To solve the problem, we need to analyze the forces acting on a 5 kg mass rotating in a vertical circle with a constant speed of 4 m/s. We will determine the position of the body when the tension in the string is 31 N.
### Step 1: Identify the Forces Acting on the Body
At any point in the vertical circle, the forces acting on the body are:
- The gravitational force (weight) acting downwards: \( F_g = mg \)
- The tension in the string acting upwards (or towards the center of the circle)
### Step 2: Calculate the Gravitational Force
Given:
- Mass \( m = 5 \, \text{kg} \)
- Acceleration due to gravity \( g = 10 \, \text{m/s}^2 \)
The gravitational force is calculated as:
\[
F_g = mg = 5 \, \text{kg} \times 10 \, \text{m/s}^2 = 50 \, \text{N}
\]
### Step 3: Analyze Forces at Different Points
1. **At the Lowest Point:**
The net force towards the center of the circle at the lowest point is given by:
\[
T - mg = \frac{mv^2}{r}
\]
Rearranging gives:
\[
T = mg + \frac{mv^2}{r}
\]
2. **At the Highest Point:**
The net force towards the center of the circle at the highest point is given by:
\[
T + mg = \frac{mv^2}{r}
\]
Rearranging gives:
\[
T = \frac{mv^2}{r} - mg
\]
### Step 4: Calculate Tension at the Lowest Point
Using the radius \( r = 1 \, \text{m} \) and speed \( v = 4 \, \text{m/s} \):
\[
T = mg + \frac{mv^2}{r} = 50 \, \text{N} + \frac{5 \times (4)^2}{1}
\]
Calculating:
\[
T = 50 \, \text{N} + \frac{5 \times 16}{1} = 50 \, \text{N} + 80 \, \text{N} = 130 \, \text{N}
\]
### Step 5: Calculate Tension at the Highest Point
Using the same values:
\[
T = \frac{mv^2}{r} - mg = \frac{5 \times (4)^2}{1} - 50 \, \text{N}
\]
Calculating:
\[
T = 80 \, \text{N} - 50 \, \text{N} = 30 \, \text{N}
\]
### Step 6: Conclusion
Since the tension in the string is given as 31 N, we find that the body is most likely at the **highest point** of the vertical circle where the tension is approximately 30 N, which is close to the given value of 31 N.
### Final Answer
The body will be at the **highest point** of the vertical circle.
