The drive propeller of a ship starts from rest and accelerates at 2.85 x 10-3 rad/s² for 2.44 x 10³ s. For the next 1.65 x 10³ s the propeller rotates at a constant angular speed. Then it decelerates at 2.30 x 10-3 rad/s² until it slows (without reversing direction) to an angular speed of 2.66 rad/s. Find the total angular displacement of the propeller. Number i ! Units rad
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.
please make sure answer is in 3 sig figs
![### Problem Statement
The drive propeller of a ship starts from rest and accelerates at \(2.85 \times 10^{-3} \, \text{rad/s}^2\) for \(2.44 \times 10^3 \, \text{s}\). For the next \(1.65 \times 10^3 \, \text{s}\) the propeller rotates at a constant angular speed. Then it decelerates at \(2.30 \times 10^{-3} \, \text{rad/s}^2\) until it slows (without reversing direction) to an angular speed of \(2.66 \, \text{rad/s}\). Find the total angular displacement of the propeller.
### Solution
To find the total angular displacement, we need to consider three distinct phases:
1. **Acceleration Phase:** The propeller accelerates from rest.
2. **Constant Speed Phase:** The propeller rotates at a constant angular speed.
3. **Deceleration Phase:** The propeller slows down until it reaches a final angular speed.
Let's analyze each phase in detail.
#### Phase 1: Acceleration from Rest
- Initial angular speed, \( \omega_0 = 0 \, \text{rad/s} \)
- Angular acceleration, \( \alpha = 2.85 \times 10^{-3} \, \text{rad/s}^2 \)
- Time, \( t_1 = 2.44 \times 10^3 \, \text{s} \)
Using the kinematic equation:
\[ \omega_1 = \omega_0 + \alpha t_1 \]
\[ \omega_1 = 0 + (2.85 \times 10^{-3})(2.44 \times 10^3) \]
\[ \omega_1 = 6.954 \, \text{rad/s} \]
Total displacement during this phase:
\[ \theta_1 = \omega_0 t + \frac{1}{2} \alpha t_1^2 \]
\[ \theta_1 = 0 + \frac{1}{2} (2.85 \times 10^{-3})(2.44 \times 10^3)^2 \]
\[ \theta_1 = 8499 \, \text{rad} \]
#### Phase 2: Constant Angular Speed
- Angular speed](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fda579a50-4d79-460d-b1ac-a80a17754259%2F1ed6d9a6-4ae6-4f6a-865b-62368f9afe60%2Fn9ytdx_processed.png&w=3840&q=75)
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