A particle performs harmonic oscillations along the x -axis, according to the law x = a cos ω t . Assuming the probability P of the particle, to fall within an interval from - a to + a , to be equal to unity, find how the probability density d P d x depends on x . Here, d P denotes the probability of the particle falling within an interval from x to x + d x . Plot d P d x as a function of x .

Simple harmonic motion is a periodic and oscillatory motion. It repeatedly passes through a given region in the range of - a ≤ x ≤ a . The probability that it lies in the range ( x , x + d x ) is defined as the fraction Δ t t (as t → ∞ ) , where Δ t is the time that the particle lies in the range ( x , x + d x ) , out of the total time t . Because of periodicity, this is, d P = d t T = 2 d x v T (since v = d x d t ⇒ d t = d x v ) Where, the factor 2 is needed to take account of the fact that the particle is in the range ( x , x + d x ) , during both up and down phases of its motion. Now, in simple harmonic motion, v = d x d t = ω a cos ω t = ω a 2 - x 2 Here, ω T = 2 π ( T is the time period) We get, d P = d P d x d x = d t T = 2 d x v T = 2 d x T ω a 2 - x 2 = 1 π d x a 2 - x 2 It is given that, ∫ - a + a d P d x d x = 1 So, d P d x = 1 π 1 a 2 - x 2 is the required expression.

HARD

JEE Main

IMPORTANT

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Find the angular frequency and the amplitude of harmonic oscillations of a particle if at distances $x_{1}$ and $x_{2}$ from the equilibrium position its velocity equals $v_{1}$ and $v_{2}$ respectively.

Important Questions on OSCILLATIONS AND WAVES

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 4 - OSCILLATIONS AND WAVES>MECHANICAL OSCILLATIONS>Q 4.5.

A point performs harmonic oscillations along a straight line with a period

T = 0.60 s

and an amplitude

a = 10.0 cm

. Find the mean velocity of the point averaged over the time interval during which it travels a distance

\frac{a}{2}

, starting from

(a)

the extreme position,

(b)

the equilibrium position.

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 4 - OSCILLATIONS AND WAVES>MECHANICAL OSCILLATIONS>Q 4.6.

At the moment

t = 0

, a point starts oscillating along the

x

-axis, according to the law

x = a \sin ω t

. Find:
(a) the mean value of its velocity vector projection

<v_{x}>

,
(b) the modulus of the mean velocity vector

| ⟨ v ⟩ |

,
(c) the mean value of the velocity modulus

⟨ v ⟩

, averaged over

\frac{3}{8}

of the period after the start.

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 4 - OSCILLATIONS AND WAVES>MECHANICAL OSCILLATIONS>Q 4.7.

A particle moves along the

x

-axis according to the law

x = a \cos ω t

. Find the distance that the particle covers during the time interval from

t = 0

t

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 4 - OSCILLATIONS AND WAVES>MECHANICAL OSCILLATIONS>Q 4.8.

At the moment

t = 0

, a particle starts moving along the

x

-axis so that, its velocity projection varies as

v_{x} = 35 \cos (π t) cm s^{- 1}

, where

t

is expressed in

seconds

. Find the distance that this particle covers during,

t = 2.80 s

after the start. Use

\sin (2.8 π) = 0.59

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 4 - OSCILLATIONS AND WAVES>MECHANICAL OSCILLATIONS>Q 4.9.

A particle performs harmonic oscillations along the

x

-axis, according to the law

x = a \cos ω t

. Assuming the probability

P

of the particle, to fall within an interval from

- a

+ a

, to be equal to unity, find how the probability density

\frac{d P}{d x}

depends on

x

. Here,

d P

denotes the probability of the particle falling within an interval from

x

x + d x

. Plot

\frac{d P}{d x}

as a function of

x

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 4 - OSCILLATIONS AND WAVES>MECHANICAL OSCILLATIONS>Q 4.10.

Using graphical means, find the amplitude

a

of oscillations resulting from the superposition of the following oscillations of the same direction:
(a)

x_{1} = 3.0 \cos (ω t + \frac{π}{3})

x_{2} = 8.0 \sin (ω t + \frac{π}{6})

(b)

x_{1} = 3.0 \cos ω t

x_{2} = 5.0 \cos (ω t + \frac{π}{4})

x_{3} = 6.0 \sin (ω t)

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 4 - OSCILLATIONS AND WAVES>MECHANICAL OSCILLATIONS>Q 4.11.

A point participates simultaneously in two harmonic oscillations of the same direction:

x_{1} = a \cos ω t

and

x_{2} = a \cos 2 ω t

. Find the maximum velocity of the point.

HARD

JEE Main

IMPORTANT

Problems in General Physics>Chapter 4 - OSCILLATIONS AND WAVES>MECHANICAL OSCILLATIONS>Q 4.12.

The superposition of two harmonic oscillations of the same direction results in the oscillation of a point according to the law,

x = a \cos (2.1 t) \cos (50.0 t)

, where

t

is expressed in second. Find the angular frequencies of the constituent oscillations and the period with which they beat.

Find the angular frequency and the amplitude of harmonic oscillations of a particle if at distances x1 and x2 from the equilibrium position its velocity equals v1 and v2 respectively.

Important Questions on OSCILLATIONS AND WAVES

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Find the angular frequency and the amplitude of harmonic oscillations of a particle if at distances $x_{1}$ and $x_{2}$ from the equilibrium position its velocity equals $v_{1}$ and $v_{2}$ respectively.