Q: Given $r(t) = \alpha_1 - \beta_1 \sin(\gamma_1 \pi t)$ and $\theta(t) = \alpha_2 - \beta_2 \cos(\gamma_2 \pi t)$, what is the velocity vector of point A in polar coordinates?

$\vec{v}_A = \dot{r} \hat{e}_r + r \dot{\theta} \hat{e}_\theta$

Q: For the crane system, what is the expression for the acceleration vector of point A in polar coordinates?

$\vec{a}_A = (\ddot{r} - r \dot{\theta}^2) \hat{e}_r + (r \ddot{\theta} + 2 \dot{r} \dot{\theta}) \hat{e}_\theta$

Question 1

For the industrial robot shown in Figure 1, the vertical shaft rotates at a constant angular velocity $\omega$. The length $h$ of the shaft and $l$ of the arm are functions of time. What is the expression for the velocity magnitude of point P?

Accepted Answer

$v_P = \sqrt{\dot{h}^2 + \dot{l}^2 + (l + l_0)^2 \omega^2}$

Question 2

Which of the following correctly represents the centripetal acceleration of point P in the robot arm system?

Accepted Answer

$a_{c} = (l + l_0) \omega^2$

Question 3

In the robot arm system, what is the expression for the tangential acceleration of point P?

Accepted Answer

$a_{t} = (l + l_0) \dot{\omega}$

Question 4

For the robot arm, what is the Coriolis acceleration experienced by point P?

Accepted Answer

$a_{cor} = 2 \dot{l} \omega$

Question 5

Given $r(t) = \alpha_1 - \beta_1 \sin(\gamma_1 \pi t)$ and $	heta(t) = \alpha_2 - \beta_2 \cos(\gamma_2 \pi t)$, what is the velocity vector of point A in polar coordinates?

Accepted Answer

$\vec{v}_A = \dot{r} \hat{e}_r + r \dot{	heta} \hat{e}_	heta$

Question 6

For the crane system in Figure 2, if $\gamma_1 = \sqrt{2}$ and $\gamma_2 = 1$, what can be said about the periodicity of the trajectory of point A?

Accepted Answer

The trajectory is not periodic because $\gamma_1/\gamma_2$ is irrational.

Question 7

If $\alpha_1 = 1$, $\beta_1 = 1$, $\gamma_1 = 1$, $\alpha_2 = 1$, $\beta_2 = 1$, $\gamma_2 = 1$, what is the initial position of point A at $t = 0$?

Question 8

For the pendulum whose suspension point $OP$ moves as $\vec{OO_P} = f(t) \hat{x} + g(t) \hat{y}$, what is the velocity of the mass point P in the fixed frame $(O; \hat{x}, \hat{y})$?

Accepted Answer

$\vec{v}_P = \dot{f}(t) \hat{x} + \dot{g}(t) \hat{y} + l \dot{	heta} \hat{e}_	heta$

Question 9

In the pendulum system, what is the acceleration of the mass point P in the fixed frame $(O; \hat{x}, \hat{y})$?

Accepted Answer

$\vec{a}_P = \ddot{f}(t) \hat{x} + \ddot{g}(t) \hat{y} + l \ddot{	heta} \hat{e}_	heta - l \dot{	heta}^2 \hat{e}_r$

Question 10

For the robot arm, if $h$ and $l$ are constant and only $\omega$ is changing, which acceleration components are nonzero?

Accepted Answer

Tangential acceleration only

Question 11

In the crane system, if $\gamma_1 = 2$ and $\gamma_2 = 1$, how does the trajectory of point A compare to the case where $\gamma_1 = \gamma_2 = 1$?

Accepted Answer

The trajectory is periodic but with a different period.

Question 12

For the pendulum, if the suspension point moves with constant velocity, what effect does this have on the pendulum's motion in the fixed frame?

Accepted Answer

The pendulum's mass point experiences an additional constant velocity component.

Question 13

In the robot arm system, if $\dot{l}$ is nonzero and $\omega$ is constant, which acceleration components are present?

Accepted Answer

Coriolis and centripetal acceleration

Question 14

For the crane system, what is the expression for the acceleration vector of point A in polar coordinates?

Accepted Answer

$\vec{a}_A = (\ddot{r} - r \dot{	heta}^2) \hat{e}_r + (r \ddot{	heta} + 2 \dot{r} \dot{	heta}) \hat{e}_	heta$

Question 15

In the pendulum system, if $f(t)$ and $g(t)$ are sinusoidal functions, what is the nature of the motion of the mass point P?

Accepted Answer

The motion is a superposition of pendulum oscillation and periodic translation.

Question 16

For the robot arm, if $l$ increases linearly with time and $\omega$ is constant, what happens to the centripetal acceleration?

Accepted Answer

It increases linearly with time.

Question 17

In the crane system, if $\beta_1 = 0$, what is the nature of the trajectory of point A?

Accepted Answer

The trajectory is a circle with radius $\alpha_1$.

Kinematics and Dynamics of Rotating and Moving Systems

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