Tutorial Sheet 1: Planar Kinematics
Topics covered are:
- Types of motion
- Rotation around a fixed axis
- Relative velocity
- Instantaneous centres
Tips
- Always draw the situation!
- The order of the cross product matters, $\omega\times r \neq r\times\omega$
Question 1
At the instant shown, the disk has angular velocity is 2 rad/s counter clockwise and angular acceleration 6 rad/s $^2$. Its radius is 0.2 m.
What are the magnitudes of the velocity and acceleration of point A?
Answer
\[1.44m/s^2\]Question 2
The mass A starts from rest at t=0 and falls with a constant acceleration of 8 m/s $^2$. When the mass has fallen one meter, determine the magnitudes of:
(a) The angular velocity of the pulley.
(b) The tangential and normal components of acceleration of a point at the outer edge of the pulley.
Answer
(a)
\[40 rad/s\](b)
\[8m/s^2\]Question 3
(a) If the bicycle’s 120 mm radius sprocket wheel rotates through one revolution, through how many revolutions does the 45 mm gear turn?
(b) If the angular velocity of the sprocket wheel is 1 rad/s, what is the
angular velocity of the gear?
Answer
(a)
\[x=2.67 \text{ revs}\](b)
\[2.67 rad/s\]Question 4
The disk is rotating about the origin with a constant clockwise angular velocity of 100 rpm. Determine the 𝑥 and 𝑦 components of velocity of points 𝐴 and 𝐵 (in cm/s).
Answer
\[v_B=-167.55j cm/s\]Question 5
The bar is moving in the x–y plane and is rotating in the counterclockwise direction. The magnitude of the velocity of point A relative to point B is 8 m/s, and the velocity of point A relative to the origin is 0. Relative to a nonrotating reference frame with origin A, what is the
(a) Angular velocity of the bar.
(b) Velocity of B relative to the reference frame in vector form.
Answer
(a)
\[4 rad/s\](b)
\[-4i + 6.93j\]Question 6
The bar is rotating in the counterclockwise direction with angular velocity ω. The magnitude of the velocity of point A relative to point B is 6 m/s. Determine the velocity of point B (relative to the origin).
Answer
\[v_B = 1.2i+2.4j \text{ m/s}\]Question 7
The helicopter is in planar motion in the x–y plane. At the instant shown, the position of its center of mass, G, is x=2m, y=2.5m, and its velocity is $v_G=12i+4j$ (m/s). The position of point T, where the tail rotor is mounted, is x= −3.5m, y=4.5m. The helicopter’s angular velocity is 0.2 rad/s clockwise. What is the velocity of point T?
Answer
\[12.4i+5.1j \text{ m/s}\]Question 8
At the instant shown, the piston’s velocity is $v_C = −14i$ m/s. What is the angular velocity of the crank AB, which rotates around A?
Answer
\[\omega_{BA}=218 \text{ rad/s}\]Question 9
Points A and B of the 2 m bar slide on the plane surfaces. Point B is moving to the right at 3 m/s. What is the velocity of the midpoint G of the bar?
Answer
\[1.5i- 0.547j \text{ m/s}\]Question 10
Bar AB rotates in the counterclockwise direction at 6 rad/s. Determine the angular velocity of bar BD and the velocity of point D.
Answer
\[v_D=6.4i-1.28j\]Question 11
The horizontal member ADE supporting the scoop is stationary. If the link BD is rotating in the clockwise direction at 1 rad/s,what is the angular velocity of the scoop?
Answer
\[\omega_{CE}=-1.47 \text{ rad/s}\]Question 12
The velocity of point O of the bat is $v_O$ =−1.83i− 4.27j m/s, and the bat rotates about the z axis with a counterclockwise angular velocity of 4 rad/s. What are the x and y coordinates of the bat’s instantaneous center?
Answer
\[(1.07,-0.46) \text{ m}\]Question 13
Points A and B of the 1m bar slide on the plane surfaces. The velocity of B is $v_B$ = 2i m/s.
(a) What are the coordinates of the instantaneous center of the bar?
(b) Use the instantaneous center to determine the velocity at A.
Answer
(a)
\[(0.34, 0.94) m\](b)
\[-0.724j \text{ m/s}\]