Motion of a Particle

Abstract

In this chapter, we will learn how to depict the motion of one particle in space. We will consider the mechanics quantities in the Cartesian coordinate system and the natural coordinate system, respectively.

Exercises

• 7.1 As shown in Fig. 7.4, a particle is in a circular motion, from the origin O to an arbitrary point A. The radius of the circle is R = 2 mm, and the angular displacement $\varphi = 3t - \frac{1}{2}t^{2}$. When t = 1 s, please calculate the values of the angular velocity $\omega$, the angular acceleration $\varepsilon$, the arc length s, the velocity, and acceleration of point A. Please also schematize the velocity and acceleration of point A in the fiugre.

  Fig. 7.4

  

  A particle in a circular motion

• 7.2 Please derive the expressions of the tangential and normal accelerations in the natural coordinate system.

• 7.3 As shown in Fig. 7.5, a bar AB moves upward with the uniform velocity u, and at the initial state, the angle $\phi = 0$. The bar OC rotates with respect to point O, and there is a jacket linking the bar AB at point A. When the angle $\phi = 30^\circ$, please calculate the velocity and acceleration for point C. The angular velocity and angular acceleration of the bar are denoted by $\omega$ and $\varepsilon$, respectively.

  Fig. 7.5

  

  A mechanical system with two bars

Answers

• 7.1 $\omega = 3 - t\,{\text{rad/s}},\;\varepsilon = - 1\,{\text{rad/s}}$.

• 7.3 $v_{c} = \omega L_{OC} ,a_{c}^{n} = \omega^{2} L_{OC} ,a_{c}^{\tau } = \varepsilon L_{OC}$.