= Solution
{c}
The <massless inextensible string> has a common <tension> magnitude $T$. Since $\mathbf r_1=r_1(\mathbf e_r-\mathbf e_z)$, the <unit vector> from the first particle towards the apex is $(-\mathbf e_r+\mathbf e_z)/\sqrt2$. The second particle lies below the apex, so
$$
\boxed{\mathbf T_1=\frac{T}{\sqrt2}(-\mathbf e_r+\mathbf e_z),\qquad \mathbf T_2=T\mathbf e_z.}
$$
The <string length constraint> is
$$
l=\sqrt2r_1-z_2=-\sqrt2z_1-z_2,\qquad \dot r_1=-\dot z_1,\qquad \dot z_2=-\sqrt2\dot z_1.
$$
The <velocity> of the first particle in <cylindrical coordinates> is $\dot r_1\mathbf e_r+r_1\dot\theta\mathbf e_\theta+\dot z_1\mathbf e_z$. Therefore the <powers> delivered by the two <tension> <forces> are
$$
\mathbf T_1\cdot\dot{\mathbf r}_1=\frac T{\sqrt2}(-\dot r_1+\dot z_1)=\sqrt2T\dot z_1,\qquad \mathbf T_2\cdot\dot{\mathbf r}_2=T\dot z_2=-\sqrt2T\dot z_1.
$$
Thus
$$
\boxed{\mathbf T_1\cdot\dot{\mathbf r}_1=-\mathbf T_2\cdot\dot{\mathbf r}_2.}
$$
The <tension> can transfer <energy> between the particles, but its total <power> on the pair is zero.
Back to article page
