Force and Energy Method in SHM

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JEE Main 2015

LEVELJEE Main

For a simple pendulum, a graph is plotted between its Kinetic Energy (KE) and Potential Energy (PE) against its displacement (d) Which one of the following represents these correctly? (graphs are schematic and not drawn to scale)

Starting from the origin, a body oscillates simple harmonically with a period of $2 s$ . After what time will its kinetic energy be $75%$ of the total energy?

(A)

\frac{1}{6} s

(B)

\frac{1}{4} s

(C)

\frac{1}{3} s

(D)

\frac{1}{12} s

LEVELBoard

In a simple harmonic oscillator, at the mean position

(A)

kinetic energy is minimum, potential energy is maximum

(B)

both kinetic and potential energies are maximum

(C)

kinetic energy is maximum, potential energy is minimum

(D)

both kinetic and potential energies are minimum

JEE Main 2019

LEVELJEE Main

A particle is executing simple harmonic motion (SHM) of amplitude $A$ , along the X-axis, about $x = 0$ . when its potential energy (PE) equals kinetic energy (KE), the position of the particle will be

(A)

A

(B)

\frac{A}{2}

(C)

\frac{A}{2 2}

(D)

\frac{A}{2}

JEE Main 2021

LEVELJEE Main

A particle of mass $1 kg$ is hanging from a spring of force constant $100 Nm^{- 1}$ . The mass is pulled slightly downward and released, so that it executes free simple harmonic motion with time period $T$ . The time when the kinetic energy and potential energy of the system will become equal, is $T / x$ . The value of $x$ is.

JEE Main 2021

LEVELJEE Main

In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.

(A)

\frac{1}{2}

(B)

\frac{3}{4}

(C)

\frac{1}{3}

(D)

\frac{1}{4}

JEE Main 2021

LEVELJEE Main

For what value of displacement the kinetic energy and potential energy of a simple harmonic oscillation become equal?

(A)

x = 0

(B)

x = \pm A

(C)

x = \pm \frac{A}{2}

(D)

x = \frac{A}{2}

JEE Main 2004

LEVELBoard

The total energy of a particle, executing simple harmonic motion is where, $x$ is the displacement from the mean position.

(A)

\propto x

(B)

\propto x^{2}

(C)

independent of

x

(D)

\propto x^{1/2}

JEE Main 2021

LEVELJEE Main

A particle starts executing simple harmonic motion (SHM) of amplitude $a$ and total energy $E$ . At any instant, its kinetic energy is $\frac{3 E}{4}$ , then its displacement $y$ is given by

(A)

y = a

(B)

y = \frac{a}{2}

(C)

y = \frac{a 3}{2}

(D)

y = \frac{a}{2}

LEVELJEE Main

A body executes simple harmonic motion. The potential energy (PE), the kinetic energy (KE) and total energy (TE) are measured as function of displacement $x$ . Which of the following statements is true?

(A)

KE is maximum when

x = 0

(B)

TE is zero when

x = 0

(C)

KE is maximum when

x

is maximum

(D)

PE is maximum when

x = 0

JEE Advanced 2011

LEVELJEE Advanced

Comprehension Passage

Phase space diagrams are useful tools in analysing all kinds of dynamical problems. They are especially useful in studying the motion where position and momentum are changed. Here we consider some simple dynamical systems in one-dimension. For such systems, phase space is a plane in which position is plotted along horizontal axis and momentum is plotted along vertical axis. The phase space diagram is

x (t)

p (t)

curve in this plane. The arrow on the curve indicates the time flow. For example, the phase space diagram for a particle moving with constant velocity is a straight line as shown in the figure. We use the sign convention in which position or momentum upwards (or to right) is positive and downwards (or to left) is negative.

Question 1:

The phase space diagram for a ball thrown vertically up from ground is

The phase space diagram for simple harmonic motion is a circle centered at the origin. In the figure, the two circles represent the same oscillator but for different initial conditions, and $E_{1}$ and $E_{2}$ are the total mechanical energies respectively. Then,

(A)

E_{1} = 2 E_{2}

(B)

E_{1} = 2 E_{2}

(C)

E_{1} = 4 E_{2}

(D)

E_{1} = 16 E_{2}

Question 3:

Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is

A wooden block performs SHM on a frictionless surface with frequency $ν_{0}$ . The block carries a charge $+ Q$ on its surface. If now a uniform electric field $E$ is switched-on as shown, then the SHM of the block will be

(A)

of the same frequency and with shifted mean position

(B)

of the same frequency and with the same mean position

(C)

of changed frequency and with shifted mean position

(D)

of changed frequency and with the same mean position

JEE Advanced 2015

LEVELJEE Advanced

Two independent harmonic oscillators of equal masses are oscillating about the origin with angular frequencies $ω_{1}$ and $ω_{2}$ and have total energies $E_{1}$ and $E_{2}$ , respectively. The variations of their momenta $p$ with positions $x$ are shown in the figures. If $\frac{a}{b} = n^{2}$ and $\frac{a}{R} = n$ , then the correct equation(s) is/are

* Multiple Correct Options

(A)

E_{1} ω_{1} = E_{2} ω_{2}

(B)

\frac{ω _{2}}{ω _{1}} = n^{2}

(C)

ω_{1} ω_{2} = n^{2}

(D)

\frac{E _{1}}{ω _{1}} = \frac{E _{2}}{ω _{2}}

JEE Advanced 2000

LEVELJEE Advanced

The period of oscillation of simple pendulum of length $L$ suspended from the roof of the vehicle which moves without friction, down an inclined plane of inclination $α$ , is given by

(A)

2 π \frac{L}{g c o s α}

(B)

2 π \frac{L}{g s i n α}

(C)

2 π \frac{L}{g}

(D)

2 π \frac{L}{g t a n α}

JEE Advanced 1988

LEVELJEE Main

Two bodies $M$ and $N$ of equal masses are suspended from two separate massless springs of spring constants $k_{1}$ and $k_{2}$ respectively. If the two bodies oscillate vertically such that their maximum velocities are equal, the ratio of the amplitude of $M$ to that of $N$ is

(A)

\frac{k _{1}}{k _{2}}

(B)

\frac{k _{2}}{k _{1}}

(C)

\frac{k _{2}}{k _{1}}

(D)

\frac{k _{1}}{k _{2}}

JEE Advanced (1990)

LEVELJEE Main

A uniform cylinder of length $L$ and mass $M$ having cross-sectional area $A$ is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half-submerged in a liquid of density $ρ$ at equilibrium position. When the cylinder is given a small downward push and released it starts oscillating vertically with a small amplitude. If the force constant of the spring is $k$ , the frequency of oscillation of the cylinder is

(A)

\frac{1}{2 π} (\frac{k - A ρ g}{M})^{1/2}

(B)

\frac{1}{2 π} (\frac{k + A ρ g}{M})^{1/2}

(C)

\frac{1}{2 π} (\frac{k + ρ g L ^{2}}{M})^{1/2}

(D)

\frac{1}{2 π} (\frac{k + A ρ g}{A ρ g})^{1/2}

JEE Advanced 1999

LEVELJEE Advanced

A particle free to move along the $X$ -axis has potential energy given by $U (x) = k [1 - exp (- x^{2})]$ for $- \infty \leq x \leq + \infty$ , where $k$ is a positive constant of appropriate dimensions. Then,

(A)

at points away from the origin, the particle is in unstable equilibrium

(B)

for any finite non-zero value of

x

, there is a force directed away from the origin

(C)

if its total mechanical energy is

k /2

, it has its minimum kinetic energy at the origin

(D)

for small displacements from

x = 0

, the motion is simple harmonic

JEE Advanced 2003

LEVELJEE Main

For a particle executing SHM the displacement $x$ is given by $x = A cos ω t$ . Identify the graph which represents the variation of potential energy (PE) as a function of time $t$ and displacement $x$ .

A uniform rod of length $L$ and mass $M$ is pivoted at the centre. Its two ends are attached to two springs of equal spring constants $k$ . The springs are fixed to rigid supports as shown in the figure, and rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle $θ$ in one direction and released. The frequency of oscillation is

(A)

\frac{1}{2 π} \frac{2 k}{M}

(B)

\frac{1}{2 π} \frac{k}{M}

(C)

\frac{1}{2 π} \frac{6 k}{M}

(D)

\frac{1}{2 π} \frac{24 k}{M}

JEE Advanced (1981)

LEVELJEE Main

Two masses $m_{1}$ and $m_{2}$ are suspended together by a massless spring of spring constant $k$ as shown in the figure. When the masses are in equilibrium, $m_{1}$ is removed without disturbing the system. Find the angular frequency and amplitude of oscillation of $m_{2}$ .

JEE Advanced 1998

LEVELJEE Advanced

A particle of mass $m$ is executing oscillation about the origin on the $X$ -axis. Its potential energy is $U (x) = k ∣ x ∣^{3}$ , where $k$ is a positive constant. If the amplitude of oscillation is $a$ , then its time period $T$ is

(A)

proportional to

1/ a

(B)

independent of

a

(C)

proportional to

a

(D)

proportional to

a^{3/2}

JEE Advanced 1993

LEVELJEE Main

One end of a long metallic wire of length $L$ is tied to the ceiling. The other end is tied to a massless spring of spring constant $k$ . A mass $m$ hangs freely from the free end of the spring. The area of cross-section and the Young's modulus of the wire are $A$ and $Y$ respectively. If the mass is slightly pulled down and released, it will oscillate with a time period $T$ equal to

(A)

2 π (m / k)^{1/2}

(B)

2 π \frac{m ( Y A + k L )}{Y A k}

(C)

2 π [(mY A / k L)]^{1/2}

(D)

2 π (m L / Y A)^{1/2}

JEE Advanced 1993

LEVELJEE Advanced

Comprehension Passage

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 figure. The pipe is fixed in a horizontal plane. The centres of the balls can move in a circle of radius

0.06 m

. Each spring has a natural length of

0.06 π m

and spring constant

0.1 N/m

. Initially, both the balls are displaced by an angle

θ = π /6 rad

with respect to the diameter

P Q

of the circle (as shown in figure) and released from rest.

Question 1:

A $0.1 kg$ mass is suspended from a wire of negligible mass. The length of the wire is $1 m$ and its cross-sectional area is $4.9 \times 1 0^{- 7} m^{2}$ . If the mass is pulled a little in the vertically downward direction and released, it performs simple harmonic motion of angular frequency $140 rad s^{- 1}$ . If the Young's modulus of the material of the wire is $n \times 1 0^{9} Nm^{- 2}$ , the value of $n$ is.

JEE Advanced 1996

LEVELJEE Advanced

A thin rod of length $L$ and uniform cross-section 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 $d_{1}$ of the material of the rod is smaller than the density $d_{2}$ of the liquid. The rod is displaced by 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.

For a simple pendulum, a graph is plotted between its Kinetic Energy (KE) and Potential Energy (PE) against its displacement (d) Which one of the following represents these correctly? (graphs are schematic and not drawn to scale)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Starting from the origin, a body oscillates simple harmonically with a period of 2 s. After what time will its kinetic energy be 75% of the total energy?

In a simple harmonic oscillator, at the mean position

In a simple harmonic oscillation, what fraction of total mechanical energy is in the form of kinetic energy, when the particle is midway between mean and extreme position.

For what value of displacement the kinetic energy and potential energy of a simple harmonic oscillation become equal?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The total energy of a particle, executing simple harmonic motion is where, x is the displacement from the mean position.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A particle starts executing simple harmonic motion (SHM) of amplitude a and total energy E. At any instant, its kinetic energy is 43E​, then its displacement y is given by

Comprehension Passage

The phase space diagram for a ball thrown vertically up from ground is

Consider the spring-mass system, with the mass submerged in water, as shown in the figure. The phase space diagram for one cycle of this system is

Comprehension Passage

Calculate the frequency of oscillation of ball B.

Find the speed of ball A when A and B are at the two ends of the diameter PQ.

What is the total energy of the system?

Starting from the origin, a body oscillates simple harmonically with a period of $2 s$ . After what time will its kinetic energy be $75%$ of the total energy?

The total energy of a particle, executing simple harmonic motion is where, $x$ is the displacement from the mean position.

A particle starts executing simple harmonic motion (SHM) of amplitude $a$ and total energy $E$ . At any instant, its kinetic energy is $\frac{3 E}{4}$ , then its displacement $y$ is given by