Using the reference circle derive the expression of the particle executing SHM

EASY

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle executes S.H.M., the graph of velocity as a function of displacement is :

HARD

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle executes simple harmonic motion with an amplitude of

5 cm

. When the particle is at

4 cm

from the mean position, the magnitude of its velocity in

SI

units is equal to that of its acceleration. Then, its periodic time in seconds is:

EASY

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle executes S.H.M. with amplitude

A

and time period

T

. The displacement of the particle when its speed is half of maximum speed is

\frac{\sqrt{x} A}{2} .

The value of

x

is

MEDIUM

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A body undergoing simple harmonic motion has a maximum acceleration of

8 m s^{- 2}

and maximum speed of

1.6 m s^{- 1} .

What is the time period

T ?

MEDIUM

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

Two particles are performing simple harmonic motion in a straight line about the same equilibrium point. The amplitude and time period for both particles are same and equal to

A

and

T

, respectively. At time

t = 0

one particle has displacement

A

while the other one has displacement

- \frac{A}{2}

and they are moving towards each other. If they cross each other at time

t

, then

t

is:

EASY

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle executing S.H.M. has velocities

v_{1}

and

v_{2}

at distance

x_{1}

and

x_{2}

respectively from mean position. The angular velocity

(ω)

of the particle is given by

MEDIUM

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

In case of a simple harmonic motion, if the velocity is plotted along the $X$ -axis and the displacement (from the equilibrium position) is plotted along the $Y$ -axis, the resultant curve happens to be an ellipse with the ratio:
$\frac{major axis (along X)}{minor axis (along Y)} = 20 π$

What is the frequency of the simple harmonic motion?

EASY

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A simple pendulum performing harmonic motion has an amplitude $A$ and period $T .$ The speed of the bob at $x = \frac{A}{2}$ is

HARD

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle is performing a linear simple harmonic motion of amplitude

A

. When it is midway between its mean and extreme position, the magnitudes of its velocity and acceleration are equal. What is the periodic time of the motion?

EASY

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A simple pendulum oscillates harmonically about

x = 0

with an amplitude

A

and time period

T

. Its speed at

x = A / 2

is

MEDIUM

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

The velocity of a body while passing through the mean position is

10 m s^{- 1}

. What will be the frequency of it, if it is executing SHM of amplitude

1 m

?

EASY

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle executes linear simple harmonic motion with an amplitude of 3 cm. When the particle is at 2 cm from the mean position, the magnitude of its velocity is equal to that of its acceleration. Then its time period in seconds is

EASY

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle is oscillating according to the equation

V (t) = 4 \sin 0.4 π t

metres. In how much time after the motion begins will that particle reach the extreme position from the mean position?

EASY

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle is executing SHM along a straight line. Its velocities at distances

x_{1}

and

x_{2}

from the mean position are

V_{1}

and

V_{2}

respectively. Its time period is:

EASY

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A body of mass

10 g

is executing simple harmonic motion about a point with an amplitude

20 cm .

If its maximum velocity is

100 cm \cdot s^{- 1},

its velocity will be

50 cm . s^{- 1}

at a distance of

cm

from its mean position.

HARD

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle performing linear S. H. M. has maximum velocity of $25 c m / s$ and maximum acceleration of $100 c m / s^{2}$ . Find the amplitude and period of oscillation. $(π = 3.142)$

EASY

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A simple harmonic motion has amplitude

a

and time period

T

. The maximum velocity will be

EASY

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle performs

S . H . M .

with amplitude

A

. Its speed is tripled at the instant when it is at a distance of

\frac{2 A}{3}

from the mean position. The new amplitude of the motion is

MEDIUM

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle performs simple harmonic motion with amplitude A. Its speed is tripled at the instant that it is at a distance

\frac{2 A}{3}

from equilibrium position. The new amplitude of the motion is:

HARD

Physical Sciences>Energy>Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.>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.

A particle performs linear SHM. At a particular instant, the velocity of the particle is

u

and acceleration is

α

(both having the same direction). At another instant velocity is

v

and acceleration is

β (0 < α < β)

(both in opposite direction to each other).The distance between the two positions is

Using the reference circle, derive the expression of the particle executing SHM.

Important Questions on Create a computational model to calculate the change in the energy of one component in a system when the change in energy of the other component(s) and energy flows in and out of the system are known.

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