All questions of Driven (Forced) Oscillations and Resonance for EmSAT Achieve Exam

In an ideal environment where there is no resistance to oscillatory motion, that is, damping is zero, when we oscillate a system at its resonant frequency, since there is no opposition to oscillation, a very large value of amplitude will be recorded. Forced oscillation is when you apply an external oscillating force.

**Explanation:**

When a particle is executing Simple Harmonic Motion (SHM), its displacement from the mean position follows a sinusoidal function. The maximum displacement of the particle is called the amplitude (A) of the motion. The particle oscillates back and forth between two extreme positions, which are equal in magnitude but opposite in direction.

The time taken for one complete oscillation of the particle is called the time period (T) of the motion. It is the time taken for the particle to go from one extreme position to the other and back again.

The maximum velocity (vmax) of the particle occurs when the particle passes through the mean position (equilibrium position). At this point, the velocity is at a maximum and is in the same direction as the acceleration. The maximum acceleration (amax) of the particle occurs when the particle is at its extreme positions. At these points, the acceleration is at a maximum and is in the opposite direction to the displacement.

If the maximum velocity and acceleration of the particle are equal in magnitude, it means that the amplitude of the motion is equal to the acceleration. Mathematically, this can be represented as:

amax = vmax

Using the equations of SHM, we can relate the amplitude, time period, maximum velocity, and maximum acceleration as follows:

amax = ω^2 A

vmax = ω A

where ω is the angular frequency of the motion.

Since amax = vmax, we can equate the above two equations:

ω^2 A = ω A

Simplifying the equation:

ω^2 = ω

ω = 1

The angular frequency ω is related to the time period T by ω = 2π/T. Substituting this value into the equation:

(2π/T)^2 = 2π/T

Simplifying the equation:

4π^2/T^2 = 2π/T

2π/T = 4π^2/T^2

T = 2

Therefore, the time period of the motion is T = 2 seconds, which is equivalent to 6.28 seconds (approximately). Hence, the correct answer is option A: 6.28 s.

Harmonic motion is the natural motion of a body(we consider no air friction) under no force where as damped oscillation are under force hence the iscilation are different

Damping force is a resistance force that opposes the motion of an object in a medium, such as air or water. It is responsible for reducing the amplitude of oscillations or the speed of a moving object. In simple models, the damping force is directly proportional to velocity.

Explanation:
The damping force is caused by the interaction between the object and the medium it is moving through. As the object moves, it displaces the medium particles, causing them to collide and exert a resistance force on the object. This resistance force is proportional to the velocity of the object.

Mathematical Representation:
The mathematical representation of damping force can be given by the equation:
F_damping = -bv

Where F_damping is the damping force, b is the damping coefficient, and v is the velocity of the object.

Directly Proportional to Velocity:
According to the equation, the damping force is directly proportional to the velocity of the object. This means that as the velocity increases, the damping force also increases, resulting in a greater resistance to the object's motion. Conversely, if the velocity decreases, the damping force decreases as well.

Relationship with Acceleration:
Although the damping force is not directly proportional to acceleration, it does have an indirect relationship with it. Since acceleration is the rate of change of velocity, an increase in acceleration results in a greater change in velocity over time. Therefore, a larger acceleration would lead to a larger damping force, and vice versa.

Conclusion:
In simple models, the damping force is directly proportional to velocity. This relationship is important in understanding the behavior of objects in the presence of damping forces, as it helps explain the gradual decrease in amplitude or speed of oscillations or motion.

The Period of a Simple Pendulum
The period of a simple pendulum is determined by the formula:
T = 2π√(L/g)
where:
- T is the period
- L is the length of the pendulum
- g is the acceleration due to gravity
Understanding the Influence of Length on Period
- The period T is directly proportional to the square root of the length L.
- This means if you increase the length L, the period T increases.
Explaining the Correct Answer (Option C)
- If the length of the pendulum is increased four times (L becomes 4L), the new period T' can be calculated as follows:
T' = 2π√(4L/g) = 2π√(4)√(L/g) = 2 * 2π√(L/g) = 2T
- This shows that the period T' is now doubled.
Analysis of Other Options
- Option A: Doubling mass and length does not double the period. The mass does not impact the period; it only affects the amplitude.
- Option B: Doubling the mass of the bob does not change the period at all; it remains T.
- Option D: Doubling the length (L becomes 2L) results in:
T' = 2π√(2L/g) = √2 * T (not doubled).
Conclusion
Thus, the only scenario where the period of a simple pendulum is doubled is when its length is increased four times, making Option C the correct answer.

Periodic motion is a type of motion that repeats itself at regular intervals of time. It is characterized by the recurring pattern of the motion, where the object or system returns to its initial state after a certain period of time. This motion can be observed in various natural and man-made phenomena, such as the swinging of a pendulum, the rotation of the Earth around its axis, the vibration of a guitar string, or the oscillation of a spring.

Characteristics of periodic motion:

- Repetition: Periodic motion involves the repetition of a certain pattern or motion over time. The object or system returns to its initial position, velocity, and other properties after completing one cycle of motion.

- Regular intervals: The time taken to complete one cycle of motion is constant and is known as the period. This means that the motion occurs at regular intervals of time. For example, the period of a pendulum is the time taken to swing from one extreme position to the other and back again.

- Consistent amplitude: In periodic motion, the amplitude remains constant throughout the motion. The amplitude represents the maximum displacement or distance from the mean position.

- Harmonic motion: Many periodic motions, such as the swinging of a pendulum or the vibration of a guitar string, can be described by simple harmonic motion. Simple harmonic motion is a type of periodic motion where the restoring force is directly proportional to the displacement from the mean position and acts in the opposite direction.

- Periodic function: The mathematical representation of periodic motion is often described by a periodic function, such as a sine or cosine function. These functions repeat themselves after a certain interval, corresponding to one cycle of the motion.

- Examples of periodic motion: Some common examples of periodic motion include the swinging of a pendulum, the rotation of the Earth around its axis, the vibration of a guitar string, the oscillation of a spring, the motion of a bouncing ball, and the orbit of planets around the sun.

In conclusion, periodic motion is a type of motion that repeats itself at regular intervals of time. It is characterized by the recurring pattern of the motion, where the object or system returns to its initial state after completing one cycle of motion.