Point A of the circular disk is at the angular position 0 = 0 at time t = 0. The disk has angular velocity wo = 0.29 rad/s at t = 0 and subsequently experiences an angular acceleration a = 1.8t where t is in seconds, and a is in radians per second squared. Determine the velocity and acceleration of point A in terms of fixed i and j unit vectors at time t = 2.7 s. Assumer = 145 mm. α Answers: VA = aд = 90 (i ( i 0 6.806 i + i + i i 1 0.705 j) m/s j) m/s²
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.
![### Circular Disk Kinematics Analysis
#### Problem Statement
Point A of the circular disk is at the angular position θ = 0 at time t = 0. The disk has angular velocity ω₀ = 0.29 rad/s at t = 0 and subsequently experiences an angular acceleration α = 1.8t where t is in seconds, and α is in radians per second squared. Determine the velocity and acceleration of point A in terms of fixed i and j unit vectors at time t = 2.7 s.
Assume r = 145 mm.
#### Diagram Explanation
A circular disk is depicted with:
- Point \( O \) at its center.
- A radius line \( r \) extending from \( O \) to point \( A \) on the circumference.
- Angular position \( \theta \) measured counter-clockwise from the positive y-axis.
- Angular acceleration \( \alpha \) shown counter-clockwise.
#### Considerations
1. The radius \( r \) is given as 145 mm, which is equal to 0.145 m (after conversion to meters).
2. Initial angular velocity \( \omega_0 = 0.29 \) rad/s.
3. Angular acceleration \( \alpha = 1.8 t \).
#### Formulas
1. **Angular Velocity** at time \( t \):
\[ \omega(t) = \omega_0 + \int_0^t \alpha(t') dt' \]
Given \( \alpha(t') = 1.8 t' \):
\[ \omega(t) = 0.29 + \int_0^t 1.8 t' dt' = 0.29 + 1.8 \left( \frac{t^2}{2} \right) \]
\[ \omega(t) = 0.29 + 0.9 t^2 \]
2. **Angle** at time \( t \):
\[ \theta(t) = \omega_0 t + \int_0^t \omega(t') dt' \]
Since:
\[ \omega(t') = 0.29 + 0.9 t'^2 \]
\[ \theta(t) = \int_0^t (0.29 + 0.9 t'^2) dt' = 0.29 t + 0.3 t^3 \]
3. **Velocity** of point \( A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b7dd8f1-dbab-4607-8b33-bde9974f18c9%2Fccbb28d5-1ab3-496b-ba3b-10a40d77079c%2Fpvb96so_processed.png&w=3840&q=75)
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