114 年 國立成功大學工程科學系碩士班丁組《工程力學（含靜力及動力）》

A collar that slides along a vertical rod is subjected to the three forces illustrated. Find:

(a) the value of $\alpha$ that makes the resultant of the three forces horizontal, and
(b) the magnitude of the resultant force under these conditions. (20%)

(hint: $\sin^2\alpha+\cos^2\alpha=1$)

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Determine the force in each member of the given truss using the method of joints. Indicate whether each member is experiencing tension or compression. (30%)

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(a) Consider a particle $P$, described by polar coordinates $r$ and $\theta$, moving in a plane under the influence of multiple forces (Fig. 3). Demonstrate how the forces acting on the particle can be resolved into radial and transverse components. (5%)

(b) Let particle $P$, with mass $m$, move with respect to a Newtonian reference frame $Oxyz$ (Fig. 4). Prove that the angular momentum of the particle can be fully expressed by the scalar $H_O$. (10%)

(c) Show that the sum of the moments about point $O$ of the forces acting on the particle is equal to the time rate of change of the particle's angular momentum (or moment of momentum) about $O$. (5%)

(d) If the only force acting on particle $P$ is a central force $\mathbf{F}$ directed toward or away from a fixed point $O$ (Fig. 5), prove that the angular momentum of the particle remains constant in both magnitude and direction. (5%)

(e) For a particle moving under the influence of a central force, demonstrate that its areal velocity remains constant. (5%)

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Consider a rigid body rotating about a fixed axis $AA'$. Let $P$ be a point on the body, and $\mathbf{r}$ represent its position vector relative to a fixed reference frame. Assume the reference frame is centered at point $O$ on axis $AA'$, with the $z$-axis aligned along $AA'$ (Fig. 6). Let $B$ be the projection of $P$ onto $AA'$. Since $P$ must remain at a constant distance from $B$, it traces a circular path with center $B$ and radius $r\sin\phi$, where $\phi$ is the angle between $\mathbf{r}$ and $AA'$.

(a) Derive the acceleration $\mathbf{a}$ of particle $P$, given by the expression:
$\vec{a} = \vec{\alpha} \times \vec{r} + \vec{\omega} \times \vec{\omega} \times \vec{r}$
(where $\vec{\alpha} \times \vec{r} =$ tangential acceleration component, $\vec{\omega} \times \vec{\omega} \times \vec{r} =$ radial acceleration component) (10%)

(b) To describe the rotation of a rigid body about a fixed axis, analyze the motion of a representative slab located in a reference plane perpendicular to the axis of rotation. Let the $xy$-plane serve as the reference plane, coinciding with the plane of the figure, and assume the $z$-axis points out of the page (Fig. 7). Show the given relationship:
$\mathbf{a} = \alpha\vec{k} \times \vec{r} - \omega^2\vec{r}$
(10%)

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