Do only first $3$

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the applied torque on a particle about any arbitrary axis is zero, what can you say about its angular momentum about that axis?

Calculate the net torque $($magnitude and direction$)$ on the beam in Figure about $($a$)$ an axis through $O$ perpendicular to the page and $($b$)$ an axis through $C$ perpendicular to the page.

In an amusement park, a merry$-$go$-$round of radius $2\mathrm{.}0$ m and $150-kg$ is turning at a rate of $32\mathrm{.}0$ rpm in horizontal plane. Assuming the merry$-$go$-$round is a uniform disk find its angular momentum and kinetic energy.

Which has greater angular momentum: a solid sphere of mass $m$ rotating at a constant angular frequency ${}_0$ about the z $-$axis, or a solid cylinder of same mass and rotation rate about the z $-$axis? $(1)$

An artificial satellite is placed into an elliptical orbit about the earth, as illustrated in Figure. Telemetry data indicate that its point of closest approach $($called the perigee$)$ is $r_p=8\mathrm{.}37{10}^6$ m from the center of the earth, and its point of greatest distance $($called the apogee$)$ is $r_{A}25\mathrm{.}1{10}^{6}m$ from the center of the earth. The speed of the satellite at the perigee is $v_p=8450\frac{m}{s}.$ Find its speed $v_A$ at the apogee.

The drawing shows a wire tooth brace used by orthodontists. The topmost tooth is protruding slightly, and the tension in the wire exerts two forces $T^{\text{'}}$ and $T$ on this tooth in order to bring it back into alignment. If the forces have the same magnitude of $21\mathrm{.}0$ N $,$ what is the magnitude of the net force exerted on the tooth by these forces?

A light, rigid rod of length, $l=1\mathrm{.}00m$ joins two particles, with masses $m1=4\mathrm{.}00kg$ and $m2=3\mathrm{.}00$ kg $,$ at its ends. The combination rotates in the $xy$ plane about a pivot through the center of the rod $($Figure below$).$ Determine the angular momentum of the system about the origin when the speed of each particle is $5\mathrm{.}00\frac{m}{s}.$

$(3)$