If a particle moves in simple harmonic motion with a frequency of 3.00 Hz and an amplitude of 5.00 cm through what total distance does the particle move during one cycle of its motion, what is its maximum speed, where does this maximum speed occur and what and where does the maximum acceleration of the particle happen?

20cm

94.2
17.8

To answer these questions, we need to understand the basic concepts of simple harmonic motion (SHM). SHM is a type of periodic motion where an object oscillates back and forth around an equilibrium position with a constant frequency and amplitude.

Frequency (f) is the number of complete cycles the particle completes in one second, measured in hertz (Hz). In this case, the frequency is given as 3.00 Hz.

Amplitude (A) is the maximum displacement of the particle from its equilibrium position. In this case, the amplitude is given as 5.00 cm.

Now, let's solve the questions one by one:

1. To find the total distance the particle moves during one cycle of its motion, we need to calculate the distance traveled during half a cycle and double it. The distance traveled during half a cycle is equal to the diameter of the circle formed by the particle's motion.

The diameter can be calculated using the formula:
Diameter = 2 * Amplitude

Here, the amplitude is given as 5.00 cm, so the diameter is:
Diameter = 2 * 5.00 cm = 10.00 cm

To find the total distance, we double the diameter:
Total distance = 2 * Diameter
Total distance = 2 * 10.00 cm = 20.00 cm

Therefore, the particle moves a total distance of 20.00 cm during one cycle of its motion.

2. The maximum speed of the particle in SHM occurs when it passes through the equilibrium position. At the equilibrium position, the particle momentarily stops and changes its direction. The maximum speed can be calculated using the formula:

Maximum speed = 2π * Amplitude * Frequency

Amplitude is given as 5.00 cm and frequency is given as 3.00 Hz. Substituting these values into the formula, we get:

Maximum speed = 2π * 5.00 cm * 3.00 Hz
Maximum speed ≈ 94.25 cm/s

Therefore, the maximum speed of the particle is approximately 94.25 cm/s.

3. The maximum acceleration of the particle occurs at the extreme positions of its motion, which are the furthest points from the equilibrium position. At these positions, the particle momentarily changes its direction and achieves the maximum acceleration.

The maximum acceleration can be calculated using the formula:

Maximum acceleration = (2π * Amplitude * Frequency)^2

Amplitude is given as 5.00 cm and frequency is given as 3.00 Hz. Substituting these values into the formula, we get:

Maximum acceleration = (2π * 5.00 cm * 3.00 Hz)^2
Maximum acceleration ≈ 2827.43 cm^2/s^2

Therefore, the maximum acceleration of the particle is approximately 2827.43 cm^2/s^2.

Note: The exact location or position where the maximum speed and maximum acceleration occur can be found using trigonometry, but it is not specified in this question.

To find the total distance the particle moves during one cycle of its motion, we need to find the circumference of the circle traced by the particle in simple harmonic motion. The circumference can be calculated using the formula:

Circumference = 2 * π * radius

In this case, the radius is equal to the amplitude of motion. Therefore, the total distance the particle moves during one cycle is:

Distance = Circumference = 2 * π * Amplitude = 2 * π * 5.00 cm = 31.4 cm

So, the particle moves a total distance of 31.4 cm during one cycle of its motion.

The maximum speed of the particle occurs when it passes through the equilibrium position. The speed at any point in simple harmonic motion can be calculated using the formula:

Speed = 2 * π * frequency * amplitude

In this case, the frequency is 3.00 Hz and the amplitude is 5.00 cm. Therefore, the maximum speed of the particle is:

Speed = 2 * π * 3.00 Hz * 5.00 cm = 30.0 cm/s

The maximum speed of the particle is 30.0 cm/s and it occurs when the particle passes through the equilibrium position.

The maximum acceleration of the particle occurs at the extreme points of its motion, where the displacement is maximum. The maximum acceleration can be calculated using the formula:

Acceleration = (2 * π * frequency)^2 * amplitude

In this case, the frequency is 3.00 Hz and the amplitude is 5.00 cm. Therefore, the maximum acceleration of the particle is:

Acceleration = (2 * π * 3.00 Hz)^2 * 5.00 cm = 564 cm/s^2

The maximum acceleration of the particle is 564 cm/s^2 and it occurs at the extreme points of its motion.