A friend of mine asked me to help him with this exercise. He has to deliver it tomorrow:
A small child with mass equal to m is on a swing with length l and has the ability to change the length of the swing l. In the beginning he starts from the point 0 (see in the figure) where the length of the swing is l+b and the angle is \(\displaystyle \phi_0\). While he is moving to the right when he pass the vertical point 1, the length changes to lb. Calculate the maximum angle \(\displaystyle \phi_f\) where the child is on the highest point of the orbit.
It is given that for small angles \(\displaystyle (1cos(\phi_0)=\frac{{\phi_0}^2}{2} \) .Also \(\displaystyle \frac{b}{l}<<1\) and
\(\displaystyle (1+\frac{b}{l})^3 \approx (1+3\frac{b}{l}) \)
This is my solution:
Since there is no information about the mass of the swing I assume its weight is negligible. Therefore the only returning force is the component of the weight.
The point 1 is where the potential energy is zero.
From the 0 point to 1 we have
\(\displaystyle mgy=\frac{1}{2}mu^2\)
From point 1 to 3 we have
\(\displaystyle mgy'=\frac{1}{2}mu^2\)
And because the second parts are equal so are the first parts.
\(\displaystyle gy=gy' \Rightarrow gxtan\phi_0 = gx' tan \phi_f \)
Because the angles are small \(\displaystyle tan\phi \approx sin \phi\)
Also \(\displaystyle x=(l+b)sin\phi x'=(lb)sin\phi \)
Therefore
\(\displaystyle gxtan\phi_0 = gx' tan \phi_f \Rightarrow (l+b) sin^2 \phi_0=(lb) sin^2 \phi_f \Rightarrow \)
\(\displaystyle \frac{1cos2\phi_0}{2} (l+b) = \frac{1cos2\phi_f}{2} (lb) \Rightarrow \)
\(\displaystyle \cdots \Rightarrow \phi_f=\sqrt{\frac{\phi_0^2(l+b)}{lb}}\)
Is everything ok?
I also have to point out that he cannot use Lagrange dynamics.
A small child with mass equal to m is on a swing with length l and has the ability to change the length of the swing l. In the beginning he starts from the point 0 (see in the figure) where the length of the swing is l+b and the angle is \(\displaystyle \phi_0\). While he is moving to the right when he pass the vertical point 1, the length changes to lb. Calculate the maximum angle \(\displaystyle \phi_f\) where the child is on the highest point of the orbit.
It is given that for small angles \(\displaystyle (1cos(\phi_0)=\frac{{\phi_0}^2}{2} \) .Also \(\displaystyle \frac{b}{l}<<1\) and
\(\displaystyle (1+\frac{b}{l})^3 \approx (1+3\frac{b}{l}) \)
This is my solution:
Since there is no information about the mass of the swing I assume its weight is negligible. Therefore the only returning force is the component of the weight.
The point 1 is where the potential energy is zero.
From the 0 point to 1 we have
\(\displaystyle mgy=\frac{1}{2}mu^2\)
From point 1 to 3 we have
\(\displaystyle mgy'=\frac{1}{2}mu^2\)
And because the second parts are equal so are the first parts.
\(\displaystyle gy=gy' \Rightarrow gxtan\phi_0 = gx' tan \phi_f \)
Because the angles are small \(\displaystyle tan\phi \approx sin \phi\)
Also \(\displaystyle x=(l+b)sin\phi x'=(lb)sin\phi \)
Therefore
\(\displaystyle gxtan\phi_0 = gx' tan \phi_f \Rightarrow (l+b) sin^2 \phi_0=(lb) sin^2 \phi_f \Rightarrow \)
\(\displaystyle \frac{1cos2\phi_0}{2} (l+b) = \frac{1cos2\phi_f}{2} (lb) \Rightarrow \)
\(\displaystyle \cdots \Rightarrow \phi_f=\sqrt{\frac{\phi_0^2(l+b)}{lb}}\)
Is everything ok?
I also have to point out that he cannot use Lagrange dynamics.
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