Tutorial Sheet 2: Planar Kinematics with Acceleration
Topics covered are
- Acceleration in 2D
- Relative accelerations
Tips
- The questions start tp get quite wordy. Drawing it all out helps!
- Because we are dealing with 2D planar motion, it does not matter which relative acceleration formula we use - the one using $\omega\times\omega$ or $\omega^2$. When expanding to 3D, only $\omega\times\omega$ works, but for the moment $\omega^2$ is faster.
Question 1
The rigid body rotates about the z axis with counterclockwise angular velocity ω = 4 rad/s and counterclockwise angular acceleration α = 2 rad/s $^2$. The distance $r_{A/B}$ = 0.6m.
(a) What are the rigid body’s angular velocity and angular acceleration vectors?
(b) Determine the acceleration of point A relative to point B.
Answer
(a)
\[\alpha = 2k\](b)
\[a_{A/B} = -9.6i+1.2j \text{ m/s}^2\]Question 2
The helicopter is in planar motion in the xy plane. At the instant shown, the position of its center of mass G is x=2 m, y=2.5 m, its velocity is $v_G$ = 12i + 4j m/s, and its acceleration is $a_G$ = 2i + 3j m/s $^2$. The position of point T where the tail rotor is mounted is x=-3.5 m, y=4.5 m. The helicopter’s angular velocity is 0.2 rad/s clockwise, and its angular acceleration is 0.1 rad/s $^2$ counter-clockwise.
What is the acceleration of point T?
Answer
\[a_T= 2.02i+2.37j \text{ m/s}^2\]Question 3
The bar rotates with a counterclockwise angular velocity of 5 rad/s and a counterclockwise angular acceleration of 30 rad/s $^2$. Determine the acceleration of A using $ a_{A}=a_B+\alpha\times r_{A/B}+\omega\times(\omega\times r_{A/B}) $.
Answer
\[-73.3i+27j \text{ m/s}^2\]Question 4
If $\omega_{AB}$=2rad/s, $\alpha_{AB}$=2rad/s $^2$, $\omega_{BC}$=−1rad/s, and $\alpha_{BC}$=−4rad/s $^2$, what is the acceleration of point C where the scoop of the excavator is attached?
Answer
\[-24.1i-18.3j \text{ m/s} ^2\]Question 5
The length of the bar is L = 4 m and the angle $\theta$ = 30°. The bar’s angular velocity is $\omega$ = 1.8 rad/s and its angular acceleration is $\alpha$ = 6 rad/s $^2$. The endpoints of the bar slide on the plane surfaces. Determine the acceleration of the midpoint G.
Answer
\[a_G = 7.15i-11.6 \text{ m/s} ^2\]Question 6
The angular velocity’s magnitude $\omega_{AB}$ = 6 rad/s. If the acceleration of the slider C is zero at the instant shown, what is the angular acceleration $\alpha_{AB}$?
Answer
\[19\text{ m/s} ^2 \text{ clockwise}\]Question 7
At the instant shown, the piston’s velocity and acceleration are $v_C$ =−14i m/s and $a_C$ = −2200i m/s $^2$. What is the angular acceleration of the crank AB?
Answer
AB rotates 3580 rad/s $^2 $ clockwise
Question 8
If arm AB has a constant clockwise angular velocity of 0.8 rad/s, arm BC has a constant angular velocity of 0.2 rad/s, and arm CD remains vertical, what is the acceleration of part D?
Answer
\[-0.135i-0.144j \text{ m/s}^2\]Question 9
Point A of the rolling disk is moving toward the right and accelerating toward the right. The magnitude of the velocity of point C is 2 m/s, and the magnitude of the acceleration of point C is 14 m/s $^2$. Determine the angular acceleration of the disk.
Answer
\[\alpha = 20 \text{ m/s}^2 \text { in the negative direction}\]Question 10
The disk rolls on the circular surface with a constant clockwise angular velocity of 1 rad/s. What are the accelerations of points A and B?
Answer
\(a_A=-0.5j\text{ m/s}^2\)
\(a_B=0.3j\text{ m/s}^2\)