A body executing uniform circular motion has its position vector and acceleration vector

Options

A. along the same direction
B. in opposite direction
C. normal to each other
D. not related to each other

Answer

To solve the problem regarding the relationship between the position vector and acceleration vector of a body executing uniform circular motion, we can follow these steps:

### Step 1: Understand Uniform Circular Motion
In uniform circular motion, a body moves in a circular path with a constant speed. Although the speed is constant, the direction of the velocity vector changes continuously.

**Hint:** Recall that in uniform circular motion, the speed remains constant, but the direction of motion changes.

### Step 2: Define the Position Vector
The position vector (\( \vec{r} \)) of the body in uniform circular motion points from the center of the circular path to the body itself. This vector is directed radially outward from the center of the circle.

**Hint:** Visualize the circular motion and identify where the position vector points relative to the center of the circle.

### Step 3: Define the Acceleration Vector
In uniform circular motion, the acceleration vector (\( \vec{a} \)) is the centripetal acceleration, which always points towards the center of the circular path. This means that the acceleration vector is directed radially inward.

**Hint:** Remember that centripetal acceleration is always directed towards the center of the circle, regardless of the position of the body on the circular path.

### Step 4: Analyze the Direction of Vectors
Now, we can analyze the directions of the position vector and the acceleration vector:
- The position vector (\( \vec{r} \)) points outward from the center.
- The acceleration vector (\( \vec{a} \)) points inward towards the center.

**Hint:** Consider the angles between the two vectors. What can you conclude about their directions?

### Step 5: Determine the Relationship
Since the position vector points outward and the acceleration vector points inward, they are in opposite directions. The angle between them is 180 degrees.

**Hint:** Think about how vectors can be related in terms of direction: same direction, opposite direction, or perpendicular.

### Conclusion
Based on the analysis, the correct relationship between the position vector and the acceleration vector for a body executing uniform circular motion is that they are in opposite directions.

**Final Answer:** The position vector and acceleration vector are in opposite directions.