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# The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m

Q. 14.14 The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of $1.0 m$ If the piston moves with simple harmonic motion with an angular frequency of  $200\; rad/min,$  what is its maximum speed ?

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Amplitude of SHM = 0.5 m

angular frequency is

$\\\omega =200\ rad/min \\\omega =3.33\ rad/s$

If the equation of SHM is given by

$\\x(t)=Asin(\omega t+\phi )\\$

The velocity would be given by

$\\v(t)=\frac{\mathrm{d} x(t)}{\mathrm{d} t}\\ v(t)=\frac{\mathrm{d} (Asin(\omega t+\phi ))}{\mathrm{d} t}\\ v(t)=A\omega cos(\omega t+\phi )$

The maximum speed is therefore

$\\v_{max}=A\omega \\ v_{max}=0.5\times 3.33 \\v_{max}=1.67ms^{-1}$

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