Take this Quiz on Energy in Simple Harmonic Motion

Explanation
The period of the kinetic energy of a particle performing simple harmonic motion is half of the period of the displacement. This means that the kinetic energy repeats itself every half of the time it takes for the displacement to complete one cycle. Therefore, the correct answer is T/2.

Explanation
The maximum kinetic energy of a body in simple harmonic motion is equal to half of the maximum potential energy. In this case, the maximum displacement of the body is given by the amplitude of the sine function, which is 6 cm. The maximum potential energy can be calculated using the formula E = (1/2)kA^2, where k is the spring constant and A is the amplitude. Since the mass is 1 kg, the spring constant can be calculated using the formula k = (2πf)^2m, where f is the frequency and m is the mass. The frequency can be calculated using the formula f = 1/T, where T is the time period. The time period can be calculated using the formula T = 2π/ω, where ω is the angular frequency. The angular frequency can be calculated using the formula ω = 2πf. Substituting the given values into the formulas, we can find the spring constant, frequency, time period, and angular frequency. Then, using the formula for potential energy, we can calculate the maximum potential energy. Finally, we can calculate the maximum kinetic energy by dividing the maximum potential energy by 2. The result is 18 J.

Explanation
The energy of Simple Harmonic Motion (SHM) depends on both the frequency and amplitude. The frequency determines how quickly the system oscillates, while the amplitude determines the maximum displacement from the equilibrium position. Both factors contribute to the total energy of the system. A higher frequency or larger amplitude will result in a greater amount of energy in the SHM system.

Explanation
In simple harmonic motion, the potential energy is maximum at the extreme position. This is because at the extreme position, the displacement from the mean position is maximum, resulting in a maximum potential energy. At the mean position, the displacement is zero and the potential energy is minimum. The mid-point is not mentioned in the context of simple harmonic motion, so it is not relevant to the question. Therefore, the correct answer is extreme position.

Explanation
When a particle is undergoing simple harmonic motion, its kinetic energy and potential energy are constantly changing. At the extremes of its motion, when the displacement is equal to the amplitude A, the potential energy is at its maximum and the kinetic energy is at its minimum. Conversely, at the equilibrium position (when the displacement is zero), the potential energy is at its minimum and the kinetic energy is at its maximum.

The question asks when the kinetic energy and potential energy are equal. This occurs when the displacement is equal to A / √2. At this point, both the potential energy and kinetic energy are equal, resulting in a balanced energy state.

Explanation
The total energy of a simple harmonic oscillator is proportional to the square of the amplitude. This is because the amplitude represents the maximum displacement from the equilibrium position, and the energy of the oscillator is directly related to how far it is from equilibrium. As the amplitude increases, the oscillator has to overcome greater forces, resulting in higher energy. Therefore, the energy is proportional to the square of the amplitude.

Explanation
The time period is the only thing that remains constant for a particle performing Simple Harmonic Motion (SHM). In SHM, the particle oscillates back and forth around an equilibrium position, and the time taken to complete one full cycle of oscillation remains constant. This is true regardless of the amplitude or frequency of the motion. Velocity, acceleration, and force vary throughout the motion, but the time period remains the same.

Explanation
When the displacement of the particle is half of its amplitude, the potential energy of the particle is zero. At this point, all the total energy of the particle is in the form of kinetic energy. Therefore, the kinetic energy of the particle will be equal to the total energy, which is E.

Explanation
In simple harmonic motion (SHM), the total energy of the body remains constant throughout its motion. The total energy (E) is the sum of the kinetic energy and potential energy. According to the question, the kinetic energy is one-third of the total energy. Let's assume the kinetic energy is K and the potential energy is U. Therefore, E = K + U. Given that K = (1/3)E, we can substitute this value into the equation. (1/3)E = (1/2)mv^2, where m is the mass of the body and v is the velocity. Rearranging the equation, we get v^2 = (2/3)(E/m). Since v = ωa, where ω is the angular frequency and a is the amplitude, we can substitute this into the equation. (ωa)^2 = (2/3)(E/m). Simplifying further, we get ω^2a^2 = (2/3)(E/m). Since ω = √(k/m), where k is the spring constant, we can substitute this into the equation. (√(k/m))^2a^2 = (2/3)(E/m). Simplifying, we get a^2 = (2/3)(E/k). Taking the square root of both sides, we get a = √(2E/3k). Therefore, the displacement of the body having amplitude (a) is √(2a/√3), which matches the given answer of √2a/√3.

Explanation
When the frequency of a body moving with simple harmonic motion is doubled, it means that the body completes twice as many oscillations in the same amount of time. Since the amplitude remains the same, it implies that the distance covered by the body in each oscillation is also the same. Therefore, the maximum velocity, which is the maximum speed reached by the body during its oscillations, is doubled. This is because the body now covers the same distance in half the time, resulting in a higher speed.