Question

The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of $1\,m$. if the piston moves with SHM with an angular frequency of $200\,rad/min$. What is its maximum speed?

Answer 1

Here the piston is undergoing a SHM, simple harmonic motion. In case of simple harmonic motion, the body moves to and fro, that is it repeats its motion after a regular interval of time. There is one position of the body which is called equilibrium position or the mean position. When the body is displaced from its mean position it tends to come back.

Complete step by step answer:
The maximum displacement from the mean position is called the amplitude. Given, the stroke is twice the amplitude, thus amplitude, A= 0.5 m. Also, the angular velocity is given in rev/min and we do not convert it into rev/s. The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1 m. if the piston moves with SHM with an angular frequency of 200 rad/min. What is its maximum speed?
$\omega =200\,rev/\min$
We know the relationship between the maximum velocity, angular velocity and amplitude for a body undergoing SHM is given by:
${{v}_{\max }}=\omega A \\\ \Rightarrow {{v}_{\max }}=200\times 0.5 \\\ \therefore {{v}_{\max }}=100m/\min$

So, the maximum speed comes out to be 100m/min.

Additional information:
Simple harmonic motion can be described as an oscillatory motion in which the acceleration of the particle at any position is directly proportional to the displacement from the mean position. It is a special case of oscillatory motion. All the Simple Harmonic Motions are oscillatory and also periodic but not all oscillatory motions are SHM. Oscillatory motion is also called the harmonic motion of all the oscillatory motions wherein the most important one is simple harmonic motion (SHM).

Note: The pendulum has two extreme positions and one mean position. At the two extreme positions, the energy is potential and in between the two extreme positions the energy is potential plus kinetic and at the mean position, the energy is kinetic. Also, at both the extreme position the body has maximum velocity and at the mean position the velocity of the body is zero.