A particle of mass 0.1 k g is executing S H M of amplitude 0.1 m . When the particle passes through the mean position, its kinetic energy is 8 × 10 – 3 J . Obtain the equation of motion of this particle if the initial phase of oscillation is 45 ° .

Let the equation of oscillating particle be y = A s i n ω t + ϕ 0 …(i) A = 0.1 m , ϕ 0 = 45 o = π 4 Now, Kinetic energy of the particle at mean position, 1 2 m ω 2 A 2 = 8 × 10 – 3 ω 2 = 16 × 10 – 3 0.1 × 0.1 × 0.1 or ω = 4 rad s – 1 From equation (i) y = 0.1 s i n 4 t + π 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 figure. Each liquid occupies one-fourth the circumference of the tube. (i) Find the angle θ that the radius to the interface make with the vertical in equilibrium position. (ii) If the whole liquid column is given a small displacement from its equilibrium position, show that the resulting oscillations are simple harmonic. Find the time period of these oscillations.

(i) Let the liquid of density 1.5 ρ occupy the left portion A B and the liquid of density ρ occupy the right portion B C of the tube. The pressure at the lowest point D due to the liquid on the left is P 1 = R − R sin θ 1.5 ρ The pressure due to the liquid on the right is P 2 = R sin θ + R cos θ ρ g + R − R cos θ 1.5 ρ g Since the liquid are in equilibrium P 1 = P 2 or R − R sin θ 1.5 ρ g = R sin θ + R cos θ p g + R − R cos θ 1.5 ρ g Solving, we get tan θ = 0.2 or θ = tan − 1 0.2 (ii) Let the whole liquid be given a small angular displacement α towards. Then the pressure difference between the right and the left limbs is d P = P 2 − P 1 = R sin θ + α + R cos θ + α ρ g + [ R − R cos θ + α 1.5 ρ g − [ R − R sin θ + α 1.5 ρ g = R ρ g 2.5 sin θ + α − 0.5 cos θ + α = R ρ g 2.5 sin θ cos α + cos θ sin α − 0.5 cos θ cos α − sin θ sin α For small α sin α ≈ α , cos α ≈ 1 ∴ d P = R ρ g 2.5 sin θ + 2.5 α cos θ − 0.5 cos θ + 0.5 α sin θ As tan θ = 0.2 , sin θ 0.2 1.04 ≈ 0.2 , cos θ = 1 1.04 ≈ 1 ∴ d P = R ρ g 2.5 × 0.2 + 2.5 α − 0.5 + 0.5 × 0.2 α = R ρ g 2.6 α = 2.6 ρ g where y = R α , the linear displacement. ∴ Restoring force F = 2.6 ρ g A y This shows that F ∝ y Hence the motion is simple harmonic with force constant k = 2.6 ρ g A Now, total mass of the liquid m = 2 π R 4 A ρ + 2 π R 4 A 1.5 p = 5 π R Α ρ 4 ∴ Time period T = 2 π m k = 2 π ρ 5 π Α ρ 4 × 2.6 ρ g A = π ρ 1.93 π R 9.8 = 2.47 R seconds,

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A particle is executing SHM given by $x = A \sin (π t + ϕ)$ . The initial displacement of particle is 1 cm and its velocity is $π cm \sec^{- 1}$ . Find the amplitude of motion and initial phase of the particle.

Important Questions on Simple Harmonic Motion

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 4

A particle of mass

0.1 k g

is executing

S H M

of amplitude

0.1 m

. When the particle passes through the mean position, its kinetic energy is

8 \times 10^{- 3} J

. Obtain the equation of motion of this particle if the initial phase of oscillation is

45 °

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 5

A body of mass

1 kg

suspended by an ideal spring oscillates up and down. The amplitude of oscillation is

1

metre and periodic time is

1.57

second. Determine.

(i) Maximum speed of body.

(ii) Maximum kinetic energy.

(iii) Total energy.

(iv) Force constant of the spring.

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 6

Potential Energy

(U)

of a body of unit mass moving in a one-dimension conservative force field is given by,

U = (x^{2} - 4 x + 3)

. All units are in

S . I .

(i) Find the equilibrium position of the body.

(ii) Show that oscillations of the body about this equilibrium position is simple harmonic motion & find it time period.

(iii) Find the amplitude of oscillations if speed of the body at equilibrium position is $2 \sqrt{6} m s^{- 1}$

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 7

Calculate the time period of a uniform square plate of side '

a

' if it is suspended through a corner.

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 8

In the arrangement as shown in fig., pulleys are small and springs are ideal.

k_{1}, k_{2}, k_{3}

and

k_{4}

are force constants of the springs. Calculate period of small vertical oscillations of block of mass

m

Question Image

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 9

A mass

M

is in static equilibrium on a massless vertical spring as shown in the figure. A ball of mass

m

dropped from certain height sticks to the mass

M

after colliding with it. The oscillations they perform reach to height '

a

' above th original level of scales & depth '

b

' below it.

Question Image

(i) find the constant of force of the spring

(ii) find the oscillation frequency.

(iii) what is the height above the initial level from which the mass $m$ was dropped?

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 10

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 figure. Each liquid occupies one-fourth the circumference of the tube.

Question Image

(i) Find the angle $θ$ that the radius to the interface make with the vertical in equilibrium position.

(ii) If the whole liquid column is given a small displacement from its equilibrium position, show that the resulting oscillations are simple harmonic. Find the time period of these oscillations.

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Beta Question Bank for Engineering: Physics>Chapter 19 - Simple Harmonic Motion>Exercise-4>Q 11

A weightless rigid rod with a load at the end is hinged at point

A

to the wall so that it can rotate in all directions. The rod is kept in the horizontal position by a vertical inextensible thread of length

l

, fixed at its midpoint. The load receives a momentum in the direction perpendicular to the plane of the figure. Determine the period

T

of small oscillations of the system.

Question Image

A particle is executing SHM given by x=Asinπt+ϕ. The initial displacement of particle is 1 cm and its velocity is π cm sec-1 . Find the amplitude of motion and initial phase of the particle.

Important Questions on Simple Harmonic Motion

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A particle is executing SHM given by $x = A \sin (π t + ϕ)$ . The initial displacement of particle is 1 cm and its velocity is $π cm \sec^{- 1}$ . Find the amplitude of motion and initial phase of the particle.