If the magnitude of velocity in the previous question is decreasing with time, what is the direction of f angular acceleration `(oversetrightarrow(alpha))? `

Options

A. east
B. west
C. north
D. south

Answer

To solve the problem, we need to analyze the situation of a body executing circular motion where the magnitude of its velocity is decreasing over time. We will determine the direction of the angular acceleration \((\overset{\rightarrow}{\alpha})\).

### Step-by-Step Solution:

1. **Identify the Direction of Velocity**:
- From the previous question, we know that the velocity of the body is directed towards the west at the topmost point of the circular path.

2. **Understand the Implications of Decreasing Velocity**:
- When the magnitude of velocity is decreasing, it implies that there is a tangential acceleration acting in the opposite direction to the velocity. This is because tangential acceleration is responsible for changing the speed of the object along the circular path.

3. **Determine the Direction of Tangential Acceleration**:
- Since the velocity is directed towards the west, the tangential acceleration must be directed towards the east (opposite to the direction of velocity).

4. **Relate Tangential Acceleration to Angular Acceleration**:
- Angular acceleration \((\overset{\rightarrow}{\alpha})\) is related to tangential acceleration \((\overset{\rightarrow}{a_t})\) by the equation:
\[ \overset{\rightarrow}{a_t} = \overset{\rightarrow}{\alpha} \times \overset{\rightarrow{r}} \]
- Here, \(\overset{\rightarrow{r}}\) is the position vector from the center of the circular path to the point of interest.

5. **Determine the Direction of Angular Velocity**:
- The angular velocity \((\overset{\rightarrow}{\omega})\) is perpendicular to the plane of motion. If we curl our fingers in the direction of motion (which is westward), the thumb points upwards, indicating that the angular velocity is directed out of the plane.

6. **Find the Direction of Angular Acceleration**:
- Since the angular velocity is directed out of the plane and the tangential acceleration (due to decreasing speed) is directed eastward, the angular acceleration must be directed opposite to the angular velocity. Therefore, the angular acceleration \((\overset{\rightarrow}{\alpha})\) will be directed downwards into the plane.

### Final Answer:
The direction of angular acceleration \((\overset{\rightarrow}{\alpha})\) is downward into the plane of motion.

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