Answer:

Time period is defined as the time taken for one complete oscillation. It is equal to the inverse of the frequency of the oscillation.

The relationship between time period

and frequency of oscillation

is,

Substitute

for

in equation

.

Therefore, the period of oscillation is

.

Answer:

(a)

The expression for the total energy of the particle executing Simple harmonic motion is,

Here, m is the mass of the particle,

is the angular frequency of oscillation, and

is the amplitude of the particle.

From the above expression, keeping the other parameters constant, it is clear that the total energy of the particle in the simple harmonic motion is directly proportional to the square of the amplitude of the particle.

Therefore, if the amplitude of a particle in the simple harmonic motion is doubled, then the total energy of the particle will be four times its initial energy.

(b)

The expression for the maximum speed of the particle in simple harmonic motion is,

Since, the maximum speed of the particle is directly proportional to the amplitude of the particle. Therefore, if the amplitude of the particle is doubled then the speed of the particle will also be doubled.

Answer:

A particle exhibits horizontal simple harmonic motion with amplitude

; the total energy of the particle is equal to kinetic and potential energies of the particle.

Here,

is force constant or spring constant,

and

are the mass and speed of the particle and

is the position of the particle.

(a)

When the oscillating particle passes through its equilibrium position, then in that case the value of

is zero and thus its potential energy is zero. At that instant, the total energy is equal to the kinetic energy and the object is traveling at its maximum speed

.

Hence, total energy of an object in simple harmonic motion of a spring can be calculated as follows:

In this case, at equilibrium position only kinetic energy is available as whole potential energy is converted into kinetic energy. Thus, at equilibrium position in a horizontal SHM, the kinetic energy of the system is at maximum.

In option (a), at equilibrium position, the kinetic energy of the system is mentioned as zero but as explained above, the kinetic energy of the system is calculated to be maximum at equilibrium position.

Hence, option (a) is incorrect.

(b)

In option (b), at equilibrium position, the kinetic energy of the system is mentioned to be at a maximum and in the section (a) above also, it is calculated that the kinetic energy of the system is maximum at equilibrium position.

Hence, option (b) is correct.

(c)

In option (c), at equilibrium position, the kinetic energy of the system is mentioned as half the maximum value but in the section (a), it is calculated that the kinetic energy of the system at equilibrium position is maximum and its value is

.

Hence, option (c) is incorrect.

(d)

In option (d), it is mentioned in the problem that none of the above parts (a), (b) and (c) are correct but as explained above, the section (b) is correct.

Hence, option (d) is incorrect.