simple pendulum , T = 2π
   			l		∝ √l
   			g

   
   where l is effective length. ​
   [i.e distance between centre of suspension and centre of gravity of bob] ​
   Initially, centre of gravity is at the centre of sphere. When water leaks the centre of gravity goes down until it is half filled; then it begins to go up and finally it again goes at the centre. That is effective length first increases and then decreases. As T ∝ √l , so time period first increases and then decreases.
   

   Correct Option: A

   Time period of simple pendulum , T = 2π
   			l		∝ √l
   			g

   
   where l is effective length. ​
   [i.e distance between centre of suspension and centre of gravity of bob] ​
   Initially, centre of gravity is at the centre of sphere. When water leaks the centre of gravity goes down until it is half filled; then it begins to go up and finally it again goes at the centre. That is effective length first increases and then decreases. As T ∝ √l , so time period first increases and then decreases.
   

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