The displacement of a follower of cam in printing machine is represented by the expression

X = 10 θ + 120 θ² + 1500 θ³ + 2000 θ⁴ + 2500 θ⁵

Where θ is the angle of rotation of the cam. The jerk given by the system at any position is 

Concept:

If X = f(θ) and θ = f(t)

Then, First derivative with respect to time represents the velocity

\(\dot x = \frac{{dx}}{{dt}} = \frac{{dx}}{{d\theta }}\frac{{d\theta }}{{dt}} = \omega \frac{{dx}}{{d\theta }}\)

The second derivative with respect to time represents the acceleration

\(\ddot x = \frac{{{d^2}x}}{{d{t^2}}} = {\omega ^2}\frac{{{d^2}x}}{{d{\theta ^2}}}\)

The third derivative with respect to time gives the jerk

\(\dddot x =\frac{{{d}^{3}}x}{d{{t}^{3}}}={{\omega }^{3}}\frac{{{d}^{3}}x}{d{{\theta }^{3}}}\)

Calculation:

Given,

Displacement (x) = 10 θ + 120 θ² + 1500 θ³ + 2000 θ⁴ + 2500 θ⁵

Velocity,  \(\left( \frac{dx}{dt} \right)=10\;\omega +240\;\theta \omega +4500\;{{\theta }^{2}}\omega +8000\;{{\theta }^{3}}\omega +12500\;{{\theta }^{4}}\omega \)

Acceleration, \(\left( \frac{{{d}^{2}}x}{d{{t}^{2}}} \right)=240\;{{\omega }^{2}}+9000\;\theta\; {{\omega }^{2}}+24000\;{{\theta }^{2}}{{\omega }^{2}}+50000\;{{\theta }^{3}}{{\omega }^{2}}\)