A linkage undergoing motion as shown. The velocity of the block, VD, is 4 m/s. The length of the rods is l=0.5m. Aj l 45° 90° 45° l 121 No What are the angular velocities of links AB and BD in rad/s. Note: They both have the same angular velocity due to the symmetric geometry.
Angular Momentum
The momentum of an object is given by multiplying its mass and velocity. Momentum is a property of any object that moves with mass. The only difference between angular momentum and linear momentum is that angular momentum deals with moving or spinning objects. A moving particle's linear momentum can be thought of as a measure of its linear motion. The force is proportional to the rate of change of linear momentum. Angular momentum is always directly proportional to mass. In rotational motion, the concept of angular momentum is often used. Since it is a conserved quantity—the total angular momentum of a closed system remains constant—it is a significant quantity in physics. To understand the concept of angular momentum first we need to understand a rigid body and its movement, a position vector that is used to specify the position of particles in space. A rigid body possesses motion it may be linear or rotational. Rotational motion plays important role in angular momentum.
Moment of a Force
The idea of moments is an important concept in physics. It arises from the fact that distance often plays an important part in the interaction of, or in determining the impact of forces on bodies. Moments are often described by their order [first, second, or higher order] based on the power to which the distance has to be raised to understand the phenomenon. Of particular note are the second-order moment of mass (Moment of Inertia) and moments of force.
### Description
- Length of each rod \( l = 0.5 \) meters
- Velocity of block \( v_D = 4 \) m/s
- Linkages AB and BD form angles of 45 degrees with the horizontal and vertical respectively
- Both links AB and BD are symmetric and form a 90-degree angle at point B
### Determination of Angular Velocities
Given these parameters, determine the angular velocities of links AB and BD in radians per second (rad/s).
**Note:** Due to the symmetric geometry of the linkage system, both links AB and BD will have the same angular velocity.
### Calculation
Use the kinematic relationship for rigid body rotation to find the angular velocity \( \omega \):
\[ v_D = \omega \times l \]
Where:
- \( v_D \) = 4 m/s
- \( l \) = 0.5 m
Solving for \( \omega \):
\[ \omega = \frac{v_D}{l} \]
\[ \omega = \frac{4 \, \text{m/s}}{0.5 \, \text{m}} \]
\[ \omega = 8 \, \text{rad/s} \]
### Answer
The angular velocities of links AB and BD are:
\[ \omega_{AB} = \omega_{BD} = 8.00 \, \text{rad/s} \]
### Conclusion
By applying the basic principles of kinematics and noting the symmetric nature of the linkage, we were able to determine that the angular velocities of both links are equal and computed to be 8 rad/s.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F697ee80b-6834-4809-8325-e33cf8ac2219%2Fa86a7d55-fa22-4cbd-b958-0dd78384c473%2Fy9rhz4b_processed.png&w=3840&q=75)
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