Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

JEE: Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

The document Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Physics 35 Years JEE Main & Advanced Past year Papers.
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Q.1. A mass M attached to a spring, oscillates with a period of 2sec. If the mass is increased by 2 kg the period increases by one sec. Find the initial mass M assuming that Hook’s Law is obeyed.       (1979)

Ans. 1.6 kg

Solution. Key Concept:

The time period T of the spring is Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

In the second case

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

From (i) and (ii)

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.2. Two masses m1 and m2 are suspended together by a massless spring of spring constant k. When the masses are in equilibrium, m1 is removed without disturbing the system. Find the angular frequency and amplitude of oscillation of  m2.           (1981 - 3 marks)

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. 

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. 

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

NOTE : When mass m1 is removed then the equilibrium will get disturbed. There will be a restoring force in the upward direction. The body will undergo S.H.M. now.

Let x1 be the extension of the spring when (m1 +m2) are suspended and x2 be the extension of spring when m1 is removed.

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Amplitude of oscillation = x1-x2

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Let at any instant the mass m2 be having a displacement x from the mean position. Restoring force will be

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Comparing this with a = – ω2x,

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.3. Two light springs of force constants k1 and k2 and a block of mass m are in one line AB on a smooth horizontal table such that one end of each spring is fixed on rigid supports and the other end is free as shown in the figure. The distance CD between the free ends of the springs is 60 cms. If the block moves along AB with a velocity 120 cm/sec in between the springs, calculate the period of oscillation of the block (k1 = 1.8 N/m, k2 = 3.2 N/m, m = 200 gm)           (1985 - 6 Marks)

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. 2.83 sec.

Solution.  The mass will strike the right spring, compress it. The K.E. of the mass will convert into P.E. of the spring. Again the spring will return to its natural size thereby converting its P.E. to K.E. of the block. The time taken for this process will

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

The block will move from A to B without any acceleration.

The time taken will be

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Now the block will compress the left spring and then the spring again attains its natural length. The time taken will be

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Again the block moves from B to A, completing one oscillation. The time taken for doing so

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

∴ The complete time of oscillation will be 

= t1 + t2 + t3 + t4 
= 0.785 + 0.5 + 1.05 + 0.5

= 2.83 (app.)

Q.4. Two non-viscous, incompressible and immiscible liquids of densities ρ and 1.5 ρ are poured into the two limbs of a circular tube of radius R and small cross section kept fixed in a vertical plane as shown in fig. Each liquid occupies one fourth the circumference of the tube.         (1991 - 4 + 4 marks)

(a) Find the angle θ that the radius to the interface makes with the vertical in equilibrium position. 
(b) If the whole is given a small displacement from its equilibrium position, show that the resulting oscillations are simple harmonic. Find the time period of these oscillations.

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. 

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. 

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

At equilibrium, taking torque of liquids about O

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

m2g × QM = m1g × PN
∴ m2g R sin (45° + θ) = m1gR sin (45° – θ)
VρgR sin  (45° + θ) = 1.5VρgR sin (45° – θ)... (i)

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

NOTE THIS STEP :  Let us now displace the liquids in anticlockwise direction along the circumference of tube through an angle α.

The net torque

τ = m2gR sin (45° + θ + α) – m1gR sin (45° – θ – a)
= VρgR sin (45° + θ + α) – 1.5VρgR (45° – θ – a)
= VρgR sin (θ + 45°) cos α + VρgR cos (45° + θ) 

sin α – 1.5 VρgR sin (45° – θ) cos  α + 1.5 VρgR cos(45° – θ) sin α

Using eq. (i) we get 

τ = VρgR [cos 45° + θ sin α + 1.5 cos 45° - θ) sin α ]
τ = VρgR [cos (45° + θ) + 1.5 cos (45° – θ)] sin α 

when a is small (given) 

∴ sin a ≈ α 

t = VρgR [cos (45° + θ) + 1.5 cos (45° – θ)]α

Since, τ and α are proportional and directed towards mean position. 

∴ The motion is simple harmonic.

Moment of inertia about O is

I = VρR2 + 1.5 VρR2

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.5. Two identical balls A and B each of mass 0.1 kg, are attached to two identical massless springs. The spring-mass system is constrained to move inside a rigid smooth pipe bent in the form of a circle as shown in Fig.  The pipe is fixed in a horizontal plane.

The centres of the balls can move in a circle of radius 0.06 π meter.

Each spring has a natural length zof 0.06 π meter and spring constant 0.1 N/m. Initially, both the balls are displaced by an angle θ =  π / 6 radian with respect to the diameter PQ of the circle (as shown in Fig.) and released from rest.              (1993 - 6 marks)

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

(i) Calculate the frequency of oscillation of ball B. 
(ii) Find the speed of ball A when A and B are at the two ends of the diameter PQ. 
(iii) What is the total energy of the system

Ans. 

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. (i) As both the balls are displaced by an angle θ =π /6 radian with respect to the diameter PQ of the circle and released from rest. It results into compression of spring in upper segment and an equal elongation of spring in lower segment. Let it be x. PB and QA denote x in the figure.

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Compression = Rθ = elongation = x 

∴ Force exerted by each spring on each ball = 2 kx
∴ Total force on each ball due to two springs = 4 kx
∴ Restoring torque about origin O =- (4kx)R

∴ τ = -4k (Rθ)R, where θ = Angular displacement

or τ = -4kR2θ

Since torque (τ) is proportional to θ, each ball executes angular SHM about the centre O.

Again, τ = -4kR2θ

or  Iα = -4kR2θ where α = angular acceleration

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

(ii) Let velocity at the mean position be vmax.

Loss in elastic potential energy = Gain in kinetic energy

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.6. A thin rod of length L and area of   cross-section S is pivoted at its lowest point P inside a stationary, homogeneous and non-viscous liquid . The rod is free to rotate in a vertical plane about a horizontal axis passing through P. The density d1 of the material of the rod is smaller than the density d2 of the liquid.

The rod is displaced by a small angle θ  from its equilibrium position and then released. Show that the motion of the rod is simple harmonic and determine its angular frequency in terms of the given parameters.      (1996 - 5 Marks)

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Ans. 

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. Let θ be the angle made by the rod at any instant t.

The volume of the rod = LS Weight of the rod = LS d1g

Upthrust acting on rod = LS d2g

Since, d2 > d1 (given). Therefore net force acting at the centre of mass of the rod at tilted position is (LS d2g – LSd1g)

Taking moment of force about P 

τ = (LSd2g – LSd1g) × PN where PN = perpendicular distance of line of action of force from P

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

∵ Motion is simple harmonic.

On comparing it with τ = Cθ

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE      ... (i)

The moment of inertia I of the rod about P

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE      ... (ii)

From (i) and (ii)

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Q.7. A small body attached to one end of a vertically hanging spring is performing SHM about its mean position with angular frequency ω and amplitude a. If at a height y* from the mean position, the body gets detached from the spring, calculate the value of y* so that the height H attained by the mass is maximum. The body does not interact with the spring during its subsequent motion after detachment. (aω2 > g)

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Ans.

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Solution. If a small mass is attached to one end of a vertically hanging spring then it performs SHM.

Angular frequency = ω, Amplitude = a

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

After detaching from spring, net downward acceleration of the block = g. 

∴ Height attained by the block = h

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE
Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE  from mean position < a.

Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE

The document Subjective Type Questions: Simple Harmonic Motion (Oscillations) | JEE Advanced Notes | Study Physics 35 Years JEE Main & Advanced Past year Papers - JEE is a part of the JEE Course Physics 35 Years JEE Main & Advanced Past year Papers.
All you need of JEE at this link: JEE

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