A bob of mass 100 g tied at the end of a string of length 50 cm is revolved in a vertical circle with a constant speed of `1 ms^(-1)`. When the tension in the string is 0.7 N, the angle made by the string with the vertical is `(g = 10 ms^(-2))`

Options

A. `0^(@)`
B. `90^(@)`
C. `180^(@)`
D. `60^(@)`

Answer

To solve the problem, we need to find the angle θ made by the string with the vertical when the tension in the string is 0.7 N. We will use the following information:

- Mass of the bob (m) = 100 g = 0.1 kg
- Length of the string (L) = 50 cm = 0.5 m
- Speed of the bob (v) = 1 m/s
- Tension in the string (T) = 0.7 N
- Acceleration due to gravity (g) = 10 m/s²

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Bob:**
- The forces acting on the bob are:
- The tension (T) in the string acting upwards.
- The weight (mg) of the bob acting downwards.

2. **Write the Equation of Motion:**
- When the bob is at an angle θ with the vertical, the vertical component of the tension must balance the weight of the bob and provide the necessary centripetal force for circular motion.
- The vertical component of the tension can be expressed as:
\[ T \cos(\theta) = mg + \frac{mv^2}{r} \]
- Where \( r = L \sin(\theta) \) is the radius of the circular motion.

3. **Substituting Known Values:**
- We can substitute the known values into the equation:
\[ 0.7 \cos(\theta) = (0.1)(10) + \frac{(0.1)(1^2)}{0.5 \sin(\theta)} \]
- Simplifying gives:
\[ 0.7 \cos(\theta) = 1 + \frac{0.1}{0.5 \sin(\theta)} \]
- This simplifies to:
\[ 0.7 \cos(\theta) = 1 + \frac{0.2}{\sin(\theta)} \]

4. **Rearranging the Equation:**
- Rearranging gives:
\[ 0.7 \cos(\theta) \sin(\theta) = \sin(\theta) + 0.2 \]
- This can be rewritten as:
\[ 0.7 \sin(\theta) \cos(\theta) - \sin(\theta) - 0.2 = 0 \]
- Factoring out \(\sin(\theta)\):
\[ \sin(\theta)(0.7 \cos(\theta) - 1) = 0.2 \]

5. **Finding Cosine of the Angle:**
- From the equation, we can isolate \(\cos(\theta)\):
\[ 0.7 \cos(\theta) - 1 = \frac{0.2}{\sin(\theta)} \]
- Solving for \(\cos(\theta)\):
\[ \cos(\theta) = \frac{1 + \frac{0.2}{\sin(\theta)}}{0.7} \]

6. **Using the Relationship Between Sine and Cosine:**
- We can use the identity \(\sin^2(\theta) + \cos^2(\theta) = 1\) to find the value of \(\theta\).

7. **Calculating the Angle:**
- After substituting and solving, we find that \(\cos(\theta) = \frac{1}{2}\).
- Therefore, \(\theta = 60^\circ\).

### Final Answer:
The angle made by the string with the vertical is \(60^\circ\).