Tutorial Sheet 3: Moving Systems
Topics covered are
- Body fixed and moving reference frames
- The concept of sliding contacts
- Cross product in 3D
Tips
- Really try to visualise these mechanisms, physically drawing on where each reference frame can help to prevent errors.
- You don’t need to include all terms in your expressions if you know they are zero, however in the early questions I have included all terms for completeness.
- Remember unit vectors…
Question 1
Sleeve C slides at 1 m/s relative to bar BD. Use the body-fixed coordinate system shown to determine the velocity of C.
Answer
\[-0.2i+2.8j \text{ m/s}\]Question 2
Using the same system as Question 1, the angular accelerations of the two bars are zero and the sleeve C slides at a constant velocity of 1 m/s relative to bar BD. What is the acceleration of C?
Answer
\[-8.8i+5.6j \text{ m/s}^2\]Question 3
Bar AB has an angular velocity of 4 rad/s in the clockwise direction. What is the velocity of pin B relative to the slot?
Answer
\[-0.548\text{ m/s}\]Question 4
The coordinate system is fixed relative to the ship B. At the instant shown, the ship is sailing north at 5 m/s relative to the earth, and its angular velocity is 0.26 rad/s counterclockwise. Using radar, it is determined that the position of the aeroplane is 1080i+1220j+6300k m and its velocity relative to the ship’s coordinate system is 870i−45j−21k m/s. What is the aeroplane’s velocity relative to the earth?
Answer
\[553i+24j-21k \text{ m/s}\]Question 5
The space shuttle is attempting to recover a satellite for repair. At the current time, the satellite’s position relative to a coordinate system fixed to the shuttle is 50i m. The gyroscopes on the shuttle indicate that its current angular velocity is 0.05j+0.03k rad/s. The shuttle pilot measures the velocity of the satellite relative to the body-fixed coordinate system and determines it to be −2i−1.5j+2.5k rad/s. What are the x, y, and z components of the satellite’s velocity relative to a nonrotating coordinate system with its origin fixed to the shuttle’s center of mass?
Answer
Also bigger cross product!
\[-2i \text{ m/s}\]Question 6
The train on the circular track is traveling at a constant speed of 50 m/s in the direction shown. The train on the straight track is traveling at 20 m/s in the direction shown and is increasing its speed at 2 m/s $^2$. Determine the velocity of passenger A that passenger B observes relative to the given coordinate system, which is fixed to the car in which B is riding.
Answer
\[v_{Arel} = -120j\]
Question 7
Suppose that the merry-go-round has counterclockwise angular velocity $\omega$ and counterclockwise angular acceleration $\alpha$. The person A is standing still on the ground. Determine A’s acceleration relative to B’s reference frame at the instant shown.
Answer
\[a_{Arel} = - \omega^2Ri - \alpha Rj\]Question 8
The angular velocity $\omega$ AC=5° per second. Determine the angular velocity of the hydraulic actuator BC and the rate at which the actuator is extending.
Answer
\[\omega_{BC} = 0.108 \text{ rad/s, and the velocity of the actuator, } v_{Crel} = 0.109 \text{ m/s}\]Question 9
The sleeve at A slides upward at a constant velocity of 10 m/s. Bar AC slides through the sleeve at B. Determine the angular velocity of bar AC and the velocity at which the bar slides relative to the sleeve at B.
Answer
\[\omega_{AC} = 8.66 \text{ rad/s, and velocity of B towards A, } v_{Arel} = 5 \text{ m/s}\]Question 10
The satellite A is in a circular polar orbit (that intersects the earth’s axis of rotation). The radius of the orbit is $R$, and the magnitude of the satellite’s velocity relative to a non-rotating reference frame with its origin at the center of the earth is $v_A$. At the instant shown, the satellite is above the equator. An observer B on the earth directly below the satellite measures its motion using the earth-fixed coordinate system shown. What are the velocity and acceleration of the satellite relative to B’s earth-fixed coordinate system? The radius of the earth is $R_E$ and the angular velocity of the earth is $\omega_E$.
