A block of mass mis attached to a massless spring of force constant k the other end of which is fixed to the wall of a truck as shown The block is placed over a smooth surface of the truck and initially the spring is unstretched Suddenly the truck starts moving towards right with a constant acceleration aas seen from the truck

MEDIUM

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

A simple harmonic oscillator of frequency

1 Hz

has a phase of

1

radian. By how much should the origin be shifted in time so as to make the phase of the oscillator vanish. (time in seconds).

EASY

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

A particle of mass

m

is moving along the

x - a x i s

under the potential

V (x) = \frac{k x^{2}}{2} + \frac{λ}{x^{'}}

where

k

and

x^{'}

are positive constants of appropriate dimensions. The particle is slightly displaced from its equilibrium position. The particle oscillates with the angular frequency

(ω)

given by

MEDIUM

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

Frequency of oscillation of a body is

5 Hz

when a force

F_{1}

is applied and

12 Hz

when another force

F_{2}

is applied. If both forces

F_{1}

and

F_{2}

are applied together, then frequency of oscillation of the body will be

MEDIUM

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

An object was found to make oscillations governed by the equation

y = \cos^{2} (ω t + \frac{π}{4}) - 0.5

. The amplitude and angular frequency of these oscillations, respectively, are

EASY

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

For a particle performing linear SHM, its average speed over one oscillation is (

A =

amplitude of S.H.M.,

n =

frequency of oscillation)

EASY

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

A mass

M

is suspended from a spring of negligible mass. The spring is pulled a little and then released so that the mass executes

S . H . M .

of period

T

. If the mass is increased by

m

, the time period becomes

\frac{5 T}{3} .

What is the ratio

(\frac{M}{m})

?

HARD

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

A load of mass $m$ falls from a height $h$ on the scale pan hung from a spring as shown. If the spring constant is $k$ and the mass of the scale pan is zero and the mass $m$ does not bounce relative to the pan, then the amplitude of vibration is

Question Image

MEDIUM

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

A wave along a string has the following equation

y = 0.05 \sin (28 t - 1.78 x) m

(where,

t

is in seconds and

x

is in meters). What are the amplitude

(A),

frequency

(f)

and wavelength

(λ)

of the wave ?

MEDIUM

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

A body of mass

M

and charge

q

is connected to a spring of spring constant

k

. It is oscillating along

x

-direction about its equilibrium position in the horizontal plane, taken to be at

x = 0

, with an amplitude

A

. An electric field

E

is applied along the

x

-direction. Which of the following statements is correct?

HARD

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

An ideal gas enclosed in a vertical cylindrical container supports a freely moving piston of mass $M$ . The piston and the cylinder have equal cross-sectional area $A$ . When the piston is in equilibrium, the volume of the gas is $V_{0}$ and its pressure is $M_{0}$ . The piston is slightly displaced from the equilibrium position and released. Assuming that the system is completely isolated from its surrounding, the piston executes a simple harmonic motion with frequency
[Assume the system is in space.]

EASY

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

Assume that a tunnel is dug along a chord of the earth, at a perpendicular distance

\frac{R}{2}

from the earth's centre, where

R

is the radius of the earth. The wall of the tunnel is frictionless. If a particle is released in this tunnel, it will execute a simple harmonic motion with a time period:

HARD

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

State the differential equation of linear simple harmonic motion.

EASY

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

The function of time representing a simple harmonic motion with a period of

\frac{π}{ω}

is :

MEDIUM

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

The function

(\sin ω t - \cos ω t)

represents the S.H.M having a time period

T

MEDIUM

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

A simple pendulum suspended from the ceiling of a stationary lift has a time period $T_{0}$ . When the lift descends at uniform speed, the time period is $T_{1}$ . When the lift descends with constant acceleration, the time period is $T_{2}$ . Which of the following is correct?

MEDIUM

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

A body of mass

1 kg

is made to oscillate on a spring of force constant

15 N / m

. Calculate the angular frequency

MEDIUM

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

A body of mass

1 kg

is made to oscillate on a spring of force constant

15 N / m

. Calculate the frequency of vibrations.

HARD

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

A particle executes simple harmonic motion and it is located at

x = a, b

and

c

at time

t_{0}, 2 t_{0} a n d 3 t_{0}

respectively. The frequency of the oscillation is:

HARD

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

Two point-like objects of masses $20 gm$ and $30 gm$ are fixed at the two ends of a rigid massless rod of length $10 cm$ . This system is suspended vertically from a rigid ceiling using a thin wire attached to its center of mass, as shown in the figure. The resulting torsional pendulum undergoes small oscillations. The torsional constant of the wire is $1.2 \times 10^{- 8} N m {rad}^{- 1}$ . The angular frequency of the oscillations in $n \times 10^{- 3} rad s^{- 1}$ . The value of $n$ is _____.

Question Image

HARD

Physical Sciences>Energy>Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).>Definitions of Energy - Energy is a quantitative property of a system that depends on the motion and interactions of matter and radiation within that system. That there is a single quantity called energy is due to the fact that a system’s total energy is conserved, even as,within the system, energy is continually transferred from one object to another and between its various possible forms.At the macroscopic scale, energy manifests itself in multiple ways, such as in motion, sound, light, and thermal energy These relationships are better understood at the microscopic scale, at which all of the different manifestations of energy can be modeled as a combination of energy associated with the motion of particles and energy associated with the configuration(relative position of the particles). In some cases the relative position energy can be thought of as stored in fields (which mediate interactions between particles). This last concept includes radiation, a phenomenon in which energy stored in fields moves across space.The availability of energy limits what can occur in any system.

A hollow sphere of radius

R

is suspended from a thin rod. If the sphere is twisted by a small angle about the wire axis and released, simple harmonic oscillation (SHM) will ensure with a period of

τ_{1}

. Now, if the hollow sphere is replaced by a solid sphere of radius and mass equal to that of a hollow sphere, the SHM will have a period

τ_{2}

. The ratio of time periods

(τ_{1} / τ_{2})

is

Important Questions on Develop and use models to illustrate that energy at the macroscopic scale can be accounted for as a combination of energy associated with the motions of particles (objects) and energy associated with the relative position of particles (objects).

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