Dynamics Final SA

What is moment of inertia? What's its unit?

$I=\int_{m}^{}\!I^2\,dm$ [kg*m²]

What is parallel-axis theorem?

$I=I_{G}+md^2$ [kg*m²]

What is radius of gyration?

$$k=\sqrt{\frac{I}{m}}$$

Write the equations of motion when the object is subjected to rectilinear translation motion.

$$\Sigma F_{x}=m\left(a_{G}\right)_{x}$$

$\Sigma F_{y}=m\left(a_{G}\right)_{y}$  

$$\Sigma M_{G}=0$$

Write the equations of motion when the object is subjected to curvilinear translation.

$$\Sigma F_{n}=m\left(a_{G}\right)_{n}$$

$\Sigma F_{t}=m\left(a_{G}\right)_{t}$   

$$\Sigma M_{G}=0$$

Write the equations of motion when the object is subjected to rotation about a fixed axis.

$$\Sigma F_{n}=m\left(a_{G}\right)_{n}=m\omega^2r_{G}$$

$$\Sigma F_{t}=m\left(a_{G}\right)_{t}=m\alpha r_{G}$$

$$\Sigma M_{G}=I_{G}\alpha$$

Write the equations of motion when the object is subjected to general plane motion.

$$\Sigma F_{n}=m\left(a_{G}\right)_{n}$$

$\Sigma F_{t}=m\left(a_{G}\right)_{t}$   

$$\Sigma M_{G}=I_{G}\alpha$$

Explain the concept of kinetic energy for a planar rigid body in general motion.

$T=\frac12mv_{G}^2+\frac12I_{G}\omega^2$; the kinetic energy is equal to the sum of translational and rotational kinetic energy.

Describe the work done by a variable force.

$$U_{F}=\int_{}^{}\!F\cos\theta\,dS$$

Describe the work done by a constant force.

$$U_{F}=\left(F_{c}\cos\theta\right)\Delta S$$

Describe the work done by gravity.

$$U_{W}=-W\Delta y$$

Describe the work done by a spring.

$$U_{S}=-\left(\frac12ks_2^2-\frac12ks_1^2\right)$$

Explain the concept of work done by a couple moment.

$W=M\cdot\theta$; the work done is the energy transferred when the couple moment causes an object to rotate.

Discuss the principle of work and energy.

$$T_1+\Sigma U_{1-2}=T_2$$

Discuss conservation of energy.

$$T_1+V_1=T_2+V_2$$

Discuss the key differences between a particle and planar body in the context of the principle of work and energy and conservation of energy.

Planar bodies include translational and kinetic energy, and work can be done by torques (moments), while particles only include translational kinetic energy.

Define angular momentum for a particle, including its unit and vector form.

$\vec{H_{G}}=\vec{I_{G}}\vec{\omega}$ ; $\vec{H_{G}}=\vec{r}\times\vec{p}$ ; [kg*m²']

Explain the concepts of linear momentum and angular momentum for a planar rigid body undergoing general plane motion.

General plane motion combines both linear momentum $p=m{v_G}$, measures translational motion, and angular momentum ${H_G}=I_{G}\omega$, measures rotational motion.

Describe the relationship between moment of a force and angular momentum.

$$\Sigma M_{O}=\dot{H_{O}}$$

Explain the principle of angular impulse and momentum for both 1-D and 2-D scenarios.

1-D: $\Sigma\int_{}^{}Mdt=I\omega_2-I\omega_1$

2-D: $\Sigma\int_{}^{}M_{O}dt=\left(H_{O}\right)_2-\left(H_{O}\right)_1$  

Discuss the concept of conservation of angular momentum.

$\Sigma\left(H_{O}\right)_1=\Sigma\left(H_{O}\right)_2$; from t₁ to t₂ the particles angular momentum stays constant.

Interpret equations ${H_p}=-\bar{y}m({v_G})_{x}+\overline{x}m({v_G})_{y}+I_{G}\omega$; $\Sigma{M_p}=-\bar{y}m({a_{G}})_{x}+\overline{x}m({a_{G}})_{y}+I_{G}\alpha$. What's the relationship between ${H_p}$ and $\Sigma{M_p}$?

$\Sigma M_{p}=\dot{H_{p}}$; ${a_G}=\dot{v_G}$, $\alpha=\dot{\omega}$