Spring pendulum, Kinetic Energy

In the included picture, I don't get how they get to the kinetic energy part. I would say, that the travelled distance is equal to (l + x(t))*θ. Then I would take the time derivative, resulting in dx(t)/dt * θ + (l + x(t))* dθ/dt. Then I would square this result and multiply that with 1/2 m. But then I would get a totally different kinetic energy.

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mfb
Mentor
I would say, that the travelled distance is equal to (l + x(t))*θ
What does that mean? The travelled distance depends on the trajectory. You'll need a position (as vector) to get a meaningful derivative.

Vishwaas
You must also consider the radial component. The velocity component in the radial direction is the time derivative of x(t) (since it is a spring, it can compress or expand). Hence the total kinetic energy is the sum of energies in both radial and angular directions.

vanhees71
$$\vec{x}=r \begin{pmatrix} \sin \varphi \\ \cos \varphi \end{pmatrix}.$$
$$T=\frac{m}{2} \dot{\vec{x}}^2=\frac{m}{2} (\dot{r}^2+r^2 \dot{\varphi}^2).$$
$$r=l+x,$$
$$V=-m g x_2+\frac{k}{2} x^2 = -m g (x+l) \cos \varphi+\frac{k}{2} x^2.$$