The motion of a particle on the surface of a right circular cylinder is defined by the relations R=A, 2,tθπ=and sin 2,zBntπ=where A and Bare constants and n is an integer. Determine the magnitudes of the velocity and acceleration of the particle at any time t.

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Accepted Answer

[latex] \begin{array}{lll} R=A & heta=2 \pi t & z=B \sin 2 \pi n t \ \dot{R}=0 & \dot{ heta}=2 \pi & \dot{z}=2 \pi n B \cos 2 \pi n t \ \ddot{R}=0 & \ddot{ heta}=0 & \ddot{z}=-4 \pi^2 n^2 B \sin 2 \pi n t \end{array} [/latex]

Velocity (Eq. 11.49)

[latex] \begin{aligned} &\begin{aligned} & \mathbf{v}=\dot{R} \mathbf{e}_R+R \dot{ heta} \mathbf{e}_ heta+\dot{z} \mathbf{k} \ & \mathbf{v}=\quad+A(2 \pi) \mathbf{e}_ heta+2 \pi n B \cos 2 \pi n t \mathbf{k} \end{aligned}\ &v=2 \pi \sqrt{A^2+n^2 B^2 \cos ^2 2 \pi n t} \end{aligned} [/latex]

Acceleration (Eq. 11.50)

[latex] \begin{aligned} &\begin{aligned} & \mathbf{a}=\left(\ddot{R}-R \dot{ heta}^2 ight) \mathbf{e}_R+(R \ddot{ heta}+2 \dot{R} \dot{ heta}) \mathbf{e}_ heta+\ddot{z} \mathbf{k} \ & \mathbf{a}=-4 \pi^2 A \mathbf{e}_k-4 \pi^2 n^2 B \sin 2 \pi n t \mathbf{k} \end{aligned}\ &a=4 \pi^2 \sqrt{A^2+n^4 B^2 \sin ^2 2 \pi n t} \end{aligned} [/latex]

Question 11.177: The motion of a particle on the surface of a right...

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