26. Out of the following functions, representing motion of a particle, which represents SHM?​​
    ​(A)​ y = sin ωt - cos ωt
    ​(B) ​y = sin³ ωt
    (C)

    y = 5 cos		3π	- 3ωt
    		4

    
    (D) y = 1 + ωt + ω² t²

1. 1. Only (A)

   
   
   

   2. Only (D) does not represent SHM

   
   
   

   3. ​Only (A) and (C)

   
   
   

   4. Only (A) and (B)

   
   
   

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1. View Hint View Answer Discuss in Forum

   Only functions given in (A) & (C) represent SHM.

   Correct Option: C

   Only functions given in (A) & (C) represent SHM.

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27. A particle of mass m is released from restand follows a parabolic path as shown. Assuming that the displacement of the mass from the origin is small, which graph correctlydepicts the position of the particle as a function of time​​​
    

1. 1. 

   
   
   

   2. 

   
   
   

   3. 

   
   
   

   4. 

   
   
   

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1. View Hint View Answer Discuss in Forum

   The given velocity-position graph depicts that the motion of the particle is SHM. ​
   In SHM, at t = 0, v = 0 and x = x_max
   ​So, option (a) is correct.

   Correct Option: A

   The given velocity-position graph depicts that the motion of the particle is SHM. ​
   In SHM, at t = 0, v = 0 and x = x_max
   ​So, option (a) is correct.

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28. A particle of mass m oscillates along x-axis according to equation x = a sin ωt. The nature of the graph between momentum and displacement of the particle is​​
    

1. 1. ​straight line passing through origin

   
   
   

   2. circle

   
   
   

   3. hyperbola

   
   
   

   4. ellipse

   
   
   

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1. View Hint View Answer Discuss in Forum

   As	v2	+	y2	= 1 This is the equation of ellipse.
   	a2 ω2		a2

   
   Hence the graph is an ellipse. P versus x graph is similar to V versus x graph.
   

   Correct Option: D

   As	v2	+	y2	= 1 This is the equation of ellipse.
   	a2 ω2		a2

   
   Hence the graph is an ellipse. P versus x graph is similar to V versus x graph.
   

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29. The oscillation of a body on a smooth horizontal surface is represented by the equation , ​​ X = A cos (ωt) ​
    where   X = displacement at time t ​​ 
    ω = frequency of oscillation ​
    Which one of the following graphs shows correctly the variation of ‘a’ with ‘t’?

1. 1. 

   
   
   

   2. 

   
   
   

   3. 

   
   
   

   4. 

   
   
   

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1. View Hint View Answer Discuss in Forum

   Displacement, x = A cos (ωt) (given) ​
   

   Velocity, v =	dx	= -Aω sin (ωt)
   	dt

   
   

   Acceleration, a =	dv	= -Aω2 cos ωt
   	dt

   
   Hence graph (c) correctly dipicts the variation of a with t.
   

   Correct Option: C

   Displacement, x = A cos (ωt) (given) ​
   

   Velocity, v =	dx	= -Aω sin (ωt)
   	dt

   
   

   Acceleration, a =	dv	= -Aω2 cos ωt
   	dt

   
   Hence graph (c) correctly dipicts the variation of a with t.
   

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30. When two displacements represented by y₁ = a sin(ωt) and y₂ = b cos(ωt) are super-imposed the motion is :​

1. 1. simple harmonic with amplitude	a
      	b

   
   
   

   2. simple harmonic with amplitude √a² - b²

   
   
   

   3. simple harmonic with amplitude	(a + b)
      	2

   
   
   

   4. ​not a simple harmonic

   
   
   

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1. View Hint View Answer Discuss in Forum

   The two displacements equations are y₁ = a sin(ωt) ​
   

   and y2 = b cos(ωt) = b sin		ωt +	π
   			2

   
   y_eq = y₁ + y₂
   

   = a sinωt + b cosωt = a sinωt + b sin		ωt +	π
   			2

   
   Since the frequencies for both SHMs are same, resultant motion will be SHM.
   

   Now Aeq = √a2 + b2 + 2ab cos	π
   	2

   
   ⇒ A_eq = √a² + b²
   

   Correct Option: B

   The two displacements equations are y₁ = a sin(ωt) ​
   

   and y2 = b cos(ωt) = b sin		ωt +	π
   			2

   
   y_eq = y₁ + y₂
   

   = a sinωt + b cosωt = a sinωt + b sin		ωt +	π
   			2

   
   Since the frequencies for both SHMs are same, resultant motion will be SHM.
   

   Now Aeq = √a2 + b2 + 2ab cos	π
   	2

   
   ⇒ A_eq = √a² + b²
   

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