the piston in the cylinder head of a locomotive has a stroke twice the amplitude of 1m.if the piston moves with shm with an angular frequenzy of 200rev./min.,what is its maximum speed

To find the maximum speed of the piston in the locomotive's cylinder head, we need to break down the information given and apply some principles of simple harmonic motion (SHM). The key points to note are the amplitude of the motion, the angular frequency, and how these relate to maximum speed.

Understanding the Components

In SHM, the maximum speed (\(v_{\text{max}}\)) of an object can be calculated using the formula:

v_max = Aω

Where:

• A is the amplitude of the motion.
• ω is the angular frequency in radians per second.

Determining the Amplitude

The problem states that the stroke of the piston is twice the amplitude of 1 meter. Therefore, the amplitude (A) is:

A = 1 m

Calculating Angular Frequency

The angular frequency (ω) is given in revolutions per minute (rev/min). To use it in our formula, we need to convert it to radians per second. Since there are \(2π\) radians in one revolution, we can convert as follows:

ω = 200 rev/min × (2π rad/rev) × (1 min/60 s)

Calculating this gives:

ω = 200 × (2π/60) = (400π/60) rad/s ≈ 20.94 rad/s

Finding Maximum Speed

Now that we have both the amplitude and the angular frequency, we can substitute these values into the maximum speed formula:

v_max = Aω = 1 m × 20.94 rad/s

Calculating this results in:

v_max ≈ 20.94 m/s

Final Thoughts

Thus, the maximum speed of the piston in the locomotive's cylinder head is approximately 20.94 meters per second. This calculation illustrates how the principles of SHM can be applied to real-world mechanical systems, such as locomotives, where understanding motion is crucial for performance and efficiency.